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what-is-logic.html
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<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>
<link rel="shortcut icon" href="https:husnaf.github.io/logical-world-s/favicon.ico">
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<title>What is logic?</title><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; }
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<div id='write' class=''><h1><a name="what-is-logic" class="md-header-anchor"></a><span>What is logic?</span></h1><h3><a name="earlier-i-said-that" class="md-header-anchor"></a><span>Earlier I said that,</span></h3><blockquote><p>“<span>You rest at the outset of an expanding </span><a href='https://www.st-andrews.ac.uk/~slr/MONISM.PDF'><span>logical world</span></a><span>. Or of </span><a href='https://plato.stanford.edu/entries/logical-pluralism/'><span>worlds of worlds of worlds</span></a><span> over and again.</span>”</p></blockquote><p><span>Therein I have placed the core strains of this present </span><i><u><span>search</span></u></i><span>. Most obviously, we are caught between a </span><strong><em><span>world</span></em></strong><span> and </span><strong><em><span>worlds of worlds of worlds</span></em></strong><span>. The solitude of the </span><strong><span>One</span></strong><span> pit against the clamor of the </span><strong><span>Many</span></strong><span> reappears here, as it always has in human thought, the question of choosing either being our perennial friend and enemy. To choose today, if ever our choice, it was sought to clarify the fog around that thing which is called </span><strong><em><span>logical world</span></em></strong><span>. What </span><strong><span>space</span></strong><span> does it occupy, and wherein is it kept? In the head, on the page, or somewhere beyond our impoverished </span><strong><span>frame of perspective</span></strong><span>? And also—what are the </span><strong><span>logical world</span></strong>’<span>s constituents? Who are its dwellers? I am asking really, or I think,</span></p><div style="font-size:150%; text-align:center;">
What is <b>the logical world</b>?<br>
(and is there an answer or many?)
</div> <p> </p><p><img src="assets\dawn-small.jpg"/></p><center><small><i>Dawn</i> by Ei-Q. The National Museum of Modern Art, Tokyo.</small></center><p> </p><p> </p><p><span>I have said as well that this </span><strong><span>world(s)</span></strong><span> is expanding. How so, and what of it? Are we to be so impressed if there are more things but have yet gained </span><strong><span>nothing</span></strong><span>? That is for you to answer. I will insist only that its multifariousness, its explosive expansion </span><strong><span>destroys as it creates</span></strong><span>, and we gain precious much. </span></p><p><del><span>Returning to the </span><strong><em><span>world</span></em></strong><span> alone, what </span><strong><span>relation</span></strong><span> does the </span><strong><span>logical world</span></strong><span> bear on the </span><strong><span>world</span></strong><span>, or vice versa? Does the </span><strong><span>world</span></strong><span> </span><strong><span>define</span></strong><span> the </span><strong><span>logical</span></strong><span>? Does the </span><strong><span>logical</span></strong><span> speak of the </span><strong><span>world</span></strong><span>?</span></del><span> </span></p><p><span>Can there ever be an outset to the </span><strong><span>logical world</span></strong><span>? Are there </span><strong><span>bounds</span></strong><span>, and if so, where do they lie? Or else, are we to think of it as all-encompassing and forever boundless?</span></p><p><span>At the end of the day, in magnificent reversal, what of </span><em><span>yourself</span></em><span>? What is </span><em><span>your</span></em><span> </span><strong><span>relation</span></strong><span> to the logical? How do you see it, how do you use it, and how can it so impress upon you the way it does?</span></p><p> </p><div style="text-align:right; font-size:75%">And so you know that my excesses are not arbitrary,<br>even if my words hide my head and heart in plain sight.<br>In truth, I have no wish for you to understand me,<br>or any extension of myself in thought, <i>easily</i>.<br>I shall be read, but I shall not surrender and be freely possessed.<br><br>
<h6><b>This is my oath against you.</b></h6></div><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><h2><a name="monism--pluralism" class="md-header-anchor"></a><span>Monism & Pluralism</span></h2><p> </p><p><span>The One and the Many: what do I mean? Of what is there One or Many or? I mean only to ask the question</span></p><p><span>Could there be/</span><a href='https://plato.stanford.edu/archives/sum2019/entries/logical-pluralism/' target='_blank'><span>one true logic</span></a><span>/to rule them all?</span></p><p><span style="background-color:#333;"><span>Surely you understand the difference between a tyrant and many.</span></span><span> But you must wonder at the curiosity that is </span>‘<span>true logic</span>’<span>. If you were Frege or Russell, </span>‘<span>true logic</span>’<span> might have been some kind of redundant redundancy. One kind of curious curiosity. To Frege and Russell the logical at its barest, in all its unadorned nakedness, is a </span><strong><span>body of truths</span></strong><span>.</span><sup class='md-footnote'><a href='#dfref-footnote-1' name='ref-footnote-1'>1</a></sup><span>The logical fundamentally structures and embodies itself around a particular class of truths, namely the laws of truths or even, the truths about truths.</span><sup class='md-footnote'><a href='#dfref-footnote-2' name='ref-footnote-2'>2</a></sup><span>So then logic cannot help if it is true. Truth is simply its reason for </span><strong><span>being</span></strong><span>. </span></p><h5><a name="but-this-is-such-an-old-conception-of-logic-and-we-have-long-left-that-frame" class="md-header-anchor"></a><span>But this is such an old conception of logic, and we have long left that frame. </span></h5><p> </p><p><span>do not mourn its obsolescence. everything goes that way in time. at least the frame still hangs on a wall somewhere and remains open for viewing like all venerated and displaced historical treasures.</span></p><p> </p><p><img src="assets\Reversed Frames-MoMa.jpg" alt="Four frames of pictures"></p><center><small><i>Reversed Frames</i> by Mario Restife. MoMA.</small></center><p> </p><p><span>But moving now to stand before the present, there are </span><em><span>four</span></em><span> things I should like to tell you. </span></p><p><span>Four things of consequence:</span></p><ol start='' ><li><span>We are currently situated in that frame where logic</span>’<span>s most fundamental core is </span><em><span>consequence</span></em><span>.</span><sup class='md-footnote'><a href='#dfref-footnote-3' name='ref-footnote-3'>3</a></sup></li><li><em><span>Consequence</span></em><span> — </span><em><span>that primordial, chaotic concept from which the logical world is forged and cools over</span></em><span>. </span></li><li><span>So, we are currently situated in that frame where logic</span>’<span>s most fundamental core is </span><em><span>that primordial, chaotic concept from which the logical world is forged and cools over</span></em><span>. </span></li></ol><div style="text-align:center;">Within the fourth where the three are</div> <div style="margin: 0px 50% 0px 50%;"><b>strung</b> together <b>in</b> linear <b>union</b>,</div><div style="text-align:center">I make no singular assertion. In passing <i>above</i> over all three I have sought to speak to you of a hypothetical, a dream. And so in the fourth, I ceased to merely speak of consequence: I demonstrated it. </div><p> </p><p><span>This last thing cannot be seen with simple </span><span style="font-size:1000%"><b><span>blackened words.</span></b></span><span> Be assured that it is there as I am here. I am eluding you, of course, but this frame eludes only because a frame becomes indiscernible when you yourself sit inside its room. Let me put it another way. Churning 1 and 2, which alone do nothing, and then to mingle them—to disturb their natural nothingness—to form 3: this is the creation of consequence. It is an inventive act and purely constructive as it builds new facts upon the old; or at least, it reveals the old</span>’<span>s new faces. </span></p><p><span>Yet for all that it creates, we stand not amazed. For the drawing of consequence is to draw in midair. The process is unseen, though its product is seen, and there is little time to hold sacred forms we cannot even imagine. </span><del><span>though little time has not kept some from trying.</span></del><span> Conjoined with consequence</span>’<span>s invisibility is the fact of its inevitability. We are moved, pushed even, to find thought</span>’<span>s end, otherwise known as its </span><em><span>conclusion</span></em><span>, once we are on that road. Can you escape 3 were it the case 1 and 2? </span><b><span>There is simply nowhere for you, this day or tomorrow. </span></b><span> </span></p><p><span>Thus is the matter: so </span><strong><span>ordinary</span></strong><span>, so </span><strong><span>banal</span></strong><span> to us is consequence that we usually do not note its procession, and we leave its mark off the page. </span></p><p><span>But as I said, consequence is logic</span>’<span>s most fundamental core, and so it must mark consequence</span>’<span>s revered presence one way or another. Of course the logician cannot help doing so. She would have done this in many ways:</span></p><p> </p><div style="text-align:right; font-size:120%;background-color: #E9F5F8">
3. So, we are currently situated in that frame where logic’s most fundamental core is <i>that primordial, chaotic concept from which the logical world is forged and cools over</i>.<b> (from 1 & 2)</b><br>
OR<br>
<b>(MP 1, 2)</b><br>
OR<br>
<b>Application of Modus Ponens using 1 and 2</b><br>
OR what have you.</div><p> </p><p> </p><p><span>Here already is one thing you may learn. Logic seeks to make </span><strong><span>visible</span></strong><span> what was latent, ignored, and taken for granted. It seeks to clarify and uncover what was crouched and shrouded. No wonder the logician has few admirers though she helps many.</span></p><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><div style="font-size:95%;line-spacing:0;letter-spacing: 0.05em;">WHY?<br>WHO SHOULD LIKE THAT THEY ARE EXACTLY CLEAR TO ALL? </div><h5><a name="that-their-thoughts-or-that-they-themselves-are-bare-in-daylight" class="md-header-anchor"></a><span>That their thoughts, or that they themselves, are bare in daylight? </span></h5><div style="font-size:95%;line-spacing:0;letter-spacing: 0.05em;">NOT ME AT LEAST.<br>YOU KNOW I HAVE ALREADY SWORE AGAINST YOU.</div><h5><a name="but-clarity-is-the-very-design-of-logic-to-know-what-follows-from-what-and-how-so" class="md-header-anchor"></a><span>But clarity is the very design of logic: to know what </span><em><span>follows</span></em><span> from what and how so,</span></h5><div style="font-size:95%;line-spacing:0;letter-spacing: 0.05em;">WHERE THIS <i>HOW SO</i> IS ANSWERED AND DETERMINED BY UNCOVERING THE PRECISE LOGICAL STRUCTURE OF OUR WORDS<br>OR OTHER SUCH MEANINGFUL CONTEXTS.</div><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><h4><a name="it-leaves-little-room-to-hide-and-this-is-undesirable-to-us-we-people-who-like-to-hide-as-i-hide-now-within-a-bracket" class="md-header-anchor"></a><span>(it leaves little room to hide, and this is undesirable to us, we people who like to hide, as i hide now within a bracket.)</span></h4><p> </p><p><img src="assets\small-black-dot.png" alt="dot" /></p><p> </p><p><span>Why have I taken these pains to speak to you of consequence? </span></p><p><span>BECAUSE</span></p><p><span>It is the core concept of the logical: yes</span><del><span>, </span><strong><span>how many times am I to convey this to you</span></strong></del><span>?</span></p><p><span>Consequently, this is what we mean when we say </span>“<span>there is only one true logic,</span>”<span> or </span>“<span>there are many logics.</span>”</p><p><span>I mean that </span>“<span>logical pluralism is a </span><em><span>pluralism</span></em><span> about consequence,</span>”<span> and logical monism is the contrary thesis.</span><sup class='md-footnote'><a href='#dfref-footnote-4' name='ref-footnote-4'>4</a></sup></p><p><span>Now in words that are more mine:</span></p><div style="text-align:left; font-size:200%;">Logical worlds forged and defined<br>by their own unique cores of consequence.</div><p> </p><div class="row">
<div class="column"><div style="text-align:center;background-color:#333;margin: 0px 45% 0px 45%;">It is<br>clear<br>to you<br>now,<br>at least<br>I hope,<br>or<br>maybe I<br>hope not,<br>what our<br>basic<br>conundrum<br>is.<br>You<br>are<br> waiting<br>for<br>my<br>verdict?<br>Pity<br>that<br>you<br>should<br>wait<br>for<br>nothing.<br>I<br>suppose<br>you<br>are<br>no<br>different<br>from<br>the<br>rest<br>of<br>us.</div></div>
