Is there any formal logic system that considers tautologies to not be well-formed?

Question

I have been thinking about some pretty speculative material that has been hard to wrap my mind around, but I feel like there is more to discover in this train of thought.

I can't remember exactly how the topic came up, but I was thinking about how sometimes we might be inclined to ask, Why does 1 + 3 = 4?

Actually, maybe this came about because I was thinking about integers. They have very curious, particular properties. It is interesting that integers are defined mathematically as rings. They have one operation, +, which creates group structure, and a second operator, *, which I think with integers is a semi-group, maybe (there is an identity element, but there are not inverses for every element).

I guess I was wondering if the properties of the integers are kind of random, in a way. They come up so often in daily life, but when you look at it as a mathematical object, it just comes across as particular. There are more general, symmetrical objects. Even just the fact that you have two operations but they don't have the same properties. It would feel cleaner and simpler to have two operations that both forms groups, for example. But there are many ways to think about why the integers have the particular properties they do.

I think it led me back to this idea I've been thinking about, which is how it's possibly that mathematical truths are "necessarily necessary". To summarise some other thoughts I've had, it's basically about how if you try to imagine a counterfactual, an alternative, hypothetical reality where there are some different laws of math, there seems to be a philosophical paradox.

It is, that if there is a valid, logical consistent, formulable alternative to some mathematical theorem, then that doesn't really mean we've discovered a "different" kind of math, because the truth of math does not depend on circumstances or states of a particular world. So long as one can recognise, logically, that given certain premises, a certain theorem would follow, then both the original and the opposite form of that theorem would both be valid mathematical theorems, in our world, in one and the same. They are both logically sound. They just start with different premises. In other words, there cannot be a hypothetical alternative math, because anything that is, is just normal math anyway. If a math theorem is true, it is true in all possible worlds, because it is a product of inherent reason, intrinsically necessary logical self-consistency. Logic, whatever it is, by its own definition, has to be valid, to be logic at all. You cannot make an illogical logic, otherwise, it is not a logic.

I believe Wittgenstein especially may have spent some time thinking about this, and I think there's an interesting passage possibly in the Tractates where he says that it is not really a coherent idea to consider what a geometric figure might look like, outside of a geometric "plane". That is, all shapes exist within some... parameters which give rise to them. If you plot 2D space with an X and Y coordinate, you can draw myriad geometric figures in there, any one you like. But it is not merely false to consider a geometric figure which does not occur in the geometric plane. It is more than false. It is more like incoherent. It simply doesn't make sense.

Maybe it's something to do with co-creation, semantic, mutual inter-dependence. If the rules of the geometric plane are themselves what constitute geometric figures, maybe, the issue with thinking of a geometric figure that exists outside of a geometric plane, is maybe, going further than "nonsensical", it is not even meaningful because you don't realise that they are actually the same thing.

I am not sure if I stand by it, but it was a new way to think about it that intrigued me.

Instead of merely saying, that the question of if 3 + 1 could equal something other than 4, in some world, is... impossible because it would be impossible to reconcile such a "fact" with the rest of any logically consistent world (i.e., how do you count to 4, given the properties of arithmetic? if 3 + 1 does not equal 4, then what does it equal? A different number? Or nothing?), if you try to flesh out your view of such a world, it likely seems to fall apart immediately. It cannot be reconciled with a logical world without reinterpreting it so heavily that it loses the original meaning you were aiming for. In that case, you end up back to where you started - you have processed, trimmed, finagled, converted a proposition which is not only false - it is not even well-formed, because you had no idea what it even meant, and you had to think extensively about it, and basically change core aspects of its meaning to suddenly not merely make it true, but to make it even interpretable, even coherent enough to ask whether it's true or not. Before then, you haven't really attained that right yet. It's like Schrodinger's paradox where it's as of yet undecidable what the state of a particle is before observation. A logically incoherent, ill-formed assertion, can not even be examined for its truth content before it is determined if it is even a member of the domain of logic itself, some system that we use. Just like the geometric picture, in some conception of what logic necessarily is, the fundamental form it must take to even be logic, we cannot have a logical assertion that is not of the form of a logical assertion. So it's possible that 1 + 3 does not equal 4 fails to even qualify, not merely as true, but rather, as meaningful.

