How does Logic define "true" and "false"?

Question

"Truth", "Falsehood" are pretty axiomatic expressions, but even axioms need to be defined in common language terms.

What are the "official" definitions of these in Informal logic, Formal logic, Symbolic logic and Mathematical logic respectively?

(please, no non-constructive deliberations. If there are a few conflicting definitions, please just present the most prevalent ones. If one is missing, tell it's missing, don't try to design one on the spot.)

Answer 1

No logics ever really "define" truth, they use it. It is assumed that there is some pre-theoretic understanding of what "truth" is.

But you don't even need a notion of truth. You can get by with any designated values. In mathematical logic the truth values are typically "1" and "0". Now, these are generally taken to code truth and falsity but that is not required. All that is required is that you have a designated value so that you can define a notion of a valid inference as one that preserves designated values.

In many valued logics they will often have more than one designated value. Also, it is hard to see what the values in fuzzy logic would be. Are they "degrees" of truth? Does truth come in degrees?

A quote from Russell's Principles of Mathematics seems appropriate here:

In addition to these [indefinable primitives of mathematics], mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.

I think that much the same can be said of logic, especially math logic. The study of truth is the domain of truth theory. See the SEP article on Truth.

I really can't state with confidence that informal logic is the same, using rather than defining truth. But a quick scan of the SEP article on Informal Logic makes me think that what I've said probably holds of informal logic as well.

Answer 2

The Haskell programming language simply defines True and False as members (technically, 0-argument data constructors) of the algebraic data type Bool. Fundamentally, they're symmetric and have no meaning in their definition. The physical representation that the computer uses is irrelevant within the system; all you know is that True and False are distinct. The usefulness comes in the definitions of the relational operators (1 > 0 is True, 0 > 0 is False, etc.), and of the boolean operators (&&, ||, not).

This is analogous to axiomatic definitions in pure math logic. Truth-values are primitives, and their usefulness comes when you define axioms for inference.

Answer 3

truth and falsity are values given to propositions. these values, once determined, have a bearing on the truth values for other propositions. the more general the concept the greater the difficuly in defining it. what is certain is that meaningful propositions must be capable of being ascribed a truth value in a given context.

Answer 4

True and False are neither axioms nor expressions, they are results of the calculus. An axiom is an expression that forms a basis of a logical construction. An expression is set of symbols/words within an (implicitly or explicitly agreed-upon) grammar. True and False, then, are the (potential) results of an expression that follows a "compatible" (i.e. agreed-upon), logical grammar. In that grammar, I would take the words "true" and "false" both as symbols and as the common conceptual meaning (despite the ability to make a twisted grammar which would create the opposite). In any case, they are the end point, not the beginning.

The source of your confusion is the internalization of (the premises of) science as the beginnings of your reason -- a historical construction that you've accepted as your point of departure in relating to the world. There it just assumes True the "object", and more tacitly, the objects of reason. Ditch science for a moment, if you are able to, and you'll find that you float in something more primordial, the mystery itself, of which logic came.

Answer 5

1. Question Restatement:

"Truth", "Falsehood" are pretty axiomatic expressions, but even axioms need to be defined in common language terms. What are the "official" definitions of these in Informal logic, Formal logic, Symbolic logic and Mathematical logic respectively?

Or Rather: How should "Truth" be defined within the context of logic or causality?

---

2. Answer - the Simplest, Consistent and Valid Definitions of "Truth" and "Falsity":

Logic is a computational process, examining the validity of a context of reasoning to infer a conclusion that has a "Truth Value".

So, "Truth" is to "Reasoning", as "PI" is to "Circles" - an inviolable constraint.

"Truth" defined in Terms of Logic, (premises, conclusions and implications):

1. Truth: That which "is entailed", necessarily.
   • 2 + 2 must = 4, because 1 + 1 = 2, and 1 + 1 + 1 + 1 = 4.
2. Falsity: That which "cannot be entailed", necessarily.
   • 2 + 1 = 4 = false; 2 + 1 cannot = 4, because 1 + 1 = 2, and 1 + 1 + 1 + 1 = 4;

Note 1: In Deductive, Inductive, and Abductive Logic, the term "Necessarily" changes to "Probably" and then "Plausibly".

Note 2: This "definition" applies within the contexts of logic and even causality, (it breaks down in metaphysics, etc.; In metaphysics it is not necessary that "a square must have 4 equal sides"). In the context of logic, this definition could be falsified if these definitions somehow do not apply to a valid conclusion following a valid syllogism.

Disproving a "Truth":

1. A > D;
2. C > B;
3. D > C;
4. D.
5. Therefore A;

In that case, it is not necessary for "A" to be true, as "B" is another explanation. Although, both A and B are "Truths" in form - the type of "Truth" that "A" is, (a plausible truth) is not interchangeable with deductive truth. Even if "A" IS deductively true, within our own "Causal Domain", it is NOT the case that it MUST be true within this "Logic Domain".

So, "A" is not "True" [deductively true], because it is not necessarily true - nor false - in this context/domain.

---

3. The Distinction Between Truth and Existence:

The fallacy that leads to equating "Existence" and "Truth" is that only the "Truth Value" of "Existence" is compared to "Truth", not the "holistic" value of the qualities that make up existence, (time, location, form, nature, etc.).

1. A [Exists]
2. A > B [is True]
3. A [Exists] Therefore "B [is True]"
4. "B [is True]" does not necessarily mean "B [Exists]".

Because of this "injection", there is a lot of equivocation between the terms.

But, abstractly: even if "2 + 2 = 4" is true AND ALSO exists. ... It would appear that "2 + 2 = 4" would exist in a different "domain of reality" than where "my tomato" exists in my own "causal domain". Causality does not necessarily apply to logic, but can certainly be a factor. And, when one domain of reality is plausibly affected by other domains, it is plausible that certain domains might "override" each other.

Answer 6

Exsistence is axiomatic and just exsists.Truth is not a strong word and can be manipulated with (post hoc - add hoc) alernatives. How dose logic define true and false from mazy of out-of-context-facts? Created stories that seem valid but are not. Truth can be manipulated. Honesty is personal hard work. A person must work hard to be objective and honest. Honest logic dose not accept the false to live by. Individualy we must use honest knowledge attained. To live with self esteem logic must be used with honesty to attain the truth or the fallacy of the origination of the percept. Honesty is the logic knowledge that sets the proper part of truth and false into exsistence. The missing link is honesty.