<div class="column"><div style="text-align: left;">AGAINST</div><br><br><div style="text-align:right;font-size:200%;">A logical world forged and defined<br>by a single core of consequence.</div><br><br><div style="text-align:center;background-color:#333;margin: 0px 46% 0px 46%;">No<br>matter.<br>You<br>may<br>pursue<br>these<br>two<br>roads<br>alone<br>and<br>obtain<br>your<br>own<br>end<br>and<br>conclusion.<br>How<br>I<br>leave<br>you<br>something<br>still.</div> </div></div><p> </p><p> </p><p><span>Dread is to choose between two roads that appear always to promise something when, in fact, they promise nothing. Here, two lines are monuments to fragility and limitation, as well as the hope we hold on the way. </span><strong><span>Can you see now what else may two lines become?</span></strong><span> You may imagine a form, a word, or a world. And when we are extraordinarily lucky, two begets three, an offspring set of points to tread:</span></p><p style="text-align:center;">In a yellow wood,<br>the gap where two roads diverge,<br>there waits another question.<br>Do you follow me there?</p><p> </p><h2><a name="on-following" class="md-header-anchor"></a><span>On Following</span></h2><p> </p><div style="text-align:right; font-size:75%;">“What these logicians claimed to have done is to free our ordinary language from unclarities and ambiguities.”<br>— Jaakko Hintikka, “Which Logic is the Logic of Mathematics?” 3.</div><p> </p><p><span>“<span>Thus,</span>”<span> </span>“<span>consequently,</span>”<span> </span>“<span>as such,</span>”<span> </span>“<span>therefore,</span>”<span> and </span>“<span>it follows that</span>”</span><span> are all phrases that denote roughly the same thing: </span><span style="font-size:200%;"><b><span>consequence.</span></b></span><span> </span></p><p><span>It seems that concept of consequence is definite and precise, just as definite and precise its token is underlined there. However, this seeming does not coincide with reality (or at least what I think is reality, but I reserve for myself the thought that reality is just what I think).</span></p><div style="text-align:right;">After all, the concept of consequence is primordial and chaotic. Wherefore primordial? </div><p><span>Because though the concept has a formal use, it first lived in the domain of our ordinary and everyday intuitions.</span><sup class='md-footnote'><a href='#dfref-footnote-5' name='ref-footnote-5'>5</a></sup><span>These intuitions were given to us just as they were given to our forebears and they their forebears; no one can know just how old they are. </span></p><div style="text-align:right;">And why chaotic?</div><p><span>Because its foray into the formal domain was pushed on by </span><span style="font-size:450%"><span>people</span>&<span>people</span></span><span> are such that they often make </span><strong><span>arbitrary choices</span></strong><span>. And besides, these are people who sought to reconcile the content of the </span><em><span>everyday concept</span></em><span> with the </span><em><span>formal, precise one</span></em><span>.</span><sup class='md-footnote'><a href='#dfref-footnote-6' name='ref-footnote-6'>6</a></sup><span> </span></p><p><img src="assets\small-black-dot.png" alt="dot"></p><blockquote><p>“<span>The concept of following is not distinguished from other concepts of everyday language by a clearer content or more precisely delimited denotation, the way it is used is unstable, the task of capturing and reconciling all the murky, sometimes contradictory intuitions connected with that concept has to be acknowledged a priori as unrealizable, and one has to reconcile oneself in advance to the fact that every precise definition of the concept under consideration will to a greater or lesser degree bear the mark of arbitrariness.</span>”<sup class='md-footnote'><a href='#dfref-footnote-7' name='ref-footnote-7'>7</a></sup></p></blockquote><p><img src="assets\small-black-dot.png" alt="dot"></p><p><span style="background-color:#333;font-size:150%"><span>There. I have used the weight of another’s word: am I to be more or less trusted? For if I am helped and am more trustworthy to you, then that means my own word is not enough and should not be trusted at all. </span></span><span> Against </span><strong><span>arbitrariness</span></strong><span> and </span><strong><span>murk</span></strong><span>, the </span><strong><span>bedrock safety</span></strong><span> of a theory may give you short peace.</span></p><p> </p><p><img src="assets\dead-mother-small.jpg" /></p><center><small><i>Tote Mutter I (Dead Mother I)</i> by Egon Schiele. Leopold Museum.</small></center><p> </p><h3 style="text-align:center;">A Theory of Following Logically</h3><p>“<span>We say that the sentence </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.979ex" height="1.827ex" viewBox="0 -741.3 852 786.8" role="img" focusable="false" style="vertical-align: -0.106ex;"><defs><path stroke-width="0" id="E17-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E17-MJMATHI-58" x="0" y="0"></use></g></svg></span><script type="math/tex">X</script><span> follows logically from the sentences of the class </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.923ex" height="1.928ex" viewBox="0 -741.3 828 830.3" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E16-MJFRAK-52" d="M27 496Q31 569 102 627T234 685Q236 685 241 685T251 686Q287 686 318 672T367 638T399 598T418 564L423 550Q424 554 434 567T463 601T505 639T561 671T626 685Q672 685 688 659T710 572Q713 533 721 523T766 513Q781 513 787 514T794 516Q796 512 798 509T801 504T802 501T787 493Q702 461 624 401L607 389Q655 383 688 358L697 352V342Q699 330 699 297Q704 209 710 173T734 103Q751 69 765 69Q769 69 806 83L824 90V74Q823 73 759 24T693 -26Q692 -26 660 32L628 90L629 111Q631 159 631 177Q631 278 614 300Q584 340 523 340Q500 340 467 333T431 325Q429 325 429 322Q428 321 426 308T420 275T410 230T392 178T366 125L358 112L342 99Q306 70 269 38T213 -10T193 -26Q192 -26 163 0T116 26Q82 26 50 -8L42 -16L35 -8L27 0L35 10Q43 21 58 38T104 80T158 106Q179 106 218 65L235 48Q238 48 255 60T295 99T329 158Q352 231 352 359Q352 555 242 614Q210 628 187 628Q140 628 116 600T91 548Q91 522 138 464T185 382V376Q185 345 158 313T103 263L76 246Q74 244 64 253L54 260L65 267Q91 285 100 302Q111 318 111 337Q111 355 69 410T27 496ZM562 628Q504 628 443 507L435 491L436 479Q437 471 437 446Q437 396 432 351L529 389L602 426Q673 462 673 463H672Q644 470 637 483T622 553Q608 628 562 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E16-MJFRAK-52" x="0" y="0"></use></g></svg></span><script type="math/tex">\mathfrak{R}</script><span> if and only if every model of the class </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.923ex" height="1.928ex" viewBox="0 -741.3 828 830.3" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E16-MJFRAK-52" d="M27 496Q31 569 102 627T234 685Q236 685 241 685T251 686Q287 686 318 672T367 638T399 598T418 564L423 550Q424 554 434 567T463 601T505 639T561 671T626 685Q672 685 688 659T710 572Q713 533 721 523T766 513Q781 513 787 514T794 516Q796 512 798 509T801 504T802 501T787 493Q702 461 624 401L607 389Q655 383 688 358L697 352V342Q699 330 699 297Q704 209 710 173T734 103Q751 69 765 69Q769 69 806 83L824 90V74Q823 73 759 24T693 -26Q692 -26 660 32L628 90L629 111Q631 159 631 177Q631 278 614 300Q584 340 523 340Q500 340 467 333T431 325Q429 325 429 322Q428 321 426 308T420 275T410 230T392 178T366 125L358 112L342 99Q306 70 269 38T213 -10T193 -26Q192 -26 163 0T116 26Q82 26 50 -8L42 -16L35 -8L27 0L35 10Q43 21 58 38T104 80T158 106Q179 106 218 65L235 48Q238 48 255 60T295 99T329 158Q352 231 352 359Q352 555 242 614Q210 628 187 628Q140 628 116 600T91 548Q91 522 138 464T185 382V376Q185 345 158 313T103 263L76 246Q74 244 64 253L54 260L65 267Q91 285 100 302Q111 318 111 337Q111 355 69 410T27 496ZM562 628Q504 628 443 507L435 491L436 479Q437 471 437 446Q437 396 432 351L529 389L602 426Q673 462 673 463H672Q644 470 637 483T622 553Q608 628 562 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E16-MJFRAK-52" x="0" y="0"></use></g></svg></span><script type="math/tex">\mathfrak{R}</script><span> is at the same time a model of the sentence </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.979ex" height="1.827ex" viewBox="0 -741.3 852 786.8" role="img" focusable="false" style="vertical-align: -0.106ex;"><defs><path stroke-width="0" id="E17-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E17-MJMATHI-58" x="0" y="0"></use></g></svg></span><script type="math/tex">X</script><span>.</span>”<sup class='md-footnote'><a href='#dfref-footnote-8' name='ref-footnote-8'>8</a></sup></p><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><p><span>We call such a theory </span><em><span>semantical</span></em><span>, the heads to the tails of </span><em><span>syntax</span></em><span> on the coin of </span><em><span>meaning</span></em><span>. </span></p><ol start='' ><li><p><span>HEADS: TAILS</span></p><p><span>A) OBVERSE: REVERSE</span></p><p><span>B) VERSO: RECTO</span></p><p><span>C) CONTENT: FORM </span></p><p><span>D) SEMANTICS: SYNTAX</span></p></li></ol><p><span>Shall we flip our coin to make the choice?</span></p><div style="text-align: center; margin: 0px 20% 0px 20%;"> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i> <b>Syntax</b><br><i>Semantics</i> <b>Syntax</b> <i>Semantics</i> <b>Syntax</b><br>
<i>Semantics</i> <b>Syntax</b> <i>Semantics</i> <b>Syntax</b><br>
<i>Semantics</i> <b>Syntax</b> <i>Semantics</i> <b>Syntax</b><br>
<i>Semantics</i> <b>Syntax</b> <i>Semantics</i> <b>Syntax</b><br>
<i>Semantics</i> <b>Syntax</b> <i>Semantics</i> <b>Syntax</b><br>
<i>Semantics</i> <b>Syntax</b> <i>Semantics</i> <b>Syntax</b><br>
<i>Semantics</i> <b>Syntax</b> <i>Semantics</i> <b>Syntax</b><br>
<i>Semantics</i> <b>Syntax</b> <i>Semantics</i> <b>Syntax</b><br>
<i>Semantics</i> <b>Syntax</b> <i>Semantics</i> <b>Syntax</b><br>
<i>Semantics</i> <b>Syntax</b> <i>Semantics</i> <b>Syntax</b><br>
<b>Syntax</b> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i><br>
<b>Syntax</b> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i><br>
<b>Syntax</b> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i><br>
<b>Syntax</b> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i><br>
<b>Syntax</b> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i><br>
<b>Syntax</b> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i><br>
<b>Syntax</b> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i><br>
<b>Syntax</b> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i><br>
<b>Syntax</b> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i><br>
<b>Syntax</b> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i><br>
<b>Syntax</b> <i>Semantics</i> <b>Syntax</b> <i>Semantics</i><br>
<b>Syntax</b> <i>Semantics</i>
</div><p> </p><p><span>ALL, and we proceed.</span></p><p> </p><p><img src="assets\odilon-redon-small.jpg" alt="Cellule Auriculaire"></p><center><small><i>Cellule Auriculaire</i> by Odilon Redon. Museum of Fine Arts, Houston.</small></center><p> </p><p><span>I mentioned </span>‘<span>peace</span>’<span>? I am beside myself; it was a misstep. I did not mean it in sincerity even if peace were true. I could never give you such lightness of being or such heaviness of solace, so do not feel that you are my debtor. I give you my word, and this time, it is sincerely for you. That is, what I can give to you is </span><strong><span>what I express,</span></strong><span> and </span><strong><span>what I express</span></strong><span>—my meaning, my content and discontent—is held together by the </span><strong><span>bindings of form and ruled structure.</span></strong><sup class='md-footnote'><a href='#dfref-footnote-9' name='ref-footnote-9'>9</a></sup><span> If I have said something—something of meaning—it was carried to you across time and place in these wrappings and is your present </span><strong><span>BURDEN</span></strong><span>. </span><span style="background-color:#333;"><span>No doubt, I am restlessly concerned that if I have said something, it is something much like nothing. It is a worry owing to our seat in time and place, WHICH IS NONE OF YOUR CONCERN.</span></span><span> </span><span style="font-size:150%;"><span>It will push against you, the </span><b><span>weight</span></b><span> of what I mean, and I pray you shall </span><sub><span>sink</span></sub><span> there. Such is my cruelty. </span></span><span><del><span>though surely it is only human need.</span></del></span></p><p><span>It is not formless: believe me, meaning has its </span><b><span>body</span></b><span> too.</span><br><span>(how else could it have form? how else could it have weight?)</span><br><span>It is just that some bodies make themselves more known than others. But meaning must have it, for without body, there would be only abyss. Hold yourself over it now, look down, and do not fall. </span></p><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><div style="margin:0 35% 0 35%;">
<p style="text-align:justify;">What if content and meaning is never without form and body? What if the gulf between them is actually one quiet, unidirectional continuous stream? This I think when staring into a little pond of reflection.<br>Then would evaporate our two-sided coin, SEMANTICS: SYNTAX.<br> Or maybe it sinks into our stream, SEMANTICS—SYNTAX.<br>But that is only my pond; I know not others.</p>
