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motivating paraconsistent logic (pop-sci approach)
july 27th-28th, 2023hello everypony ! it's been a while because of vacations, took a bit of time for myself and all. still, i did delve into some logic-related topics, and since i've joined kane b's discord server (whose channel you can find there), i got to strengthen my logical skills even further. the plan for this article is to first motivate why we may want to define true and false (in logic) with a very famous issue that conceptually plagues most of mathematics, by giving a solution (paraconsistent logic) to cope with this problem, solution that will require us to understand what we mean true and false in logic.
part i : the issue
introduction : zf and explosion is double trouble
math is largely based on stuff like the zermelo-fraenkel set theory, commonly refered to as zf. i won't explain exactly what this is, just tell yourselves it's a huge building block for most of mathematics. the most of mathematics, obviously, seeks to figure out whether a given proposition is true or not, so that in the case that it is, it become a theorem, a lemma, or whatever. so, obviously, mathematicians strongly believe that our job couldn't be futile in the slightest. and yet, this is exactly what we put ourselves at risk by mostly basing ourselves on zermelo-fraenkel set theory. this ground is indeed fragilised by the conjunction of two mastodons in mathematical and classical logic : gödel's incompleteness theorems on the one hand, and the principle of explosion on the other. we won't look at these in too much detail for now, but we'll see what's relevant to our concerns.
the principle of explosion
first off, what is the principle of explosion ? to put it simply, it tells us that if we risk ourselves to assume something contradictory, or at least false, ie that something contradictory or at least false be treated as true, it would entail that anything follow. for example, if i assumed that santa claus exist (which is false, but by assuming it, i'm treating it as thought it were true), by the principle of explosion, i could deduce that i'm the pope. therefore, if there were only but one thing that zf allow to conclude as true whilst actually being false, zf would simply be f*cked ; all propositions in zf would be equally true, reducing zf into a so-called trivialist logic under one interpretation, and that no question can ever be treated by zf in another that makes the bridge between explosion and vacuous truth (which is when arguments start by choosing an element in the empty set, which is obviously absurd but mainly shows such an element doesn't exist[1]), making the mathematician's job utterly useless in both cases anyway. to avoid such a possibility, we would need to make sure that no statement could ever be both true and false at the same time – in technical terms, we have to ensure that zf be consistent (aka non-contradictory). and yet, if there is one thing that mathematicians fear the most, it'd rather be a proof of the consistency of zf, or at least, if it were to be thanks to zf itself. upon hearing this, you're likely asking yourself tons of questions, which are all justified : notably, why is it that a proof, of the very thing that mathematicians want, be so feared by themselves ? to understand this very odd behaviour, we'll have to pay a visit to uncle göddy, as i had warned in the previous paragraph.
gödel and the constant threat zf's explosion
the most relevant result of gödel's incompleteness theorems in this very context is the following paradox : if zf were able to prove its own consistency, then zf would have to be inconsistent. in other words, zf can't prove its consistency, unless it's actually inconsistent. this, is why mathematicians hope from the bottom of our hearts that such a proof doesn't exist within zf, as it would spell the end of our profession. of course, nothing prevents a proof of zf's consistency ever existing within a coherent system outside of zf. however, no such proof has ever been found, meaning that we mathematicians have to live under the constant threat of a single proposition possibly ruining our whole lives. i mean, well, we could technically fall back on peano axioms, whose consistency was indeed proven, thankfully[2], but it's still a significant downgrade from zf, so we keep working under zf despite everything, meaning that a single inconsistency would automatically wipe out most of mathematics.
opening : how can we cope with this possibility ?
mathematicians thus need to establish mechanisms to cope with this possibility, the most common one being « oh come on, if such an inconsistency existed, surely we should have found it already ». but, well, that's neither rigorous nor convincing, as many mathematical results end up contradicting empirical observation.[3] but well, even without that, maths obtained through zf seem way too useful and concretely verified to imagine zf could be fragilised a single inconsistency, as inconsequential may it be ; this is obviously a purely pragmatic argument, but which shows a certain refusal to abide by the principle of explosion as said earlier. and indeed, we can perfectly stop accepting the principle of explosion, even though many results would need more rework than we would've had thought necessary. though tempting, denying this principle also requires to deny the principle of non-contradiction, which states that no statement be treated as both true and false at the same time.