I still spent a lot of time trying to think that over in my head. I was looking at the figure of four people standing near me. I tried to really realise, and apply, the idea that there is no "why" as to the observation that "1 + 3 = 4". My first way of viewing it was just by reason - I adduced that the counterfactual, 1 + 3 != 4, would be too hard to reconcile with a logical world, and so it simply could never be so.

But that is the conventional way of thinking about tautologies and contradictions. It didn't satisfy me, because even if the opposite is not reasonable, that is not enough to explain why something has the form it does, and not some other. I considered that just because something is true in all possible worlds does not imply it is a tautology. Something that is true in all possible worlds may be true by virtue of some kind of extended reasoning process. It may not be simple. It may be highly deductive, moving from one fact to another. And yet, by flawless reasoning, it is shown to always be logically mandated.

But some things are in a way "true in all possible words", but it's not like that. Like the equation "1 = 1".

Conventionally, we consider that this is a) true, and b) it surely could never not be true.

But as I thought about why 1 + 3 = 4... I considered that surely it cannot be that there is some "rule" governing our universe that decreed this order, this relationship. I was thinking about how, if my understanding of the necessary necessity of mathematical truths is to be accepted or assumed, it basically means that whatever the foundations of mathematics or logical truth are, they cannot be propositions that have a conceivable formulable alternative. Like, there is Euclidean geometry, and non-Euclidean geometry. They are two systems of geometry with different properties. They are the result of striking out only one of the axioms - a logical basis, a proposition - from Euclidean geometry. That is what I mean by contingent. The axioms of Euclidean geometry are apparently contingent, because it is mentally possible to form an alternative. As I said at the beginning, the totality of mathematics contains all logical possibilities. Any contingent statement is joined by its contingent alternatives within the bracket of "all of mathematics". Which is why if mathematics has a basis, it cannot be a contingent statement, but must be something that it is not possible to even express an alternative formulation of.

It kind of hints at this weird idea that just like a geometric figure, we should not get confused about the minor question of if a proposition happens to be the case in a particular world - this is a minor, trivial attribute, like an afterthought, to be easily ascertained through observation - but it's almost like what we conventionally call "true" is one and the same thing as "well-formed", because if logic is the domain of "truth" (under a certain interpretation of "truth"), then it can't be logic if it's not "true" (in the same slightly unconventional interpretation of that word).

So as I looked at those 4 people, I was thinking that I cannot think of 1 + 3 = 4 as being assertive at all, as having propositional content at all. If I held this to be a self-evident truth of the world, deeply difficult to justify why, then, according to my thinking process above... it's like we have to look at numbers in a way with the least prior assumptions possible - our culture, words, symbols. If we look at some form itself, like a human, its kind of like we can imagine without even the concept of "numbers", that there is a form, and then a form again, and then a form again, and then a form again. In this way of looking... in a world almost without language, there is no question "why" to even be asked. If someone said "why", we could say, "why what?" In a way, by observing some form, some additional presence of some form, I think we observe what I was trying to say, that it could never be the case that 1 + 3 could not equal 4, because if you study where those concepts even came from (the observation of recurrence), it is intrinsically necessary that after each one comes another, and there you have it. If you want to give names to them, sure, there's 1, there's 2, then 3, then 4. But this was not a phenomenon that required an explanation, to occur. It created its own existence, in a way.

So, it's almost more like humans are under an illusion of language, just because they can write an equation "1 + 3 = 4" - it makes them think that 1, 3, 4, +, and = are all highly distinct entities, each with unique identity, bound up in a complex, non-arbitrary, peculiar system of relationship to each other.

But in that non-verbal way of looking at something, it's the opposite. As I said above, it is almost (kind of a trippy, or semi-Buddhist, thought) as if they are the same thing. It is very much like how Wittgenstein says you can't have a concept of the "King" in chess, without the entire context of chess itself. The meaning of the concept of "4" necessarily derives from the precondition of some previous concepts, 1, 2 and 3. And in a weird way of thinking about it, it makes it easier to see it no longer as contingent, as distinct things requiring a "why" for why they get put together, honestly, to stop thinking of them as separate things. It's almost like the set of all numbers, all numbers together, are one single thing. You couldn't have the very fundamental concept of "numbers" if you left one of the numbers we know out. The mystery of contingency disappears. It would be impossible for 1 + 3 to not equal 4, exactly like the impossibility of a geometric figure outside the parameters of geometry. It isn't simply false. I feel like it needs a stronger word than "necessary" or "tautological" or even "not well formed"... something like, "it is intrinsically impossible given the very definitive criteria of the concept of numbers, themselves". I think I prefer that way of saying it, I think it says something more, than the other epithets.