<img src="assets\mobius-crop-small.jpg" alt="Möbius strip"><br><small><i>Möbius Strip I</i> by M. C. Escher. The National Gallery of Art.</small></div><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><p><em><span>Do you hear me? Stand if you are unhurt.</span></em></p><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><p><span>Before that semantical theory of following logically, there was a syntactical theory.</span><sup class='md-footnote'><a href='#dfref-footnote-10' name='ref-footnote-10'>10</a></sup><span> It was thought that one thing follows some others if it could be derived from those others using the rules.</span><sup class='md-footnote'><a href='#dfref-footnote-11' name='ref-footnote-11'>11</a></sup></p><p><span>BUT THE RULES DID NOT AND COULD NOT SUFFICE. </span></p><p><span style="font-size: 500%"><span>For,</span></span><span> how could </span><strong><span>inference rules</span></strong><span> give us the means to derive </span>“<span>Such and such holds for all humankind,</span>”<span> from</span></p><p>“<span>Such and such holds for the first person &</span></p><p><span>the person after the first & </span></p><p><span>the person after the person after the first &</span></p><p><span>then the person after that one &</span></p><p><span>... &</span></p><p><span>... &</span></p><p><span>... & </span></p><p><span>it holds for the last person, who has just now joined us in life and cries over his gain and loss.</span>”<sup class='md-footnote'><a href='#dfref-footnote-12' name='ref-footnote-12'>12</a></sup><span> </span></p><p><span>Yet we feel that </span>“<span>Such and such holds for all humankind,</span>”<span> </span><strong><span>must follow</span></strong><span> were we able to tell it for every human who</span>’<span>s fortune is to inhabit this Earth. </span></p><p><span>Even after adding structural rules such as induction, whose power is to obtain </span>“<span>Such and such holds for all humankind,</span>”<span> THE RULES STILL DO NOT SUFFICE.</span></p><p><span style="font-size: 500%"><span>For,</span></span><span> </span>“<span>if we abstract from certain theories with a very elementary structure, then always—no matter how we enrich the stock of rules of inference—we shall be able to construct sentences which follow in the everyday sense from the theorems of the deductive theory under consideration, but which cannot be proven in this theory on the basis of the accepted rules.</span>”<sup class='md-footnote'><a href='#dfref-footnote-13' name='ref-footnote-13'>13</a></sup><span> See here, a serendipitous meeting with the mirror: the </span><strong><span>INCOMPLETENESS</span></strong><span> of the syntactical theory follows from a certain kind of </span><span class="mirror"><b><span>INCOMPLETENESS</span></b></span><span> of our basic formal languages.</span><sup class='md-footnote'><a href='#dfref-footnote-14' name='ref-footnote-14'>14</a></sup></p><p><span style="font-size: 500%"><span>Thus,</span></span><span> we leave the syntactical theory (of) following behind us. Though the semantical theory does not fall behind in this way, and though it agrees with our old intuitions </span></p><p><span>& though for the concept of following logically it makes necessary and insufficient the property </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.74ex" height="1.827ex" viewBox="0 -741.3 749 786.8" role="img" focusable="false" style="vertical-align: -0.106ex;"><defs><path stroke-width="0" id="E13-MJMATHI-46" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E13-MJMATHI-46" x="0" y="0"></use></g></svg></span><script type="math/tex">F</script><span> that </span><strong><span>deliberate substitution of non-logical constant terms for other constant terms in </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.979ex" height="1.827ex" viewBox="0 -741.3 852 786.8" role="img" focusable="false" style="vertical-align: -0.106ex;"><defs><path stroke-width="0" id="E17-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E17-MJMATHI-58" x="0" y="0"></use></g></svg></span><script type="math/tex">X</script><span> and in the sentences of class </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.923ex" height="1.928ex" viewBox="0 -741.3 828 830.3" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E16-MJFRAK-52" d="M27 496Q31 569 102 627T234 685Q236 685 241 685T251 686Q287 686 318 672T367 638T399 598T418 564L423 550Q424 554 434 567T463 601T505 639T561 671T626 685Q672 685 688 659T710 572Q713 533 721 523T766 513Q781 513 787 514T794 516Q796 512 798 509T801 504T802 501T787 493Q702 461 624 401L607 389Q655 383 688 358L697 352V342Q699 330 699 297Q704 209 710 173T734 103Q751 69 765 69Q769 69 806 83L824 90V74Q823 73 759 24T693 -26Q692 -26 660 32L628 90L629 111Q631 159 631 177Q631 278 614 300Q584 340 523 340Q500 340 467 333T431 325Q429 325 429 322Q428 321 426 308T420 275T410 230T392 178T366 125L358 112L342 99Q306 70 269 38T213 -10T193 -26Q192 -26 163 0T116 26Q82 26 50 -8L42 -16L35 -8L27 0L35 10Q43 21 58 38T104 80T158 106Q179 106 218 65L235 48Q238 48 255 60T295 99T329 158Q352 231 352 359Q352 555 242 614Q210 628 187 628Q140 628 116 600T91 548Q91 522 138 464T185 382V376Q185 345 158 313T103 263L76 246Q74 244 64 253L54 260L65 267Q91 285 100 302Q111 318 111 337Q111 355 69 410T27 496ZM562 628Q504 628 443 507L435 491L436 479Q437 471 437 446Q437 396 432 351L529 389L602 426Q673 462 673 463H672Q644 470 637 483T622 553Q608 628 562 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E16-MJFRAK-52" x="0" y="0"></use></g></svg></span><script type="math/tex">\mathfrak{R}</script><span> </span><em><span>preserves</span></em><span> </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.979ex" height="1.827ex" viewBox="0 -741.3 852 786.8" role="img" focusable="false" style="vertical-align: -0.106ex;"><defs><path stroke-width="0" id="E17-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E17-MJMATHI-58" x="0" y="0"></use></g></svg></span><script type="math/tex">X</script>’<span>s truth so long as every sentence in </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.923ex" height="1.928ex" viewBox="0 -741.3 828 830.3" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E16-MJFRAK-52" d="M27 496Q31 569 102 627T234 685Q236 685 241 685T251 686Q287 686 318 672T367 638T399 598T418 564L423 550Q424 554 434 567T463 601T505 639T561 671T626 685Q672 685 688 659T710 572Q713 533 721 523T766 513Q781 513 787 514T794 516Q796 512 798 509T801 504T802 501T787 493Q702 461 624 401L607 389Q655 383 688 358L697 352V342Q699 330 699 297Q704 209 710 173T734 103Q751 69 765 69Q769 69 806 83L824 90V74Q823 73 759 24T693 -26Q692 -26 660 32L628 90L629 111Q631 159 631 177Q631 278 614 300Q584 340 523 340Q500 340 467 333T431 325Q429 325 429 322Q428 321 426 308T420 275T410 230T392 178T366 125L358 112L342 99Q306 70 269 38T213 -10T193 -26Q192 -26 163 0T116 26Q82 26 50 -8L42 -16L35 -8L27 0L35 10Q43 21 58 38T104 80T158 106Q179 106 218 65L235 48Q238 48 255 60T295 99T329 158Q352 231 352 359Q352 555 242 614Q210 628 187 628Q140 628 116 600T91 548Q91 522 138 464T185 382V376Q185 345 158 313T103 263L76 246Q74 244 64 253L54 260L65 267Q91 285 100 302Q111 318 111 337Q111 355 69 410T27 496ZM562 628Q504 628 443 507L435 491L436 479Q437 471 437 446Q437 396 432 351L529 389L602 426Q673 462 673 463H672Q644 470 637 483T622 553Q608 628 562 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E16-MJFRAK-52" x="0" y="0"></use></g></svg></span><script type="math/tex">\mathfrak{R}</script><span> is true,</span></strong><span> </span></p><p><span>we arrive at the yellow wood, and our question is here.</span><sup class='md-footnote'><a href='#dfref-footnote-15' name='ref-footnote-15'>15</a></sup></p><p> </p><div style="text-align:center;"><i>how do we sift the logical from the non-logical?</i></div><p> </p><p><span>The virtue of the semantical theory rests in making </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.74ex" height="1.827ex" viewBox="0 -741.3 749 786.8" role="img" focusable="false" style="vertical-align: -0.106ex;"><defs><path stroke-width="0" id="E13-MJMATHI-46" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E13-MJMATHI-46" x="0" y="0"></use></g></svg></span><script type="math/tex">F</script><span> necessary but insufficient. </span><strong><span>Yet like all virtues, here too lies vice;</span></strong><span> they, of course, sleep in the same bed and warm each other by their closeness. </span></p><div style="font-size:50%;text-align:right;">I do not exactly mean,<br>“I must be cruel, only to be kind.<br>Thus bad begins<br>and worse remains behind.”<br>I mean only that<br>my cruelty to you now<br>is also an act of self-kindness,<br>and it protects me.<br><del>May you have sufficient vice<br>to absolve my virtue.</del></div><p> </p><p><span>The vice we currently face is that </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.74ex" height="1.827ex" viewBox="0 -741.3 749 786.8" role="img" focusable="false" style="vertical-align: -0.106ex;"><defs><path stroke-width="0" id="E13-MJMATHI-46" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E13-MJMATHI-46" x="0" y="0"></use></g></svg></span><script type="math/tex">F</script><span> depends upon a division of all our language</span>’<span>s terms into the logical and non-logical. However, the possibility of division is precisely our question, </span><strong><span>AND THE ANSWER IS THAT THERE IS NO SOLID LINE BETWEEN THEM</span></strong><span>.</span><sup class='md-footnote'><a href='#dfref-footnote-16' name='ref-footnote-16'>16</a></sup><span> </span></p><p> </p><div style="background-color: #E9F5F8;"><div style="font-size:400%; text-align:center;">Your consideration with more care,<br>I ask for only a moment’s briefest time—<br></div> We are companions seeking to find an answer to the question of <i>what is logic</i>. But it is the case that <b>NO ONE KNOWS THE LINE FROM THE LOGICAL AND THE NON-LOGICAL <i>EXACTLY</i></b>. Friend, I confess though it is already obvious: there is not one thing to find.<br><br>You are furious? I can see. You condemn me? That, I long knew you would. Your abuse is nothing to me. I will always guard against it. <br><br>You are acting like a brute: did I not warn you that there is no treasure to find? Yet you still become like an imperial man. It is not even my fault that there should be such a fundamental puzzle rooted within the core of logic itself.<br><br>Try to understand that looking for treasure will be your ruin and that it is not truly worth so much. What is worthier is wonder and joy in wonder: to seek without expecting to <b>possess what you find</b> and to ask without commanding <b>an</b> answer. You would not wish to disturb the world. You would wish that the world disturbs you by bringing awe and all that could be your beloved. And know that this is not “treasure.” It is past time we saw that the set of treasure and the set of the beloved are neither co-extensional nor intersecting. This was not our error; we are not to blame for what we inherit, which is a language that confuses the two when they <i>should be</i> disjoint. True, there is no definitive answer waiting for us. But <b>consider it again</b>. Is it not better to stay, just in case something should travel our way?<br><br>Come now, I will put silence to it and forget—<br><br><span style="color: #E9F5F8;">silence to forget.</span></div> <p> </p><p><span style="font-size:1300%;"><span>Thus,</span><br><span>consequently,</span><br><span>as such,</span><br><span>therefore,</span><br><span>and it follows that</span></span><span> </span></p><div style="font-size:500%"> CONSEQUENCE,</div> <p><span>that primordial, chaotic concept from which the logical world is forged and cools over, </span></p><p><span>is not merely primordial and chaotic:</span></p><p><span>it also makes us look far beyond a yellow wood</span></p><p><em><span>where we discover that our fundamental question will never yield itself.</span></em></p><p><span>Now we may face with awe the fog that surrounds</span><br></p><p><span>and try with the means we have to approach</span><br></p><p><span>even if we should always remain at its fringes. </span></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><b><span>It is simply an astonishing inversion.</span></b></p><p><span>REMEMBER THAT IT WAS ONCE SOUGHT TO CLARIFY THE FOG THAT HANGS AROUND THE LOGICAL WORLD. </span></p><p><span>Though we have managed to utter some things, we have not said enough to dispel it completely. SO TODAY WE LET OURSELVES BE SUBSUMED BY IT. </span></p><p><span>Do not laugh. You think I am offering a false mountain mysticism. </span><br><span>I am not—you will never understand my aims. </span></p><p><span>I pity that you think to accept the surrounding fog is a humiliating defeat. I pity that you think our ending must be here. </span></p><p><span>And I pity most of all that you would rather stand helpless in trepidation when faced by expanding daze. Why have you thought that fog means there is nowhere to go? </span></p><p><span style="background-color:#333;"><span>Yours is the attitude of someone who must conquer or have nothing. If they are smarted once, they will always lie low and be the kind of meek one for whom no person could have sympathy. My friend, I am sorry for you; yours is the way of someone who will never know how to live.</span></span></p><p><span>Do not think that logicians have buried their tools and wept. No, they have kept working all this time, no matter how near ambiguity lay. In fact, clarity—the design of logic—and precision—the aim of logicians—gain their power from its nearness. </span><span style="background-color:#333;"><span>You would see that they are connected if you would only LOOK.</span></span><span> Were everything </span><strong><span>so precise</span></strong><span>, </span><strong><span>so clear to us</span></strong><span> there might have been little left for the logician to do. In the end, that would have been a much more disappointing world. </span></p><p><span>Fortunately, that world is not ours. </span><span style="background-color:#333;"><span>Have heart, a possible world may present itself in time.</span></span><span> In actuality, logicians have expanded the </span><strong><span>logical world</span></strong><span> to create </span><strong><span>logical worlds</span></strong><span>, and this they did despite surrounding </span><strong><span>fog,</span></strong><span> which is an accomplishment for any theoretical undertaking. From the great and once lone classical logic, they have multiplied not just </span><em><span>how many logics there are</span></em><span> but also </span><em><span>how many notions of logic we may have</span></em><span>. These two are bound together anyway, but sometimes we forget that </span><strong><span>what there is to see affects what we may understand</span></strong><span>. </span></p><p> </p><div style="text-align: right; margin: 0 50% 0 0;">
