# Is it fair to say truth is used more in logic than in math? If so, what are the reasons for doing so?

Right out of the gate in logic we see sentences (propositions) like "all men are mortal" and we say both that it is a true proposition (e.g. independent of being a premise) and that in a logical argument, it is a true premise. We are introduced to the idea of propositions being true, and see that logical arguments comfortably use truth for premises and conclusions.

But in math we speak more of (mathematical) definitions like points, lines, planes, functions, etc. And we speak of axioms which we rarely call true but rather are assumed or "hypothesized". And when we do speak of truth, it's usually in cases such as truth under the conditions of some axioms we are working under and accompanying inference rules. We hardly just speak of truth outright.

I have a voice in the back of my head reminding me that in math when we say true, we mean true according to some assumptions. I feel that is learned. And for logic, I feel I learned that we just say these things are true, without any reminder or proviso.

You might immediately rush to, isn't it just true that 3 is greater than 2? But then that reminder perks up--we usually found math in something like set theory, where everything is a set. And then what is 2 the set? Well it's a set of two elements (if we're doing it the Von Neumann way) {∅,{∅}} where we've defined things such as 0 = ∅, the successor function, union, powersets, etc. We know without such foundations math becomes too liable to make mistakes (naive set theory), or too underpowered--if we just took the truths like 2 < 3 without their set definitions and set theory axioms, we would barely explore most of mathematics. Have I just not gotten far enough in either subject?

1. Is it fair to say this is how logic and math are taught regarding truth? --This is the main question I would like an answer on.
2. Are there good answers for why? -- I've given a few small speculations (not pushing my own pet theory)
• It is fair to say that (elementary) logic is taught to the less experienced than mathematics and this is reflected in the choice of examples and suppression of complications in teaching. The 'unproblematic' treatment of truth is an artifact of trying to stay elementary, sample sentences are chosen to be as uncontroversial as possible to avoid distractions. It changes when modal logic is involved, say. On the other hand, arithmetic is close to logic in elementarity, and has no need for set theory 'definitions', or even Peano formalization. So 2 < 3 is taken as simply true, unlike set axioms. Commented Jul 20 at 7:30
• The top-level question is answered by Tarski's undefinability; truth is not as useful as provability in maths because it is semantically empty. The deeper insight to seek is that, similarly, truth is not as useful as theoremhood in logic, for the same reason. Your secondary question probably should be asked at Math Educators SE. Commented Jul 20 at 16:38
• Consider also that logic originated from philosophy:Aristotle, and A's logic and metaphysics are greatly involved with truth. Commented Jul 20 at 19:05
• @MauroALLEGRANZA yes thank you, that is the kind of response I am looking for. Commented Jul 20 at 19:51
• One thing to keep in mind is that for a practicing mathematician, working in some speciality field as topology or differential equations, truth is used exactly as it is used in daily life. For a very simple linguistic example, when person A, skeptical of a statement made by person B, responds by saying "Wait, is that true?", the intent is identical whether A and B are two people talking about the price of peaches or two mathematicians talking about a theorem. The difference comes about in how B convinces A: when talking about a theorem, B can use mathematical proof. Commented Jul 21 at 14:06

First, I think you're overestimating the force of 'truth' in logic. If we look at simple syllogisms like (as you suggested):

• P1: All men are mortal
• P2: Socrates is a man
• C: Therefore, Socrates is mortal

All that says is that IF P1 is true and P2 is true, C is true. It doesn't ascribe truth to P1 or P2, it merely transfers a class property to an individual member of that class. I mean, If I replace the word 'mortal' with the word 'purple', the logic still works, but C becomes false because P1 is false.

The point here is that logic and maths are formal systems: systems that transform statements according to preset rules, without changing the intrinsic meaning of the statement. In logic we worry about maintaining truth-value of pre-given statements during transformations. In maths we worry about maintaining equality; in other words, we worry about maintaining the 'truth' that the left side of the equation equals the right side. But neither logic nor math cares one whit about semantics: about whether the statements or equations have meaning.