proof. indeed, the most basic logical rule is that of the modus ponens, which simply defines what it means to imply anything : if we have that A implies B and that we have A, then we have B. we'll also write it as {A⇒B, A}⊢B (⇒ means "implies", and ⊢ means "concludes"), or even
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part ii : true, false, non-true, non-false
let's think of this practically : how do we show a proposition is true ou false, in formal logic ? to show a proposition is true, we deduce it from accepted rules of inference (eg modus ponens, modus tollens, etc.) and initial premisses. to show a proposition is false, we show it yields a contradiction. so, true practically means "one of the premisses or deduced thereof using accepted rules of inference", and false practically means "leads to a contradiction". already, we can quickly see "non-true" isn't necessarily the same as "false" : since true means "one of the premisses or deduced thereof using accepted rules of inference", it means that non-true means "neither a premiss, nor does it follow from premisses and accepted rules of inference", and that has name that some of y'all may already know : a non-truth is a non-sequitur (literally "doesn't follow"). in explosive logic, falsity is a sufficient reason for it to be non-sequitur, but in other logical frameworks, it is rather a coincidence more than anything. as for non-false, that simply means "doesn't lead to a contradiction", which i must say is probably the most complicated property to actually show lol. these definitions of "true", "non-true", "false" and "non-false" are already commonly accepted within formal logic, i didn't redefine anything there ; only what we consider compatible may differ. if i had to make a recap table of all of their possible relations, that's what you'd get :
| Coincidability | True | Non-true (non-sequitur) False | Non-false
| True | ✓ | (equal) ✗ | (impossible) Possible | (unless by non-contradiction) Possible | (unless non-sequitur) Non-true | (non-sequitur) ✓ | (equal) Possible | (and always by non-contradiction) Possible | (we call these "independent") False | ✓ | (equal) ✗ | (impossible) Non-false | ✓ | (equal) |
note that "independance" is commonly accepted among mathematicians, including in zf where some propositions happen not to be decidable like the continuum hypothesis – this also helps support the idea of abandoning explosion, since bcs of these independent problems, zf would be at best coherent, at worst paracoherent (ie there'd be propositions that be both true and false), but not necessarily trivialist (as there are independent problems under zf), preventing mathematical research under zf (ie the vast majority) from getting rekt by a single contradiction. i know, this part was significantly shorter, but it's also significantly denser so it'd be more reasonable for me to stop there lol. i hope this all at least peaked y'all's curiosity, see ya in the next article. letting y'all read the footnotes, then.
footnotes
[1] this one, you can't really understand if you didn't read the second part. using the principle of explosion, if p were both true and false, then the conjunction of all sequitur propositions (another idea loosely based on neopalm's definitions, mostly what they call t) should imply p (by truth of p). however, p⇒f by explosion since p is false, so t⇒f by transitivity, or in other words, for all proposition q, we have q⇒f, meaning all q is false, and therefore non-sequitur by non-contradiction. that's what i meant by saying zf wouldn't be able to treat any question, though i guess it could also be interpreted as "all propositions would be independent from zf" which... i mean, technically, yeah ? vacuously speaking at least.
[2] gentzen's proof for peano axioms' consistency, wikipedia.
[3] my favorite example is the following : empirically, it seems the number of (positive) prime numbers (such as 2, 3, 5, 7, 11, &c.) below an arbitrary real number x always be less than li(x) – i won't explain what this function is in detail, but let's say it approaches the number of (positive) prime numbers below x pretty well, their ratio tending to 1 as x grows ever so large. yet, there's still a moment where li(x) is being exceeded, but that only happens near x≈10316... that's a 1, follow by 316 zeroes before the decimal point. for comparison, there are only 6×1079 atoms in the observable universe, which is a 6 followed by 79 zeroes before the decimal point. that's nothing in comparison lol.
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