LOOK:<br><br>Are you grasping that I draw concepts<br>together where many would not?<br>I have joined<br><b>clarity—ambiguity</b></div><div style="text-align: left; margin: 0 0 0 50%;"><b>as well as<br><br>one—many<br>creation—destruction<br>virtue—vice<br>logical—non-logical.</b></div><div style="text-align: right; margin: 0 50% 0 0;"><b>I have intersected<br>points with lines<br>visible with invisible<br>silence with saying<br>cruelty with kindness.</b></div><div style="text-align: left; margin: 0 0 0 50%;"><b>I have housed ideas in bodies<br>and traded a coin for a stream.<br>I have made my content<br>indivisible from its form,<br>and I have spoke of ‘You’ and ‘I’<br>without saying<br>these were always different.</b></div><div style="text-align: right; margin: 0 50% 0 0;">With two words and a line between,<br>you may learn to see<br>discrete beside continuous,<br> or at least see how severing tears<br>may yet be binding sites.</div><p> </p><p><span>We are ready to meet a few of the many that wait for us ahead. We will see their differences and their commons and respect all together. We will see how their lines create new understanding from destroying old assumptions, and how a view can shape our logical world. Finally, may we forswear any remaining instinct that suggests </span><strong><span>insurmountable opposition and singular domination.</span></strong><span> For that logic is an imperial logic.</span><sup class='md-footnote'><a href='#dfref-footnote-17' name='ref-footnote-17'>17</a></sup></p><p> </p><h2><a name="logicrsquos-views" class="md-header-anchor"></a><span>Logic</span>’<span>s views</span></h2><p> </p><p><span>Sometime ago, I presented to you a theory of following logically. I am sure you have since discarded it so I shall give it to you again:</span></p><p><del>“<span>We say that the sentence </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.979ex" height="1.827ex" viewBox="0 -741.3 852 786.8" role="img" focusable="false" style="vertical-align: -0.106ex;"><defs><path stroke-width="0" id="E17-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E17-MJMATHI-58" x="0" y="0"></use></g></svg></span><script type="math/tex">X</script><span> follows logically from the sentences of the class </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.923ex" height="1.928ex" viewBox="0 -741.3 828 830.3" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E16-MJFRAK-52" d="M27 496Q31 569 102 627T234 685Q236 685 241 685T251 686Q287 686 318 672T367 638T399 598T418 564L423 550Q424 554 434 567T463 601T505 639T561 671T626 685Q672 685 688 659T710 572Q713 533 721 523T766 513Q781 513 787 514T794 516Q796 512 798 509T801 504T802 501T787 493Q702 461 624 401L607 389Q655 383 688 358L697 352V342Q699 330 699 297Q704 209 710 173T734 103Q751 69 765 69Q769 69 806 83L824 90V74Q823 73 759 24T693 -26Q692 -26 660 32L628 90L629 111Q631 159 631 177Q631 278 614 300Q584 340 523 340Q500 340 467 333T431 325Q429 325 429 322Q428 321 426 308T420 275T410 230T392 178T366 125L358 112L342 99Q306 70 269 38T213 -10T193 -26Q192 -26 163 0T116 26Q82 26 50 -8L42 -16L35 -8L27 0L35 10Q43 21 58 38T104 80T158 106Q179 106 218 65L235 48Q238 48 255 60T295 99T329 158Q352 231 352 359Q352 555 242 614Q210 628 187 628Q140 628 116 600T91 548Q91 522 138 464T185 382V376Q185 345 158 313T103 263L76 246Q74 244 64 253L54 260L65 267Q91 285 100 302Q111 318 111 337Q111 355 69 410T27 496ZM562 628Q504 628 443 507L435 491L436 479Q437 471 437 446Q437 396 432 351L529 389L602 426Q673 462 673 463H672Q644 470 637 483T622 553Q608 628 562 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E16-MJFRAK-52" x="0" y="0"></use></g></svg></span><script type="math/tex">\mathfrak{R}</script><span> if and only if every model of the class </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.923ex" height="1.928ex" viewBox="0 -741.3 828 830.3" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E16-MJFRAK-52" d="M27 496Q31 569 102 627T234 685Q236 685 241 685T251 686Q287 686 318 672T367 638T399 598T418 564L423 550Q424 554 434 567T463 601T505 639T561 671T626 685Q672 685 688 659T710 572Q713 533 721 523T766 513Q781 513 787 514T794 516Q796 512 798 509T801 504T802 501T787 493Q702 461 624 401L607 389Q655 383 688 358L697 352V342Q699 330 699 297Q704 209 710 173T734 103Q751 69 765 69Q769 69 806 83L824 90V74Q823 73 759 24T693 -26Q692 -26 660 32L628 90L629 111Q631 159 631 177Q631 278 614 300Q584 340 523 340Q500 340 467 333T431 325Q429 325 429 322Q428 321 426 308T420 275T410 230T392 178T366 125L358 112L342 99Q306 70 269 38T213 -10T193 -26Q192 -26 163 0T116 26Q82 26 50 -8L42 -16L35 -8L27 0L35 10Q43 21 58 38T104 80T158 106Q179 106 218 65L235 48Q238 48 255 60T295 99T329 158Q352 231 352 359Q352 555 242 614Q210 628 187 628Q140 628 116 600T91 548Q91 522 138 464T185 382V376Q185 345 158 313T103 263L76 246Q74 244 64 253L54 260L65 267Q91 285 100 302Q111 318 111 337Q111 355 69 410T27 496ZM562 628Q504 628 443 507L435 491L436 479Q437 471 437 446Q437 396 432 351L529 389L602 426Q673 462 673 463H672Q644 470 637 483T622 553Q608 628 562 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E16-MJFRAK-52" x="0" y="0"></use></g></svg></span><script type="math/tex">\mathfrak{R}</script><span> is at the same time a model of the sentence </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.979ex" height="1.827ex" viewBox="0 -741.3 852 786.8" role="img" focusable="false" style="vertical-align: -0.106ex;"><defs><path stroke-width="0" id="E17-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E17-MJMATHI-58" x="0" y="0"></use></g></svg></span><script type="math/tex">X</script><span>.</span>”</del><sup class='md-footnote'><a href='#dfref-footnote-18' name='ref-footnote-18'>18</a></sup></p><p><span>It is intimately related to a much wider theory, who sometimes inspires the view that </span>“<span>the relation of a sentence to its models is the cornerstone of all semantics.</span>”<sup class='md-footnote'><a href='#dfref-footnote-19' name='ref-footnote-19'>19</a></sup><span> This is </span><strong><span>model theory</span></strong><span>. </span></p><p><strong><span>Model Theory</span></strong><span> is not so esoteric as you might think: it is the study of what you have done heretofore. All this time, and at every turn of the word, you have watched me and </span><strong><span>INTERPRETED</span></strong><span> what you saw. </span><span style="background-color:#333;"><span>It is exasperating that I cannot control how you see so that you would need not interpret. And it is maddening that for what I can control, what there is to see, I may control it only negligibly. </span></span><span> </span></p><div style="text-align:center; margin:0 17% 0 17%;">
After all, would you have elected to remain by me were it the case that all these words were so fundamentally unreadable & uninterpretable?<br><br>Think of sense—significance.<br><img src="assets\ph-303-small.jpg" alt="PH-303 by Clyfford Still"><br><small><i>PH 303</i> by Clyfford Still. Clyfford Still Museum.</small><br>If you similarly understand ‘sense,’ you would agree that everything contained from ‘Think’ to now has its sense.<br><br>Using this sense of sense:<br>Imagine that every word of mine was <i>much more</i> <span style="font-size:450%">SENSELESS.</span><br>Something more <span style="font-size:450%">SENSELESS</span> than a falling leaf in Spring or a drop falling into sand. That is, something so profoundly <span style="font-size:450%">SENSELESS</span> that I cannot even talk about it or tell you its name. <span style="background-color:#333;">It should be clear that I do not mean ‘something yet to be interpreted’ like a syntactical sign.</span> I have not the words to describe something that is nothing to me. <del>Of course, I am failing now too. I may as well be blathering, <span style="background-color:#333;">or aren’t I already?</span> but my failure is necessary.</del><br><br> Were it so, there would be nothing there <i>for you</i> to <b>interpret</b>, and you would not have grasped that there was anything there to <b>interpret</b> in the first place. So do not pretend that you would have kept by me. You would only look foolish.
</div><p> </p><p><span>Sometimes you will </span><strong><span>interpret</span></strong><span> me as I intend. Other times, your </span><strong><span>interpretation</span></strong><span> will be far away. Yet no matter what it is, I will either be </span><strong><span>interpreted</span></strong><span> or </span><strong><span>reinterpreted</span></strong><span>. I, anyway, cannot prevent it. </span></p><p><span>To be clear, </span><a href='https://plato.stanford.edu/entries/model-theory/#Philosophy'><span>model theory</span></a><span> is the </span><em><span>mathematical</span></em><span> study of </span><strong><span>the interpretations</span></strong><span> of any language. Here, its interest for us is that in the historical development of logic, it gave one </span><strong><span>SHAPE</span></strong><span> to our understanding of the logical. </span><sup class='md-footnote'><a href='#dfref-footnote-20' name='ref-footnote-20'>20</a></sup><span> There are, in fact, two ways that we have more-or-less </span><strong><span>SHAPED</span></strong><span> our conception of the logical since the time of Boole and Frege.</span><sup class='md-footnote'><a href='#dfref-footnote-21' name='ref-footnote-21'>21</a></sup><span> </span></p><p><span style="font-size:200%; text-align: left;"><span>ONE is </span></span><span>logic as calculus, which is to understand logic in a </span><strong><span>model-theoretic</span></strong><span> fashion.</span></p><p><span style="font-size:200%; text-align: left;"><span>TWO is </span></span><span>logic as language, which is nothing less than a </span><strong><span>universal medium</span></strong><span>.</span><sup class='md-footnote'><a href='#dfref-footnote-22' name='ref-footnote-22'>22</a></sup></p><div style="text-align: right; font-size:300%;">I have discussed FRAMES, and<br>I have discussed BOUNDLESSNESS.</div><div style="text-align: right; margin: 0 0 0 35%;">Here is where I shall relate the two though they appear opposed. In sum, their common relation is that they <b>SHAPE</b>.<br><br>
Cognizing logic as ONE—as calculus—requires that we have our frames. What would we have to frame? <i>The logical world.</i> By doing so, we would limit the height of all our logical flights by tying each to their own proper <b>domain</b>. Our frames would set a context—<b>a universe of discourse</b>—for each logical formula that we consider, and each context may be left to our imagination. <span style="background-color:#333;">Just as it is for us now: this text will appear in your frame, which shapes your interpretation of it. Or you may consider it in the context of the literature, which will also supply an interpretation. I have not kept my notes for no reason at all.</span> Moreover, a logical formula may be placed in many different frames. Thus, our logical language becomes endlessly<br><b>REINTERPRETABLE</b>.<br><br><b>REINTERPRETABILITY</b> is not the only characteristic of ONE. When we are <b>framing</b> the logical, we make our <b>universe of discourse</b> bounded, but since we may step outside any bound and enter any other domain, a <b>metatheory</b> becomes possible. We now become able to ‘speak’ about those logical languages which we leave behind.<br><br>Cognizing logic as TWO—as language—differently <b>shapes</b> the logical world and what is possible within it. When TWO is the shape of our view, the logical language becomes a <i>universal language</i>. It has a context, but its context is the whole scope: it is <b>boundless</b>, and there is no <b>alternative frame, universe of discourse, or language</b>. Once our logical language is <i>put in a frame with no bound</i>, we will never suceed in stepping outside. A <b>metatheory</b> is impossible, nonsensical even, and though there is context, this context is the only one available, making reinterpretation similarly <b>INCONCEIVABLE.</b></div> <p> </p><p> </p><p><img src="assets\frames.jpg" alt="Frames of jumping over a boys back."></p><center><small><i>Jumping over boy’s back</i> by Eadweard Muybridge. Kingston Museum and Heritage Service. </small></center><p> </p><div style="text-align:left; margin: 0 60% 0 0;"><span style="font-size:130%;">For your eyes, I have placed two lenses so that you may understand how perception shapes a view.<br><br>Perception gives you your place in a room. But no matter how the matter really is, room or no room, your perception will shape how you move in what you perceive. The perceived world is driven by your perception, your perception determines how you move, how you move determines how your surroundings move in response and these together create your <i>view.</i> And then the process may proceed in reverse as your view alters what you see and how you move. This, in fact, it must do, for how could you have initially perceived <i>a room</i> without a view? And so the process goes, moving clockwise and counterclockwise.<br><br> These metaphors may not be true in our world, but they are true in the logical world.