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

Question

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)

Answer 1

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?

Answer 2

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.

Answer 3

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.

Answer 4

The short version

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

See use-mention distinction.

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"!

Answer 5

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.