But this is where this train of thought led me.

When you write a tautological equation in mathematics, like 1 = 1, one really interesting thing about that is that you can algebraically manipulate its form in any way you like. Subtract one from both sides, now you have 0 = 0.

Isn't it strange that apparently, the assertion 1 = 1 is the same statement as 0 = 0? When we normally do algebra, we use algebraic operations to transform the form of a proposition, but only in order to see it in a clearer form, to assess the truth of it. It is not the case that by doing algebraic operations on both sides of some conjecture, that the mathematics drips into a puddle like melting ice cream, that we can send the form in any way we want, almost like we can collapse any inherent information in the proposition by validly transforming it into any other mathematical expression we want.

This is very interesting to me. It adds to what I was saying above, about the idea that "1=1" being a tautology because it is true in all possible worlds, as possibly being deeply limiting and misleading. Actually, "1 = 1" is not meaningful. It hasn't entered the "form" of a proposition, because a proposition is (somehow, in some way I haven't thought about yet), something that has inherent differentiability. There is some intrinsically logical, unavoidable way in which if a proposition is true, it changes some aspect or characteristic of a world; some way in which each valid proposition is distinct, not indistinguishable, from all other propositions.

None of this is true for 1 = 1. We showed that somehow, 1 = 1 can be validly manipulated into any expression we want, like 3 + 4 = 3 + 4. Also, it cannot possibly change the state of information in any way. Maybe I am making a mistake here, but I feel like there could be a difference between that, vs. a tautology that is actually still meaningful - somehow, an assertion that is necessarily true, but it isn't deceptively cheating through a meaninglessly senseless form, it still says something about the world.

This is where my thought led me to.

While there are many systems of formal logic with various rules and axioms, has anyone ever developed a formal logic in which certain kinds of tautologies are not considered well-formed? Some theory of logic in which "1 = 1" is not considered part of the domain of logic, is not a logical expression, because it is actually meaningless (in the way I tried to show above), because it does not actually meet a key, fundamental criteria of the forms of true statements themselves?

Answer 1

I can't find the direct example of what I thought I'd seen about A → A being disallowed in some logic system. Still, in the SEP article on relevance logic,^S they do quickly bring up a subinstance of that scheme, which subinstance they describe as paradoxical:

[Consider] p → (q → q)... [this says] that every proposition strictly implies a tautology.

They also give an example for LEM-conditioned disjunction, here.

The SEP article on substructural logic also seems to touch upon this issue:

From the axiomatic p ⊢ p (anything follows from itself) we deduce that p follows from p together with q, and then by residuation, p ⊢ q → p. If we wish to reject the inference from p to q → p, then we either reject residuation, or reject the identity axiom at the start of the proof, or we reject the first step of the proof. [emphasis added]

A negative semantics for a free logic apparently rejects the truth of identity statements for empty terms:

A negative semantics is a bivalent semantics on which all empty-termed atomic formulas (including identity statements) are false.

A subsequent discussion of "neutral semantics" has it that "all empty-termed atomic formulas not of the form E!t [are] truth-valueless."

Regarding alternative systems of addition: for reasons that are hard to understand, Immanuel Kant thought that arithmetic was logically contingent in some way. 5 + 7 = 7 + 5 might be "analytic" but 5 + 7 = 12 supposedly is not. Saharon Shelah, in his "Logical Dreams", prays:

2.1 Dream: 1) Find a “forcing method” relative to PA which shows that PA and even ZFC does not decide “reasonable” arithmetical statements, just like the known forcing method works for showing that ZFC cannot decide reasonable set theoretic questions; even showing the unprovability of various statements in bounded arithmetic (instead of PA) is formidable.