<br></span><br><img src="assets\le-jour-small.jpg" alt="Le Jour by Odilon Redon"><br><p style="font-size:50%;"><i>Le Jour (Day)</i> by Odilon Redon.<br>The Metropolitan Museum of Art.</p></div><p> </p><p><strong><span>A way of viewing shapes, and eventually becomes shaped by, activity and understanding</span></strong><span>: that much is clearly seen in those logicians who had divided themselves between ONE and TWO.</span><sup class='md-footnote'><a href='#dfref-footnote-23' name='ref-footnote-23'>23</a></sup><span> But what do we see when we see with logic? Do we see our rock-collided, ocean-covered, and core-solidified physical world? </span></p><p><span>Or rather do we see something more like, </span></p><p><img src ="assets\pointless-abstraction.jpg" alt="Picture of a Pointless Abstraction Framed Under Glass from a Hollywood Collection by David Hockney"></p><center><small><i>Picture of a Pointless Abstraction Framed under Glass</i> from <i>A Hollywood Collection</i> by David Hockney.</small></center><div style="font-size:350%; text-align: center;">?</div><p> </p><p><span>Whether logic helps us more clearly see the world or abstractions, </span><span style="background-color:#333"><span>and there is no reason to think that it could not be both</span></span><span> all logics share the characteristic that they are </span><strong><span>principled</span></strong><span>, irrespective of whether those </span><strong><span>principles</span></strong><span> align with any liquid intuition. Moreover, we can make </span><strong><span>symbol-pieces</span></strong><span> and make them so that they are forced to </span><strong><span>move</span></strong><span> in accordance with those </span><strong><span>principles</span></strong><span>. These two things, </span><strong><span>principles</span></strong><span> and </span><strong><span>symbol-pieces</span></strong><span>, when combined with any </span><strong><span>domain of play,</span></strong><span> </span><span style="background-color:#333"><span>whether or not it is boundless</span></span><span> determine every possible configuration. Here, I mean not just every possible formula or sentence. I mean as well every possible </span><em><span>valid formula</span></em><span> AND every possible </span><em><span>valid sequence of moves</span></em><span> that </span><em><span>follow</span></em><span> one after the other in succession. </span><span style="background-color:#333"><span>This is otherwise known as an argument or derivation.</span></span><span> </span></p><p> </p><div style="text-align:center;font-size:300%;"> I DESCRIBE A GAME</div>
<div style="text-align:center;font-size:150%;"> I DESCRIBE A GAME</div>
<div style="text-align:center;font-size:75%;"> I DESCRIBE A GAME</div>
<div style="text-align:center;font-size:75%;">assimilating syntax—semantics.</div>
<div style="text-align:center;font-size:150%;">needing each part for play.</div>
<div style="text-align:center;font-size:300%;">showing logic as activity. </div><p> </p><p><span>Thus to reconstruct what we see with logic and how so, we might walk in reverse and first ask </span><i><span>what do we do?</span></i></p><p><span style="font-size:150%;"><span>We begin with principles.</span></span></p><p><span>Classical logic has its principles, and some of the most notable are</span></p><ol start='' ><li><span>LAW OF EXCLUDED MIDDLE</span></li><li><span>BIVALENCE</span></li><li><span>PRINCIPLE OF NON-CONTRADICTION</span></li><li><span>PRINCIPLE OF EXTENSIONALITY.</span></li></ol><p><span>All of its principles, along with the conventional symbol-pieces that accord to them, make valid</span></p><ol start='' ><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.829ex" height="2.636ex" viewBox="0 -828.3 6815.1 1134.7" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E53-MJMAIN-AC" d="M56 323T56 336T70 356H596Q603 353 611 343V102Q598 89 591 89Q587 89 584 90T579 94T575 98T572 102L571 209V316H70Q56 323 56 336Z"></path><path stroke-width="0" id="E53-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E53-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E53-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E53-MJMATHI-42" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path><path stroke-width="0" id="E53-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E53-MJMAIN-AC" x="0" y="0"></use><use xlink:href="#E53-MJMATHI-41" x="667" y="0"></use><use xlink:href="#E53-MJMAIN-2192" x="1694" y="0"></use><use xlink:href="#E53-MJMAIN-28" x="2972" y="0"></use><use xlink:href="#E53-MJMATHI-41" x="3361" y="0"></use><use xlink:href="#E53-MJMAIN-2192" x="4389" y="0"></use><use xlink:href="#E53-MJMATHI-42" x="5667" y="0"></use><use xlink:href="#E53-MJMAIN-29" x="6426" y="0"></use></g></svg></span><script type="math/tex">\neg A \to (A \to B)</script></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.162ex" height="2.029ex" viewBox="0 -784.8 3083.6 873.8" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E156-MJMAIN-22A5" d="M369 652Q369 653 370 655T372 658T375 662T379 665T384 667T391 668Q402 666 409 653V40H708Q723 32 723 20T708 0H71Q70 0 67 2T59 9T55 20T59 31T66 38T71 40H369V652Z"></path><path stroke-width="0" id="E156-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E156-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E156-MJMAIN-22A5" x="0" y="0"></use><use xlink:href="#E156-MJMAIN-2192" x="1055" y="0"></use><use xlink:href="#E156-MJMATHI-41" x="2333" y="0"></use></g></svg></span><script type="math/tex">\bot \to A</script></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.614ex" height="2.029ex" viewBox="0 -784.8 3278.4 873.8" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E106-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E106-MJMAIN-2228" d="M55 580Q56 587 61 592T75 598Q86 598 96 580L333 48L570 580Q579 596 586 597Q588 598 591 598Q609 598 611 580Q611 574 546 426T415 132T348 -15Q343 -22 333 -22T318 -15Q317 -14 252 131T121 425T55 580Z"></path><path stroke-width="0" id="E106-MJMAIN-AC" d="M56 323T56 336T70 356H596Q603 353 611 343V102Q598 89 591 89Q587 89 584 90T579 94T575 98T572 102L571 209V316H70Q56 323 56 336Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E106-MJMATHI-41" x="0" y="0"></use><use xlink:href="#E106-MJMAIN-2228" x="972" y="0"></use><use xlink:href="#E106-MJMAIN-AC" x="1861" y="0"></use><use xlink:href="#E106-MJMATHI-41" x="2528" y="0"></use></g></svg></span><script type="math/tex">A \lor \neg A</script></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.971ex" height="2.636ex" viewBox="0 -828.3 4723.4 1134.7" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E133-MJMAIN-AC" d="M56 323T56 336T70 356H596Q603 353 611 343V102Q598 89 591 89Q587 89 584 90T579 94T575 98T572 102L571 209V316H70Q56 323 56 336Z"></path><path stroke-width="0" id="E133-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E133-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E133-MJMAIN-2227" d="M318 591Q325 598 333 598Q344 598 348 591Q349 590 414 445T545 151T611 -4Q609 -22 591 -22Q588 -22 586 -21T581 -20T577 -17T575 -13T572 -9T570 -4L333 528L96 -4Q87 -20 80 -21Q78 -22 75 -22Q57 -22 55 -4Q55 2 120 150T251 444T318 591Z"></path><path stroke-width="0" id="E133-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E133-MJMAIN-AC" x="0" y="0"></use><use xlink:href="#E133-MJMAIN-28" x="667" y="0"></use><use xlink:href="#E133-MJMATHI-41" x="1056" y="0"></use><use xlink:href="#E133-MJMAIN-2227" x="2028" y="0"></use><use xlink:href="#E133-MJMAIN-AC" x="2917" y="0"></use><use xlink:href="#E133-MJMATHI-41" x="3584" y="0"></use><use xlink:href="#E133-MJMAIN-29" x="4334" y="0"></use></g></svg></span><script type="math/tex">\neg(A \land \neg A)</script></li></ol><p><span>among others.</span></p><p><span>Despite their origin in choice and inheritance, these principles were lauded as the ultimate principles of thought and language </span><span style="background-color:#333;"><span> sometimes even the world</span></span><span>.</span><sup class='md-footnote'><a href='#dfref-footnote-24' name='ref-footnote-24'>24</a></sup><span>However, if language—or thought—is </span><i><span>the ultimate activity,</span></i><span> then there can be no other logical game </span><span style="background-color:#333;"><span>that is, no other activity</span></span><span> that goes against its principles. Today, we can see this for what it is: an old assumption. </span><del><span>to uncover an assumption is to attempt to eliminate it.</span></del><span> The world has changed many times over since then, and I include here </span><em><span>the logical world</span></em><span>. We made, and continue to make, new logics that do not always agree with classical principles. </span></p><div style="text-align:right;font-size:150%;">CLASSICAL LOGIC IS ITS OWN GAME.<br>THERE ARE OTHERS THAT<br>HAVE THEIR OWN RIGHT<br>TO THE TITLE ‘LOGICAL GAME’.</div><p> </p><p><span>The non-classical logics are, in fact, not so differently constructed to bar their addition to the concept of logic. </span></p><p><span>IT IS JUST THAT SOME USE CERTAIN </span><strong><span>PRINCIPLES</span></strong><span> WHERE OTHERS DO NOT. Our choice of principles, of course, affects our choice of </span><strong><span>symbol-pieces</span></strong><span> that must </span><strong><span>move</span></strong><span> in accordance to them. These together will then determine our possible </span><strong><span>configurations</span></strong><span> and, ultimately, who can follow whom in </span><strong><span>sequence</span></strong><span>. </span><del><span>Here again appears consequence, another testament to its importance.</span></del><span> </span><span style="background-color:#333;"><span>Owing to this, you may think of each logic as having, in some measure, their own sense of </span><b><span>following.</span></b></span><span> IN OTHER WORDS, THEY ARE MADE IN THE SAME WAY ALBEIT OF DIFFERENT MATERIAL: THIS IS THE EXTENT OF THEIR DIFFERENCE.</span></p><p> </p><div style="text-align: right; margin:0 0 0 70%">Do not so quickly reject the others simply because you believe reason too has its principles and that those principles align exactly with classical ones. <b>If you must renounce per your principles and believe all other logics to be false, keep your brutish scorn to yourself.</b> There is still wonder in the others, even if they must be false. Or do you similarly detest <i>images and poems?</i></div><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><p><span style="font-size:150%"><span>Shall we see for ourselves what these games look like?</span></span></p><p><i><span>Here</span></i><span> is Łukasiewicz logic, and </span><i><span>there</span></i><span> is something I cannot yet make out. But it is too far, and I will not seek to disturb it.</span></p><p><span>Immediately before us:</span></p><div style="font-size:200%;">PRINCIPLES</div> <ol start='' ><li><del><span>LAW OF EXCLUDED MIDDLE</span></del></li><li><del><span>BIVALENCE</span></del></li><li><del><span>PRINCIPLE OF NON-CONTRADICTION</span></del></li><li><span>PRINCIPLE OF EXTENSIONALITY.</span></li></ol><p><span>Łukasiewicz logic, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.311ex" height="2.535ex" viewBox="0 -784.8 995.1 1091.3" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E168-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(43.49) matrix(1 0 0 -1 0 0)">Ł</text><use transform="scale(0.707)" xlink:href="#E168-MJMAIN-33" x="765" y="-324"></use></g></svg></span><script type="math/tex">Ł_3</script><span>, is a many-valued logic, and being a many-valued logic, it first and foremost ignores </span><strong><span>BIVALENCE</span></strong><span>. </span></p><div style="text-align:center;">BIVALENCE is<br>thinking that<br>there are<br>only TWO<br>values <i>true</i><br>and <i>false.</i></div><p> </p><p><span>The idea that </span><strong><span>every meaningful sentence must be either true or false as a matter of general law</span></strong><span> is one that Łukasiewicz thought to be an </span>“<span>unshakeable dogma</span>”<span> of Western thought.</span><sup class='md-footnote'><a href='#dfref-footnote-25' name='ref-footnote-25'>25</a></sup><span> Though, you may also see that the idea of a </span><strong><span>general logical law</span></strong><span> is itself dogmatic; there may only be </span><strong><span>principles for each logical world</span></strong><span>. </span><sup class='md-footnote'><a href='#dfref-footnote-26' name='ref-footnote-26'>26</a></sup><span> </span></p><p> </p><div style="text-align: right; margin:0 0 0 70%;">Yet why reject bivalence? Is it not totally impossible for a sentence to be neither true nor false? Either we have or haven’t died. Either we are or are not alive.<br><br><i>It is simple; it is never other than true or false.</i><br><br>So you say, but I ask you <i>what of tomorrow?</i> Tomorrow, is it true now that we live? Or is it false?<br><br><b>I am not asking whether you know: I am asking whether it’s true.</b><br><br>There are those who think neither and consider the matter as strictly <i>undetermined,</i> irrespective of how we may protect ourselves.