We should always keep our eye on the distinction between 'validity' and 'truth'. 1+2=3 is valid according to the formal syntactic rules of addition. Whether or not it's true is a matter of semantics: of assigning meaning to he formal ciphers 1, 2, and 3. One apple plus two apples equals three apples, sure. But what does one mongoose plus two aardvarks equal?

• I'm inclined to answer "three mammals", but nevertheless, I like the clarity of your straightforward reply to the question. Commented Jul 20 at 13:14
• I don't just want to focus on the formal derivations though. There is content in logic where we don't derive, assume, or define anything. I like the idea that equality might be at play, and is largely a mathematical concept (whereas in logic we'd use IFF). Commented Jul 20 at 15:58
• @Speakpigeon: Why 'three mammals'? Why not 'three vertebrates', or 'three fur-balls', or 'twelve paws'? See, in 'one apple plus two apples' we have an explicit dimension 'apple' that cancels from both sides of the equation, leaving us with the dimensionless equation '1+2=3'. But 'mongoose' and 'aardvark' do not represent a common dimension. We can construct a common dimension using logic in a number of different ways, sure, but some of those constructed dimensions might render the equation false. Commented Jul 20 at 21:11
• @Speakpigeon: Addition is valid because it follows a set of formal rules without regard to real-world meaning. Even you acknowledge by saying "1 really means 'one thing'" (your emphasis). Without that 'thing', '1' is a mere linguistic token manipulated by rules. I mean, c'mon… imagine this conversation: I say to you: "Do you have one." You say: "one what?" I say: "No, I mean do you have one." You say: "huh?" It's an empty symbol. Commented Jul 21 at 15:31
• @Speakpigeon Well, isn't that the salient fact about all axioms? Commented Jul 21 at 16:44

I think you overestimate the difference. The branch of logic in which truth plays an important role is model theory. We may say of a sentence that it is 'true under an interpretation' or 'false under an interpretation'. This applies both to logic and mathematics.

Even with elementary syllogisms, the premises do not need to be true. "All lions speak French; all French speakers have two heads; therefore, all lions have two heads" is a valid argument, albeit with premises that are false under what we might call the standard interpretation that speakers of the English language would naturally give to it. Logic is concerned with what follows from what. Logic doesn't tell us whether our premises are true.

Similarly with mathematics. On a formalist or structuralist understanding of mathematics we do not need to concern ourselves with whether the axioms of a theory are 'true' in some absolute sense. We use proofs to derive theorems from the axioms, and we use model theory to tell us about the interpretation of the axioms and theorems.

With many theories we have in mind an intended interpretation, so we may choose to ignore the fact that other interpretations are possible. When we talk about lions and heads we typically know what we are referring to. Equally, when we talk about natural numbers, or about lines and points, the intended interpretation is obvious. So we may say simply that it is false that lions speak French or that it is true that 3>2 or that it is true that a line segment is the shortest distance between two points. Even then, other interpretations are possible. 2+2=4 is false if we interpret + to mean addition modulo 4. Various theorems of Euclidean geometry do not hold if we extend our interpretation to other kinds of geometry.

So logic and mathematics don't treat truth differently. Of course there is the separate issue of what it means to say of a proposition that it is true. That is a subject philosophers have argued about a lot.