But so did Kant dimly foresee that number theory might fall prey to logical diversification in the limit? I read an essay long ago, I don't recall the author or title, but I think their interpretation of Kant's claim about arithmetic being "synthetic" was not that e.g. 5 + 7 could possibly add up to some other number in some other "possible world," but rather that it might not add up to anything, but would be "frozen in place" as just "5 and 7." The plus sign indicates a procedure, an action; whether an agent has performed an action is synthetic (agents cannot be objectively defined as having performed an action; they must actually perform it; we can imagine an agent acting in some way, but even then this imagination is grounded in its own transcendental form, with synthetic apriority aplenty to its name). Kant said that addition of larger numbers was more "obviously" synthetical; for example, in our day (maybe not his, I don't know how mathematics was notated at the time) adding, say, 5427 to 9931157 has to be performed by e.g. lining the numbers up as best we can and combining them step-by-step. (And then this is even more convoluted when we come to multiplying much larger numbers, to say nothing of exponentiating or tetrating them...) And an ultrafinitist, who does not accept talk of even potential infinity, has to have it that there is some n for which all positive arithmetical operations fail. (Keep in mind, then, too, the mystery of dividing nonzero numbers by 0: is it really "analytic" that such expressions are beyond undecidable? Or do we, in the act of trying to divide a nonzero number by 0, see that this action has no understandable effects?)

Another option (that I've never seen written down, this is more just my own thought, as far as I know) would be to say that addition and multiplication are analytic, but then exponentiation and tetration and so on are not. Why would this be? Because it might seem "analytic" that 0 + 0 = 0, but what about 0⁰? It is a commonplace that that expression is "undefined," although one theorist, John Conway, held (I think for aesthetic reasons?) that this expression can be interpreted as a function from 0 to itself, such that its output is somehow 1. But Conway then would have to also hold that 1¹ = 1, so the idea that a function from a number to itself always has to yield a higher number would not go through, and we might wonder why it would go through for 0. At any rate, deciding base cases over 0 and 1, for higher arithmetical operations, seems perhaps logically divergent in abstracto, so that if we "knew" that 0⁰ = 1, this would be "synthetically." And this synthesis would condition the remainder of the higher operations to some degree (arguably).

One last link: to the SEP article on hyperintensionality: this is a report on cases where A = B, but FA does not equal FB, for some propositional operation F over objectively equivalent acts of reference. Again, not so much that tautologies are "ill-formed" but that "believing in them" seems like an otherwise empty state of mind. (Or if not empty, then "trivial": e.g. in higher set theory if an elementary embedding j: M → M is the identity, they call it a trivial embedding.)

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^SActually, there's a specific system of relevance logic called S in which A → A is not an axiom, and moreover there is a theorem about S that says that A → A is never itself a theorem in S either.

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Addendum

Imagine a system of numerals where every natural (or real, or whatever) number gets assigned a unique singular glyph. So instead of going 0, 1, 2, 3, ... 10, say, we would go 0, 1, 2, 3, ... Ꭿ, and so on and on. Now, write 5 + 7 = ♅ when ♅ is set to 12. But now is it analytic, synthetic, or something else, that 5 + 7 = ♅?

But contrariwise, consider Roman numerals, or even an expression like 110. I, II, and III for 1, 2, and 3 seem to allow for something like I + II = III, which looks analytic. What about saying that 4 = IV? This requires that V be given first, and then 4 is understood as a subtraction from V. Something like this then happens again with X for 10 and IX for 9. Worse, consider that I + I + I = III, or then I + II = III, but I + III = IV, and then I + IV = V. (And then, say, V + V = X, V + IV + II = XI, and so on.) But so if I didn't know any better, I would tend to imagine that I + IV would equal IVI instead, and I would fail to grasp Roman numerals by the by. (Rome really was like America, keeping to a weird semiotics for their mathematics, like us Americans with our weird resistance to using the metric system.)

And then in the Arabic/English case of 110, we literally say "one hundred and ten." So it looks as if 100 + 10 = 110 might be analytic. (And the same could be said of many such equations.)

Perhaps, then, some cases of addition are analytic, but "trivial" in that they involve adding over 0. 0 + n = n, and sometimes they talk about 0 as having an "identity property," so going back to what was said about trivial embeddings, we might say that 100 + 10 = 110 is analytic because trivial! And so on the other hand, perhaps there are also cases of addition that are synthetic, rather than all such arithmetic being only analytic or only synthetic. (Or, going back to the case of Roman and other systems of numerals, whether an equation is analytic or synthetic might depend on what semiotics you're using to write the equations down.)