</div><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.311ex" height="2.535ex" viewBox="0 -784.8 995.1 1091.3" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E168-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(43.49) matrix(1 0 0 -1 0 0)">Ł</text><use transform="scale(0.707)" xlink:href="#E168-MJMAIN-33" x="765" y="-324"></use></g></svg></span><script type="math/tex">Ł_3</script><span> similarly ignores</span></p><p><span>THE LAW OF EXCLUDED MIDDLE:</span></p><p><strong><span>Either it is the case or it is not the case</span></strong><span> holds for every formula. It is written formally as </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.614ex" height="2.029ex" viewBox="0 -784.8 3278.4 873.8" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E106-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E106-MJMAIN-2228" d="M55 580Q56 587 61 592T75 598Q86 598 96 580L333 48L570 580Q579 596 586 597Q588 598 591 598Q609 598 611 580Q611 574 546 426T415 132T348 -15Q343 -22 333 -22T318 -15Q317 -14 252 131T121 425T55 580Z"></path><path stroke-width="0" id="E106-MJMAIN-AC" d="M56 323T56 336T70 356H596Q603 353 611 343V102Q598 89 591 89Q587 89 584 90T579 94T575 98T572 102L571 209V316H70Q56 323 56 336Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E106-MJMATHI-41" x="0" y="0"></use><use xlink:href="#E106-MJMAIN-2228" x="972" y="0"></use><use xlink:href="#E106-MJMAIN-AC" x="1861" y="0"></use><use xlink:href="#E106-MJMATHI-41" x="2528" y="0"></use></g></svg></span><script type="math/tex">A \lor \neg A</script><span>. </span><span style="background-color:#333;"><span>this is not the same as bivalence.</span></span></p><p><span>THE PRINCIPLE OF NON-CONTRADICTION:</span></p><p><strong><span>It is not the case that it is and is not the case</span></strong><span> holds for every formula. It is written formally as </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.971ex" height="2.636ex" viewBox="0 -828.3 4723.4 1134.7" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E133-MJMAIN-AC" d="M56 323T56 336T70 356H596Q603 353 611 343V102Q598 89 591 89Q587 89 584 90T579 94T575 98T572 102L571 209V316H70Q56 323 56 336Z"></path><path stroke-width="0" id="E133-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E133-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E133-MJMAIN-2227" d="M318 591Q325 598 333 598Q344 598 348 591Q349 590 414 445T545 151T611 -4Q609 -22 591 -22Q588 -22 586 -21T581 -20T577 -17T575 -13T572 -9T570 -4L333 528L96 -4Q87 -20 80 -21Q78 -22 75 -22Q57 -22 55 -4Q55 2 120 150T251 444T318 591Z"></path><path stroke-width="0" id="E133-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E133-MJMAIN-AC" x="0" y="0"></use><use xlink:href="#E133-MJMAIN-28" x="667" y="0"></use><use xlink:href="#E133-MJMATHI-41" x="1056" y="0"></use><use xlink:href="#E133-MJMAIN-2227" x="2028" y="0"></use><use xlink:href="#E133-MJMAIN-AC" x="2917" y="0"></use><use xlink:href="#E133-MJMATHI-41" x="3584" y="0"></use><use xlink:href="#E133-MJMAIN-29" x="4334" y="0"></use></g></svg></span><script type="math/tex">\neg(A \land \neg A)</script><span>.</span></p><p> </p><div style="font-size:200%;">SYMBOL-PIECES</div> <p><span>Propositional constants and connectives</span><sup class='md-footnote'><a href='#dfref-footnote-27' name='ref-footnote-27'>27</a></sup><span>:</span></p><ol start='' ><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.807ex" height="1.827ex" viewBox="0 -741.3 778 786.8" role="img" focusable="false" style="vertical-align: -0.106ex;"><defs><path stroke-width="0" id="E207-MJMAIN-22A5" d="M369 652Q369 653 370 655T372 658T375 662T379 665T384 667T391 668Q402 666 409 653V40H708Q723 32 723 20T708 0H71Q70 0 67 2T59 9T55 20T59 31T66 38T71 40H369V652Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E207-MJMAIN-22A5" x="0" y="0"></use></g></svg></span><script type="math/tex">\bot</script><span> </span><em><span>for falsity</span></em><span> &</span></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.549ex" height="1.12ex" viewBox="0 -436.9 667 482.4" role="img" focusable="false" style="vertical-align: 0.207ex; margin-bottom: -0.312ex;"><defs><path stroke-width="0" id="E27-MJMAIN-AC" d="M56 323T56 336T70 356H596Q603 353 611 343V102Q598 89 591 89Q587 89 584 90T579 94T575 98T572 102L571 209V316H70Q56 323 56 336Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E27-MJMAIN-AC" x="0" y="0"></use></g></svg></span><script type="math/tex">\neg</script><span> </span><em><span>for negation</span></em><span> &</span></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.549ex" height="1.726ex" viewBox="0 -654.4 667 743.3" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E212-MJMAIN-2227" d="M318 591Q325 598 333 598Q344 598 348 591Q349 590 414 445T545 151T611 -4Q609 -22 591 -22Q588 -22 586 -21T581 -20T577 -17T575 -13T572 -9T570 -4L333 528L96 -4Q87 -20 80 -21Q78 -22 75 -22Q57 -22 55 -4Q55 2 120 150T251 444T318 591Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E212-MJMAIN-2227" x="0" y="0"></use></g></svg></span><script type="math/tex">\land</script><span> </span><em><span>for conjunction</span></em><span> &</span></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.549ex" height="1.726ex" viewBox="0 -654.4 667 743.3" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E218-MJMAIN-2228" d="M55 580Q56 587 61 592T75 598Q86 598 96 580L333 48L570 580Q579 596 586 597Q588 598 591 598Q609 598 611 580Q611 574 546 426T415 132T348 -15Q343 -22 333 -22T318 -15Q317 -14 252 131T121 425T55 580Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E218-MJMAIN-2228" x="0" y="0"></use></g></svg></span><script type="math/tex">\lor</script><span> </span><em><span>for disjunction</span></em><span> &</span></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.323ex" height="1.524ex" viewBox="0 -567.4 1000 656.4" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E222-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E222-MJMAIN-2192" x="0" y="0"></use></g></svg></span><script type="math/tex">\to</script><span> </span><em><span>for conditional</span></em><span>.</span></li></ol><p><span>Łukasiewicz logic matrix</span><sup class='md-footnote'><a href='#dfref-footnote-28' name='ref-footnote-28'>28</a></sup><span>:</span></p><ol start='' ><li><p><span>The standard propositional language </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.656ex" height="2.333ex" viewBox="0 -784.8 1143.6 1004.3" role="img" focusable="false" style="vertical-align: -0.51ex;"><defs><path stroke-width="0" id="E249-MJCAL-4C" d="M62 -22T47 -22T32 -11Q32 -1 56 24T83 55Q113 96 138 172T180 320T234 473T323 609Q364 649 419 677T531 705Q559 705 578 696T604 671T615 645T618 623V611Q618 582 615 571T598 548Q581 531 558 520T518 509Q503 509 503 520Q503 523 505 536T507 560Q507 590 494 610T452 630Q423 630 410 617Q367 578 333 492T271 301T233 170Q211 123 204 112L198 103L224 102Q281 102 369 79T509 52H523Q535 64 544 87T579 128Q616 152 641 152Q656 152 656 142Q656 101 588 40T433 -22Q381 -22 289 1T156 28L141 29L131 20Q111 0 87 -11Z"></path><path stroke-width="0" id="E249-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E249-MJCAL-4C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E249-MJMAIN-30" x="975" y="-213"></use></g></svg></span><script type="math/tex">\mathcal{L}_0</script><span>, including in particular formulas, with </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.068ex" height="2.131ex" viewBox="0 -654.4 4335 917.3" role="img" focusable="false" style="vertical-align: -0.611ex;"><defs><path stroke-width="0" id="E276-MJMAIN-AC" d="M56 323T56 336T70 356H596Q603 353 611 343V102Q598 89 591 89Q587 89 584 90T579 94T575 98T572 102L571 209V316H70Q56 323 56 336Z"></path><path stroke-width="0" id="E276-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E276-MJMAIN-2227" d="M318 591Q325 598 333 598Q344 598 348 591Q349 590 414 445T545 151T611 -4Q609 -22 591 -22Q588 -22 586 -21T581 -20T577 -17T575 -13T572 -9T570 -4L333 528L96 -4Q87 -20 80 -21Q78 -22 75 -22Q57 -22 55 -4Q55 2 120 150T251 444T318 591Z"></path><path stroke-width="0" id="E276-MJMAIN-2228" d="M55 580Q56 587 61 592T75 598Q86 598 96 580L333 48L570 580Q579 596 586 597Q588 598 591 598Q609 598 611 580Q611 574 546 426T415 132T348 -15Q343 -22 333 -22T318 -15Q317 -14 252 131T121 425T55 580Z"></path><path stroke-width="0" id="E276-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E276-MJMAIN-AC" x="0" y="0"></use><use xlink:href="#E276-MJMAIN-2C" x="667" y="0"></use><use xlink:href="#E276-MJMAIN-2227" x="1111" y="0"></use><use xlink:href="#E276-MJMAIN-2C" x="1778" y="0"></use><use xlink:href="#E276-MJMAIN-2228" x="2223" y="0"></use><use xlink:href="#E276-MJMAIN-2C" x="2890" y="0"></use><use xlink:href="#E276-MJMAIN-2192" x="3335" y="0"></use></g></svg></span><script type="math/tex">\neg, \land, \lor, \to</script><span>.</span></p></li><li><p><span>The set of truth values </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.917ex" height="2.636ex" viewBox="0 -828.3 5991.9 1134.7" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E327-MJMATHI-56" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path stroke-width="0" id="E327-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 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35Q500 35 509 28Q518 17 509 3L502 -1H166L160 3Q149 17 160 28Q167 35 195 35Q224 37 234 39T249 52Q253 66 253 355V648L242 646Q192 642 144 617Q129 609 110 588T84 552Q69 527 69 499Q69 490 64 484T50 478Q39 478 33 490V672ZM113 639L126 648H69V597L84 612Q93 623 113 639ZM389 35Q382 46 381 86Q380 134 380 350V648H289V350Q289 199 288 131T286 53T280 35H389ZM600 597V648H542L555 639Q575 623 584 612L600 597Z"></path><path stroke-width="0" id="E327-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E327-MJAMS-55" d="M16 666Q16 677 28 683H341L348 679Q359 665 348 654Q342 648 315 648Q270 644 266 632Q262 627 262 598T261 399Q261 372 261 325T260 260Q260 149 274 99T339 30Q355 25 393 25Q430 25 457 33T494 49T519 72Q562 115 575 205Q576 219 576 379Q576 538 575 550Q568 597 550 622T506 648Q498 648 493 654T487 667T499 683H697Q709 675 709 667T704 654T690 648Q653 648 633 597Q624 573 622 546T619 377Q617 193 613 174Q596 95 544 41Q477 -19 355 -19H344Q275 -16 226 5T153 57T120 110T106 154Q101 172 99 399Q99 618 95 632Q88 644 53 648Q16 648 16 666ZM228 639L233 648H128Q128 647 133 632Q135 621 135 412Q135 197 137 185Q148 115 181 79Q209 51 235 41Q242 36 258 31T277 25Q276 27 268 38T254 59T241 92T228 145Q226 161 226 399Q226 632 228 639ZM604 621Q606 626 619 648H577L586 634Q587 632 591 625T595 614L597 608L604 621Z"></path><path stroke-width="0" id="E327-MJAMS-46" d="M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z"></path><path stroke-width="0" id="E327-MJMAIN-7D" d="M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E327-MJMATHI-56" x="0" y="0"></use><use xlink:href="#E327-MJMAIN-3D" x="1046" y="0"></use><use xlink:href="#E327-MJMAIN-7B" x="2102" y="0"></use><g transform="translate(2602,0)"><use xlink:href="#E327-MJAMS-54" x="0" y="0"></use><use xlink:href="#E327-MJMAIN-2C" x="667" y="0"></use><use xlink:href="#E327-MJAMS-55" x="1111" y="0"></use><use xlink:href="#E327-MJMAIN-2C" x="1833" y="0"></use><use xlink:href="#E327-MJAMS-46" x="2278" y="0"></use></g><use xlink:href="#E327-MJMAIN-7D" x="5491" y="0"></use></g></svg></span><script type="math/tex">V = \{\mathbb{T, U, F}\}</script><span>.</span></p></li><li><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.549ex" height="1.928ex" viewBox="0 -741.3 667 830.3" role="img" focusable="false" style="vertical-align: -0.207ex;"><defs><path stroke-width="0" id="E342-MJAMS-54" d="M33 672Q36 680 44 683H624Q632 680 635 672V490L631 483Q621 479 617 479Q611 479 606 485T600 499Q600 525 584 552Q577 567 558 588T524 617Q479 642 426 646L415 648V355Q415 62 422 52Q425 42 434 40T473 35Q500 35 509 28Q518 17 509 3L502 -1H166L160 3Q149 17 160 28Q167 35 195 35Q224 37 234 39T249 52Q253 66 253 355V648L242 646Q192 642 144 617Q129 609 110 588T84 552Q69 527 69 499Q69 490 64 484T50 478Q39 478 33 490V672ZM113 639L126 648H69V597L84 612Q93 623 113 639ZM389 35Q382 46 381 86Q380 134 380 350V648H289V350Q289 199 288 131T286 53T280 35H389ZM600 597V648H542L555 639Q575 623 584 612L600 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E342-MJAMS-54" x="0" y="0"></use></g></svg></span><script type="math/tex">\mathbb{T}</script><span> is the only designated value, i. e., </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.395ex" height="2.636ex" viewBox="0 -828.3 4475.5 1134.7" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E381-MJMATHI-56" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path stroke-width="0" id="E381-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 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672Q36 680 44 683H624Q632 680 635 672V490L631 483Q621 479 617 479Q611 479 606 485T600 499Q600 525 584 552Q577 567 558 588T524 617Q479 642 426 646L415 648V355Q415 62 422 52Q425 42 434 40T473 35Q500 35 509 28Q518 17 509 3L502 -1H166L160 3Q149 17 160 28Q167 35 195 35Q224 37 234 39T249 52Q253 66 253 355V648L242 646Q192 642 144 617Q129 609 110 588T84 552Q69 527 69 499Q69 490 64 484T50 478Q39 478 33 490V672ZM113 639L126 648H69V597L84 612Q93 623 113 639ZM389 35Q382 46 381 86Q380 134 380 350V648H289V350Q289 199 288 131T286 53T280 35H389ZM600 597V648H542L555 639Q575 623 584 612L600 597Z"></path><path stroke-width="0" id="E381-MJMAIN-7D" d="M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E381-MJMATHI-56" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E381-MJMAIN-2B" x="1166" y="513"></use><use xlink:href="#E381-MJMAIN-3D" x="1752" y="0"></use><use xlink:href="#E381-MJMAIN-7B" x="2808" y="0"></use><use xlink:href="#E381-MJAMS-54" x="3308" y="0"></use><use xlink:href="#E381-MJMAIN-7D" x="3975" y="0"></use></g></svg></span><script type="math/tex">V^{+} = \{\mathbb{T}\}</script><span>.