• But when is "all men are mortal" not true? Where in propositional logic is an equivalent move to 2+2=4 being interpreted under a modulus, which does change truth values. Here I'm not taking "all me are mortal" as part of a logical argument (It's not a premise or conclusion). Will I one day learn we can interpret such a proposition differently like I learned 2+2=4 can be interpreted differently? Commented Jul 20 at 16:06
• 'Mortal' itself has many subtle variations in meaning. There are mortal enemies, mortal sins, mortal fear, mortal hurry. Even if we fix the interpretation more precisely it could still mean "unable to live forever" or "able to die". Potentially, "All men are mortal" might fail to be true if we succeed in genetically engineering a human that does not age. People are working on it. It is irrelevant to the logic however. Logic tells us only what follows from what. Commented Jul 20 at 16:32
• @JKusin: You're overly focused on the physical world, imagining that to be the only model which we address. However, we can do logic with abstracta (maths) and thoughts (psychology) and goods (logistics) without physics. "All men are mortal" is to be taken as an axiom or presupposition, like "all orders may be canceled" or "all polygons lie in the plane". Commented Jul 20 at 16:35
• @Corbin ""All men are mortal" is to be taken as an axiom or presupposition". I flatly disagree with that statement. I'm less focused on the physical than trying to demarkate the physical from "abstracta", or less contentiously, demarkate how we talk of the physical vs "abstracta" Commented Jul 20 at 17:35
• I would say that the unchanging aspect lies within the logic, not within the concept of mortality. In the standard way model theory works, we don't interpret the logical constants: they have a fixed meaning. For example, a sentence that is an instance of the form "not (P and not-P)" is a logical truth because the meaning of 'not' and 'and' are fixed. So, it would be reasonable to say that such a truth runs deeper than mathematical theorems that do require interpretation. Commented Jul 21 at 12:37
1. Is it fair to say this is how logic and math are taught regarding truth? --This is the main question I would like an answer on.
2. Are there good answers for why? -- I've given a few small speculations (not pushing my own pet theory)

Formal logic is taught, logic is not.

And a very peculiar kind of formal logic it is.

This is really a mathematical "formal logic" (MFL). And that is formal logic in name only.

George Boole taught mathematicians to forget about truth and falsity, and only consider the algebraic notions of truth values, which he very revealingly dubbed "1" and "0". Couldn't have been more explicit.

Since then, most if not all mathematicians, and at least many if not most philosophers, have come to see formal logic as an extension of Boole's Boolean Algebra.

This doesn't go without conflicting intuitions, for these people are still human beings and therefore innately logical. Here and there, they should feel the discrepancies, the cognitive dissonance. "Vacuous truth" anyone?! The Principle of Explosion?! Proof by Contradiction?! The Empty Set?!

All these are essentially nonsensical, purely theoretic, constructs.

MFL has in effect redacted the definition of the main logical concepts, including the concepts of implication, of logical validity, deduction, proof, as well as of truth and falsity.

Academics in the 19th and 20th century opted for expediency and so voted to ignore real logic, i.e., the logic of human beings. From this point, true and false could not possibly carry on meaning the same thing as it previously did for mathematicians and as it still does in everyday life, where it cannot escape anyone that there is no notion of vacuous truth.

Yet, mathematicians used to consider the mathematical proofs they themselves produced as essentially true or false. Axioms were only hypothetically true, so theorems are only hypothetically true, i.e., true under the assumption that axioms are themselves true, which maybe they are not. But mathematicians had to think of the proofs themselves as (presumably) logical. But a logical proof is just an implication, and an implication is true or false, or it is nothing. Thus, mathematicians had to think of their own proofs as either (hopefully) true or (just conceivably possibly) false. This is all in the past now.

Today, the whole of mathematics has been reframed as an extension, a development of MFL. The idea first seriously initiated in Principia Mathematica by Bertrand Russell and Alfred N. Whitehead. The project seems to have ended in failure, but the idea, bizarrely, is more or less accepted as the current paradigm.

The consequence is that the initial notion mathematicians had that mathematical proofs were either true or false is now subjected to the truth of the axioms of MFL, whatever they are. Mathematical proofs are only true subject to the axioms of MFL being true (and they are not).

Mathematics has become a sort of massive but fluffy pie in the sky.

And this is how it is actually taught.

3 is greater than 2 not because {∅,{∅},{∅,{∅}}} has more elements than {∅,{∅}} but because we all learn very quickly that three apples is more than two apples. And this is what true means.