Answer 2

There's a lot here, but I'm going to give a response a shot. I'll latch on to this, and respond:

guess I was wondering if the properties of the integers are kind of random, in a way.

Random? No. But random means:

In common usage, randomness is the apparent or actual lack of pattern or predictability in events.1 A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable.

Arbitrary? Again no:

Arbitrariness is the quality of being "determined by chance, whim, or impulse, and not by necessity, reason, or principle". It is also used to refer to a choice made without any specific criterion or restraint... Arbitrary decisions are not necessarily the same as random decisions.

The properties of integers are largely a function of the property of the naturals but with the extension that additive inverses bring. And naturals are grounded in PA:

The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction.1 In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic.3 In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms...

So, the reason the properties aren't random or arbitrary is that they theoretically derive from a series of principles that are the axioms of arithmetic which can be fully expressed in logical terms using existential quantification and composition in the form of successor functions. These questions get asked all of the time of mathematical theories such as this post in Math Overflow which asks Addition and multiplication are commutative but exponentiation and tetration are not. Do we know why? The TLDR is that succession, addition, and multiplication are commutative, but exponentation and tetration are not. Why?

Why is a call for explanation, and so let's talk about some aspects of explanation. One aspect of explanation is rooted in mathematical logic. In other words, what properties of this formal system are sufficient or necessary for commutativity? There doesn't seem to be a clear answer given in the MO post, but there is an attempt to characterize the issue in functional language, identities, etc. Let's call this approach the shallow explanation. That is, the answer should be grounded in the semantics of mathematics and logic.

But another aspect of explanation is the deeper, more meaningful question, which is what are those semantics that answer the question themselves grounded in a broader worldview. For instance, L.E.J. Brouwer, the famous mathematical intuition would have views on how the logic and arithmetic is itself grounded in the psychological, or at the very least, in a broader notion of the philosophy of language that explicates some aspects of syntax and semantics more generally.

So, we can do both. We can show that some of the properties of integers are grounded in the principles of the naturals, and that the principles of the naturals can be grounded in logic. Thus, the commutativity of integers is essentially the commutativity of naturals, and that has to do with the logical successor function being itself commutative. This is the shallow grounding in the semantics of math and logic. But to take the next step would be to argue that there are inherent abilities in the brain that are mathematical and logical in nature. For instance, the human brain is capable of subitization or as cognitive linguistis maintain, the human mind is endowed with the capacity for conceptual metaphor, such as the metaphor of containment that lies at the bottom of our notions of collections and gives rise to set-theoretic and type-theoretic theories. In this view, mathematical notions like properties of integers are derivative from cognitive capacities rooted in an image schema, an explanation offered by philosophers who advocate embodied cognition (SEP).

So, hopefully this offers you a starting point to start exploring at different levels (both within the semantics of mathematical logic and that of linguistics and psychology) your questions more thoroughly. By the way, yes, 1=1 can be shown to be a reduction of an infinite number propositions, a fact that didn't escape Wittgenstein in his Tractatus who maintained tautologies were nonsensical.

Answer 3

About A->A logic, i usually think this is a 'context logic'. You have direction reading from right to left, and 'first A' you 'read'(or think that read) earlier then 'second A'. Also you can use time line to split them, but trouble that math abstractions not need time context.

about 1+3=4 im satisfied Piano : 1+1+1+1=4. M* can say that i can't divide 3 to 1+1+1, but i still can to count till "3". So, count ability is on the blind side too.

more about this "logic" you can read in mine unpopular post about liar paradox.

gn

Answer 4

A tautology is generally considered to be a formula of truth functional propositional logic that has a truth-table that evaluates to true under every valuation. But, truth-functional Logic doesn’t need to be bivalent, given that there are three-valued truth functional logics with truth values true, false, and other. If a formula P has value “other”, then it must not be the case that we have Pv~P, since the negation of “other” is still “other”, but Pv~P is a tautology of Classical Logic. So, as with most things in logic, your question depends on which logic you’re using.