</span></p></li><li><p><span>Truth functions are given by the following tables:</span></p><p> </p><p><img src="assets\L-3-table-small.PNG"></p><p> </p></li></ol><div style="font-size:200%;">DOMAIN OF PLAY</div><p><span>Our domain of play will define for us how to </span><strong><span>interpret</span></strong><span> the notions of </span><strong><span>formula</span></strong><span> and </span><strong><span>matrix</span></strong><span>, and it will furnish the meanings of </span><strong><span>valuation</span></strong><span> and </span><strong><span>satisfaction</span></strong><span> for </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.311ex" height="2.535ex" viewBox="0 -784.8 995.1 1091.3" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E168-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(43.49) matrix(1 0 0 -1 0 0)">Ł</text><use transform="scale(0.707)" xlink:href="#E168-MJMAIN-33" x="765" y="-324"></use></g></svg></span><script type="math/tex">Ł_3</script><span>.</span><sup class='md-footnote'><a href='#dfref-footnote-29' name='ref-footnote-29'>29</a></sup></p><p> </p><div style="font-size:200%;">MOVEMENT</div> <p><span>From these preceding sections alone, it will not be completely obvious how </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.311ex" height="2.535ex" viewBox="0 -784.8 995.1 1091.3" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E168-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(43.49) matrix(1 0 0 -1 0 0)">Ł</text><use transform="scale(0.707)" xlink:href="#E168-MJMAIN-33" x="765" y="-324"></use></g></svg></span><script type="math/tex">Ł_3</script><span> moves. That is because logic is in part </span><em><span>AN ACTIVITY</span></em><span>. You will glimpse how it moves only after you have watched someone use the </span><strong><span>symbol-pieces</span></strong><span> within </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.311ex" height="2.535ex" viewBox="0 -784.8 995.1 1091.3" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E168-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(43.49) matrix(1 0 0 -1 0 0)">Ł</text><use transform="scale(0.707)" xlink:href="#E168-MJMAIN-33" x="765" y="-324"></use></g></svg></span><script type="math/tex">Ł_3</script>’<span>s </span><strong><span>domain of play.</span></strong><span> Once in movement, </span><strong><span>consequence</span></strong><span> will certainly make itself visible as you discover which </span><strong><span>consequence relations</span></strong><span> obtain in </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.311ex" height="2.535ex" viewBox="0 -784.8 995.1 1091.3" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E168-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(43.49) matrix(1 0 0 -1 0 0)">Ł</text><use transform="scale(0.707)" xlink:href="#E168-MJMAIN-33" x="765" y="-324"></use></g></svg></span><script type="math/tex">Ł_3</script><span>, and by this process, you will gain </span><i><span>a practical</span></i><span> understanding of why consequence is the logical world</span>’<span>s core. </span></p><p><span>Watch my words carefully now friend. Do not let them glide away: I can point and prime so that you are in the best position to see. Nevertheless, regardless of which words I beg, </span><strong><span>I will never have the means to make you understand how a logic moves.</span></strong><span> It would be like describing a color: to really see it, you must try to see it for yourself.</span></p><p> </p><p><img src="assets\vii.jpg"></p><center><small><i>VII Triennale, Disegno per ricamo, Mostra dei tessuti e dei ricami, Sezione dei merletti e dei ricami</i><br>by Piero Fornasetti. Triennale Milano .</small></center><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><div style="text-align: left; margin:0 70% 0 0;">You may be imagining now how else we may reject bivalence. Have I not, since the very beginning, taken an oath against you? So why do I speak? Why have I not simply kept silent?<br><br><i>You would accuse me of being a liar.</i><br><br>I will swear another thing then: this cannot be true. For if I admitted it, <span style="font-size:60%">mumbling ‘I am a liar’</span>, I would be expressing a truth <i>and</i> a falsity, which can never go together.<br><br><i>Yet is this not the trap I warned against? Failing to notice how lines may connect even <b>true and false?</b> And aren’t I accepting, without reason, bivalence for every circumstance? I should know well that the Logic of Paradox allows a sentence to be both <b>true and false</b> and admits <b>‘both’</b> as its third value. </i><br><br><b>I am in the wrong, so allow me to make amends as is my right: I will speak as I swear silence;</b> my statement is both true and false, and thus I need not confess my ways to you.<br><del>nor will I ever.</del></div><p> </p><p><span>It is too late to reject the </span><b><span>possibility of a true contradiction</span></b><span> only because I have used it against you.</span><sup class='md-footnote'><a href='#dfref-footnote-30' name='ref-footnote-30'>30</a></sup><span> I will reject whatever case you make since I have long used contradictions that did not explode into triviality</span><sup class='md-footnote'><a href='#dfref-footnote-31' name='ref-footnote-31'>31</a></sup><span> : I have said that we may create as we destroy, and I have talked of something that is nothing. But in all that time, you did not protest.</span></p><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><div style="font-size:300%; text-align:center;">What does it matter?</div><div style="font-size:75%; text-align:center;">YOU TALK OF NOTHING WHEN YOU TALK OF POSSIBILITY.</div><p> </p><p><span>I must reject your claim again. I know not whether possibilities are real in the way that we are real. </span></p><p> </p><p><span>But my response is this: </span></p><div style="font-size:300%; text-align:center;">What <i>if</i> it does matter?</div><div style="font-size:75%; text-align:center;"><i>WOULD</i> WE STILL BE TALKING OF NOTHING?</div><p> </p><p><span>And if it is not nothing, </span><span style="font-size:75%;"><i><span>COULD</span></i> <span> YOU STILL DENY POSSIBILITY?</span></span></p><p> </p><p><span>You see, then, we often talk in terms of possibility. After all, we have considered many possibilities on our way here. Though I know you knew this: you simply wished to protest now before never. </span></p><p><span>I am not chiding you: yours was a good challenge, which I have not yet honestly met. It seems that if anything, possibility is at least real in concept. </span><strong><span>It is something that is one part of our </span><i><span>view</span></i><span> of the logical world</span></strong><span> since, as we saw in earlier steps, our view suggests what is </span><i><span>possible</span></i><span> in it. </span></p><p><span>Yet, as much as that possibility and impossibility readily imprint themselves on us, these are concepts that cannot be well-expressed using only </span><strong><span>classical principles.</span></strong><span> </span></p><p> </p> <div style="text-align:right; font-size:65%;">Where are you now,<br>bonding reason to classical principle?<br>How—tell—your disdain?<br>Or would you omit<br>what tomorrow brings today?</div><p> </p><p><span style="font-size:350%;"><span>THE LOGIC OF POSSIBILITY, A SPECIES OF </span>‘<span>MODAL LOGIC</span>’<span>, IS AN INTENSIONAL LOGIC NOT AN EXTENSIONAL LOGIC. </span></span></p><p> </p><p><span>Every intensional logic involves concepts that intuitively cannot be reconciled with THE PRINCIPLE OF EXTENSIONALITY. This principle, and it is one on which classical logic depends, has that </span><strong><span>you may substitute an object for its identical in any sentence.</span></strong><span> </span><sup class='md-footnote'><a href='#dfref-footnote-32' name='ref-footnote-32'>32</a></sup></p><p><span> </span><strong><span>It cannot be so with possibility and impossibility.</span></strong><span> Think of me, for instance:</span></p><p> </p><div style="font-size:175%;text-align: center; margin:0 20% 0 20%;">IT IS POSSIBLE THAT YOU KNOW I WROTE IT.<br>IT IS POSSIBLE YOU KNOW <span style="background-color:#333;">S—H—E</span> WROTE IT.</div><p> </p><p><span>There can be no guarantee that both sentences share the same fate in truth or falsity were </span><span style="background-color:#333;"><span>S—H—E</span></span><span> and I the same person. For even if </span><span style="background-color:#333;"><span>S—H—E</span></span><span> and I were the same, it may happen that you did not know it. Thus, you could know that I wrote it without knowing that </span><span style="background-color:#333;"><span>S—H—E</span></span><span> , whoever she is, wrote it too.</span></p><p><span>Having recognized such a problem, we know that a logic of possibility must reject the principle, and so again we must look for the </span><strong><span>appropriate symbol-pieces and domain of play.</span></strong></p><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><p><span>In our last steps taken together, we had tried to approach logical worlds, the motive force of our exploration. We saw their principles, noted their constructions, and enjoyed their passing views. We came nearest to logic as activity when we witnessed the parts constitutive of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.311ex" height="2.535ex" viewBox="0 -784.8 995.1 1091.3" role="img" focusable="false" style="vertical-align: -0.712ex;"><defs><path stroke-width="0" id="E168-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(43.49) matrix(1 0 0 -1 0 0)">Ł</text><use transform="scale(0.707)" xlink:href="#E168-MJMAIN-33" x="765" y="-324"></use></g></svg></span><script type="math/tex">Ł_3</script><span>. Though there, we still did not see fully what we do. </span></p><p> </p><p><span>Turn left to me.</span></p><p><span>Straight ahead is the short way.</span></p><p> </p><div style="text-align:center;font-size:300%;">THE PRINCIPAL AIM OF</div><p><img src="assets\proof-3.jpg"></p><center><small><i>Proof.</i></small></center>
<div style="text-align:center;font-size:300%;">LOGICAL ACTIVITY IS PROOF.</div><p> </p><p><span>A proof is simply a </span><strong><span>complete demonstration</span></strong><span> that some sentence—some formulaic configuration—follows from others or from nothing. To follow from others means we prove from a beginning configuration, which are our </span><em><span>assumptions</span></em><span>. To follow from nothing means we prove using the symbolic-pieces and domain of play alone; we would begin with no configuration.</span></p><p><span>Do we understand now what we see in logic? </span><strong><span>At least for myself, it is not yet clear.</span></strong></p><p><span>What if we instead had this?</span></p><p><img src ="assets\not-proof-3.jpg" alt="This is not a proof"></p><center><small> Proof & Refutation.</small></center><p> </p><p><span>Would that refutation be true? </span></p><p><span>I myself could not know how to respond, but we may attempt it together. </span></p><p><span>We would first need to ascertain the proper </span>‘<span>material</span>’<span> of a proof. Is a logical proof </span><strong><span>a complete demonstration in abstraction</span></strong><span>? Or is it instead </span><strong><span>a complete demonstration</span></strong><span> in the physical world? </span></p><p><span>If we mark a logical proof by its physical existence, the refutation would be false, which is desirable. However, the two preceding proofs would be </span><strong><span>different proofs.</span></strong><span> It may well be that THIS IS </span><strong><span>WHAT SHOULD BE THE CASE</span></strong><span> given other facts and aims. Yet we know that this </span><strong><span>DOES NOT RESPECT OUR PRESENT INTUITIONS,</span></strong><span> LIQUID AS THEY ARE.</span></p><p> </p><div style="font-size:175%;text-align:center;">WE WOULD NOT SAY THAT<br><i><b>BLUE</b></i> &<i><b>BLUE</b></i><br>ARE TWO DIFFERENT WORDS.<br>WHY SHOULD IT BE DIFFERENT NOW?</div><p> </p><p><span>This suggests that a </span><strong><span>proof is made of abstraction, which suggests that THE LOGICAL WORLD WE SEE IS ALSO AN ABSTRACTION.</span></strong><span> </span></p><p><span>Even though it would make the refutation true, this view of logic is not so unreasonable: we have, in some sense, </span><strong><span>constructed</span></strong><span> a language whose parts are </span><strong><span>made of logical </span>‘<span>forms</span>’<span> that are structured and related immaterially.</span></strong><sup class='md-footnote'><a href='#dfref-footnote-33' name='ref-footnote-33'>33</a></sup><span> </span></p><p> </p><div style="font-size:175%;text-align:center;">HOWEVER,<br>IT IS ULTIMATELY UNSATISFYING.<br>THIS VIEW DOES NOT REVEAL WHY<br>A PROOF MUST BE PRESENTED AS IT IS.<br><br>WE HAVE OTHER ABSTRACTIONS.<br>WHY MUST WE <b>DRAW</b> THESE AND NOT THEM?<br>MERELY TO COMMUNICATE?<br>THEN WHY AM I ABLE<br>TO SPEAK TO YOU OF<br>TRUTH<br>CONCEPT<br>MEANING<br>WITHOUT PICTURES,<br>WITHOUT VISIBLE FORM?</div><p> </p><p><span>I have long suggested that form cannot be so easily pulled from content. Form, in fact, clarifies content just as it may distort: I once said, </span>“<span>there is not one thing to find.</span>”<span> Did you consider whether I meant,</span></p><p> </p><center><b>there is not</b>(one thing) <b>to find</b><br>
OR<br><b>there is not</b>(one) <b>thing to find</b><br>