• I upvoted this. Particularly the last para. Though I would use different terminology. What you call FML I would call object language, what you call "real logic" I'd call meta language. See my answer. Commented Jul 20 at 11:21
• My hesitation with this answer is in your last paragraph. How can we state complex theorems of math, much beyond simple 3>2 theorems, with the apple analogy? Commented Jul 20 at 15:51
• @Rushi "What you call FML I would call object language, what you call "real logic" I'd call meta language" Well, I object to the notion of object language and to the notion of meta language. They are purely theoretic by-products of MFL itself. We use different languages in different contexts according to how practical their use is in each context. We could in principle express exactly the same ideas using natural language as we do using a symbolic language. (Real) logic is a cognitive process, not a language, and so not a meta language. Commented Jul 20 at 16:23
• @Speakpigeon Well again I disagree with you less than might appear! Frege of the logicism school (going on to Russell's Principia which you rightly lambast calls me less than J.S. Mill of the psychologism school. Nevertheless to say that there is only 1 langauge is as absurd as saying English is identical to French. Or even English and French are only token-different and not deeply different. Just consider genders, the multiple futures tenses etc in French. Commented Jul 20 at 16:27
• @JKusin "How can we state complex theorems of math, much beyond simple 3>2 theorems, with the apple analogy?" I didn't say we could. The apple analogy was meant to explain what we mean by truth. I'm sure we all understand it. We could in principle articulate complex theorems using natural language. This would be probably either impossible or at least not worth the trouble in practice and we prefer to use a symbolic language for mathematical expressions, although logical relations between them are still expressed in natural language in all textbooks. Commented Jul 20 at 16:29

## The short version

Math deals with truth, maybe a priori/ analytic; logic deals with truth formally ie. with "truth" not truth.

## The longer version

It is fair to say all STEM including logic, deals with truth at the meta level ie. informally;
Logic deals with truth at object level. At that level True and False, very often shortened to T and F or 1 and 0 in digital logic design, really could be any two tokens, eg up and down, black and white. This is quite like particle physicists choose completely incongruous names for particle properties like strangeness, charm etc.

eg. A basic building block for an ALU is an adder; and a basic building block for an adder is a so-called half-adder: (Adding bits A and B to get Sum and Carry)

For the purposes of this question this is a mysterious example of how logic slowly merges into arithmetic — what comes in on the left is logic units, what exits on the right is the beginnings of arithmetic.

But obviously the “logic” here means AND, OR etc, not logical reasoning.

And so in the same way, there's nothing particularly truthful about true, irrespective of whether it's spelt 1, T Or "True"!

• It could be any two tokens, and many times I wish it were, although I'm not sure if "all men are mortal" being assigned a token besides T/True would lead logic to the success it has had. But it's not in practice! It's T or F or true or false in logic I've taken (prop and predicate logic) Commented Jul 20 at 16:11
• Yes there is boolean logic with different tokens, but we spend ample time in logic where the tokens are T or F. Commented Jul 20 at 16:20
• @JKusin Look at the typical digital logic text. Its almost always 0,1 Commented Jul 20 at 16:23

Your post reviews logic and mathematics under the keywords

proposition, definition, axiom, true.

1. Propositions are statements which can be true or false. They are not necessarily true.

The standard syllogism from logic takes the proposition “all men are mortable” as an assumption - in expanded form: If the proposition “all men are mortal” is true and if “Socates is a man”, then “Socrates is mortal”. It is well-known that classical logic is not truth-expanding, only truth-conserving.

(I now see that's the same what @TedWrigley said).

2. There are several different calculi of logic. In that respect I do not see any difference, when considering axiomatized theories from mathematics and from logic. Hence my answer to your first question seems a bit different than your scetch.

Due to this difference I cannot answer your second question.

3. In my opinion, an indicator for a mature theory is: Axiomatized with undefined basic concepts, explicit axioms about the relations between these basic concepts, derived definitions and finally formal proofs for certain propositions on the basis of the axioms.

• Not a bad answer. Minor nit: Classical logic possesses LEM, which creates new truths not otherwise derivable; in particular, the elimination of double negations directly asserts the truth of some statements that otherwise would be independent. Commented Jul 20 at 16:47