?</center><p> </p><p><span>These are clearly two different meanings that form alone has brought to surface. The difference is not negligible: it would have changed your </span><strong><span>interpretation</span></strong><span> and </span><strong><span>understanding,</span></strong><span> which are changes that could have affected a part if not the whole.</span></p><p><span>If you believe this is still so little and so trite, I have a demonstration for you that is larger yet. Take everything here as your challenge: Copy my words, set them plainly elsewhere, remove spaces that divide paragraphs </span><span style="background-color:#333;"><span>or even words!</span></span><span> and black lines that obscure, eliminate dots, and eliminate precise positions. Does it still say as much? </span><strong><span>Order</span></strong><span> is a part of form so put paragraphs </span><span style="background-color:#333;"><span>forget words</span></span><span> into </span><strong><span>disarray</span></strong><span> and reorder the sequence of following in the proof. </span><b><span>Does it still say as much?</span></b><span> Should I still grant you punctuation and images? </span><strong><span>It remains disastrous even if I should grant them.</span></strong><span> But if I did not, </span><b><span>DOES IT STILL SAY AS MUCH?</span></b></p><p><span>The logical world would suffer the same</span><br><span>if you dealt it with the same violence.</span><br><span>the possibility of its meaning for humans</span><br><span>depends much on its </span><strong><span>perceivable</span></strong><span> form.</span><br><span>Hence it cannot be only abstraction.</span></p><div style="font-size:175%; text-align:right;">On our hands<br>have we proof as abstraction, or<br>have we proof as material,<br>to make the refutation true or false?</div> <p> </p><p> </p><p><span style="font-size:125%"><span>In walking on this road,</span><br><span>through an expanse of logical worlds,</span><br><span>we together saw that</span><br><i><span>between two,</span><br><span>a choice must always be made</span><br></i><span>is an old assumption. </span></span></p><p> </p><p> </p><div style="text-align: center;"><b>We know then that it can be BOTH.</b><br>abstract—material and<br>true—false.<br>And thus, once again,<br>we draw the line between—</div><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p> </p><p><span>O! How </span><i><span>there</span></i><span> has become </span><i><span>here,</span></i><span> even in fog and uncertainty! And how </span><em><span>here</span></em><span> reveals yet another </span><em><span>there</span></em><span> as a reminder of the wideness of our world and worlds. O my dear companion, you who remained, let me confess at last: </span></p><p> </p><div style="text-align:left; margin:0 15% 0 15%;"><span style="background-color:#333;">
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------<i>were you <u>searching</u> for something?</i>-------</span></div><div style="text-align:left; margin:0 15% 0 15%;"><span style="background-color:#333;"><br>---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------</span></div><p> </p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><img src="assets\small-black-dot.png" alt="dot"></p><p><span class='md-image'><img src="assets\athena-trans-resize.png" alt="Marble head of Athena: The so-called Athena Medici" align="center;"></span></p><p><a href='index.html'><span>Home</span></a><span> </span><span> </span><a href='references.html'><span>References</span></a><span> </span><span> </span><a href='about.html'><span>About</span></a><span> </span><a href='a-spare-page.html'><span>A Spare Page</span></a></p><div class='footnotes-area' ><hr/>
<div class='footnote-line'><span class='md-fn-count'>1</span> <span>John Etchemendy quoted in J. C. Beall and Greg Restall, </span><em><span>Logical Pluralism</span></em><span>. (Oxford: Oxford University Press, Incorporated, 2006), 12—13.</span> <a name='dfref-footnote-1' href='#ref-footnote-1' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>2</span> <strong>“<span>To discover truths is the task of all sciences; it falls to logic to discern the laws of truth.</span>”</strong><span> [Gottlob Frege, 1918, </span><i><span>Collected Papers</span></i><span>, p. 351 as quoted in Hao Wang, “WHAT IS LOGIC?,” </span><em><span>The Monist</span></em><span> 77, no. 3 (1994): 261.]</span> <a name='dfref-footnote-2' href='#ref-footnote-2' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>3</span> <span>J. C. Beall and Greg Restall, </span><em><span>Logical Pluralism</span></em><span>, 12.</span> <a name='dfref-footnote-3' href='#ref-footnote-3' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>4</span> <span>J. C. Beall and Greg Restall, </span><em><span>Logical Pluralism</span></em><span>, 25.</span> <a name='dfref-footnote-4' href='#ref-footnote-4' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>5</span> <span>Alfred Tarski, “On the Concept of Following Logically,” </span><em><span>History and Philosophy of Logic</span></em><span> 23, no. 3 (September 2002): 176, </span><a href='https://doi.org/10.1080/0144534021000036683' target='_blank' class='url'>https://doi.org/10.1080/0144534021000036683</a><span>.</span> <a name='dfref-footnote-5' href='#ref-footnote-5' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>6</span> <span>Alfred Tarski, “On the Concept of Following Logically,” 176.</span> <a name='dfref-footnote-6' href='#ref-footnote-6' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>7</span> <span>Alfred Tarski, “On the Concept of Following Logically,” 176.</span> <a name='dfref-footnote-7' href='#ref-footnote-7' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>8</span> <span>Alfred Tarski, “On the Concept of Following Logically,” 155.</span> <a name='dfref-footnote-8' href='#ref-footnote-8' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>9</span> <strong>“<span>For Frege, the choice was as follows: either we are dealing with ink marks on paper or else these marks are signs of </span><em><span>something</span></em><span>, and what they represent is their meaning. That these alternatives are wrongly conceived is shown by the game of chess: here we are not dealing with the wooden pieces, and yet these pieces do not represent anything—in Frege</span>’<span>s sense they have no meaning. There is still a third possibility; the signs can be used as in a game.</span>”</strong><span> [Ludwig Wittgenstein on rule-based semantics as quoted in Jaroslav Peregrin, “Semantics as Based on Inference,” in </span><em><span>The Age of Alternative Logics: Assessing Philosophy of Logic and Mathematics Today</span></em><span>, ed. Johan van Benthem et al. (Dordrecht: Springer Netherlands, 2006), 27, </span><a href='https://doi.org/10.1007/978-1-4020-5012-7_3' target='_blank' class='url'>https://doi.org/10.1007/978-1-4020-5012-7_3</a><span>.]</span> <a name='dfref-footnote-9' href='#ref-footnote-9' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>10</span> <span>Alfred Tarski, “On the Concept of Following Logically,” 176–177.</span> <a name='dfref-footnote-10' href='#ref-footnote-10' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>11</span> <span>Alfred Tarski, “On the Concept of Following Logically,” 177.</span> <a name='dfref-footnote-11' href='#ref-footnote-11' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>12</span> <span>Alfred Tarski, “On the Concept of Following Logically,” 178.</span> <a name='dfref-footnote-12' href='#ref-footnote-12' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>13</span> <span>Alfred Tarski, “On the Concept of Following Logically,” 181</span> <a name='dfref-footnote-13' href='#ref-footnote-13' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>14</span> <span>Alfred Tarski, “On the Concept of Following Logically,” 181.</span> <a name='dfref-footnote-14' href='#ref-footnote-14' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>15</span> <span>Alfred Tarski, “On the Concept of Following Logically,” 186–187.</span> <a name='dfref-footnote-15' href='#ref-footnote-15' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>16</span> <span>Alfred Tarski, “On the Concept of Following Logically,” 188.</span> <a name='dfref-footnote-16' href='#ref-footnote-16' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>17</span> <span>Val Plumwood, “Dualism: The Logic of Colonisation.” In </span><em><span>Feminism and the Mastery of Nature</span></em><span>, 51–78. Routledge, 1993. </span><a href='https://doi.org/10.4324/9780203006757-8' target='_blank' class='url'>https://doi.org/10.4324/9780203006757-8</a><span>.</span> <a name='dfref-footnote-17' href='#ref-footnote-17' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>18</span> <span>Alfred Tarski, “On the Concept of Following Logically,” 155.</span> <a name='dfref-footnote-18' href='#ref-footnote-18' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>19</span> <span>Jaakko Hintikka, “On the Development of the Model-Theoretic Viewpoint in Logical Theory,” </span><em><span>Synthese</span></em><span> 77, no. 1 (1988): 4.</span> <a name='dfref-footnote-19' href='#ref-footnote-19' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>20</span> <span>Jaakko Hintikka, “On the Development of the Model-Theoretic Viewpoint in Logical Theory,” 1.</span> <a name='dfref-footnote-20' href='#ref-footnote-20' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>21</span> <span>Jean Van Heijenoort, “Logic as Calculus and Logic as Language,” </span><em><span>Synthese</span></em><span> 17, no. 3 (1967): 324–30.</span> <a name='dfref-footnote-21' href='#ref-footnote-21' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>22</span> <span>Jean Van Heijenoort, “Logic as Calculus and Logic as Language,” </span><em><span>Synthese</span></em><span> 17, no. 3 (1967): 324–30; Hintikka, Jaakko. “On the Development of the Model-Theoretic Viewpoint in Logical Theory.” Synthese* 77, no. 1 (1988): 1–36.</span> <a name='dfref-footnote-22' href='#ref-footnote-22' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>23</span> <span>Jean Van Heijenoort, “Logic as Calculus and Logic as Language,” 324–30.</span> <a name='dfref-footnote-23' href='#ref-footnote-23' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>24</span> <span>G. H. Von Wright, </span><em><span>The Tree of Knowledge and Other Essays</span></em><span> (BRILL, 1993), 14–13.</span> <a name='dfref-footnote-24' href='#ref-footnote-24' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>25</span> <span>Elena Ficara, </span><em><span>Contradictions: Logic, History, Actuality</span></em><span> (Berlin/Boston, GERMANY: De Gruyter, Inc., 2014), 59. </span><a href='http://ebookcentral.proquest.com/lib/ucalgary-ebooks/detail.action?docID=1346942' target='_blank' class='url'>http://ebookcentral.proquest.com/lib/ucalgary-ebooks/detail.action?docID=1346942</a><span>.</span> <a name='dfref-footnote-25' href='#ref-footnote-25' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>26</span> <span>Gillian Russell, “Logical Nihilism: Could There Be No Logic?*,” </span><em><span>Philosophical Issues</span></em><span> 28, no. 1 (2018): 308–24, </span><a href='https://doi.org/10.1111/phis.12127' target='_blank' class='url'>https://doi.org/10.1111/phis.12127</a><span>.</span> <a name='dfref-footnote-26' href='#ref-footnote-26' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>27</span> <span>Audrey Yap and Richard Zach. </span><em><span>What If? An Open Introduction to Non-Classical Logics.</span></em><span> The Open Logic Project, Fall 2020, 44.</span> <a name='dfref-footnote-27' href='#ref-footnote-27' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>28</span> <span>Audrey Yap and Richard Zach. </span><em><span>What If? An Open Introduction to Non-Classical Logics,</span></em><span> 53.</span> <a name='dfref-footnote-28' href='#ref-footnote-28' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>29</span> <span>For details, see pages 45–48 in Audrey Yap and Richard Zach. </span><em><span>What If? An Open Introduction to Non-Classical Logics.</span></em> <a name='dfref-footnote-29' href='#ref-footnote-29' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>30</span> <span>Graham Priest, Francesco Berto, and Zach Weber, “Dialetheism,” in </span><em><span>The Stanford Encyclopedia of Philosophy</span></em><span>, ed. Edward N. Zalta, Fall 2018 (Metaphysics Research Lab, Stanford University, 2018), </span><a href='https://plato.stanford.edu/archives/fall2018/entries/dialetheism/' target='_blank' class='url'>https://plato.stanford.edu/archives/fall2018/entries/dialetheism/</a><span>.</span> <a name='dfref-footnote-30' href='#ref-footnote-30' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>31</span> <span>Graham Priest, Koji Tanaka, and Zach Weber, “Paraconsistent Logic,” in </span><em><span>The Stanford Encyclopedia of Philosophy</span></em><span>, ed. Edward N. Zalta, Summer 2018 (Metaphysics Research Lab, Stanford University, 2018), </span><a href='https://plato.stanford.edu/archives/sum2018/entries/logic-paraconsistent/' target='_blank' class='url'>https://plato.stanford.edu/archives/sum2018/entries/logic-paraconsistent/</a><span>.</span> <a name='dfref-footnote-31' href='#ref-footnote-31' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>32</span> <span>G. H. Von Wright, </span><em><span>The Tree of Knowledge and Other Essays</span></em><span> (BRILL, 1993), 20.</span> <a name='dfref-footnote-32' href='#ref-footnote-32' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>33</span> <span>Susanne Katherina Knauth Langer, “The Study of Forms.” In </span><em><span>An Introduction to Symbolic Logic.</span></em><span>, 2d ed. (rev.) (New York, 1953), </span><a href='http://hdl.handle.net/2027/mdp.39015048060035' target='_blank' class='url'>http://hdl.handle.net/2027/mdp.39015048060035</a><span>.</span> <a name='dfref-footnote-33' href='#ref-footnote-33' title='back to document' class='reversefootnote' >↩</a></div></div></div>
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