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"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.)

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Axioms and do not have to be defined in terms of everyday language per se. For instance Hilbert has said that it does not matter if I am talking about chairs tables and beermugs, as long as the relations between the chairs tables and beer mugs is internally consistent. In terms od defining true and false, you might simply define them as boolean values of zero and one. No more no less. –  Baby Dragon Feb 8 '13 at 20:31
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5 Answers 5

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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.

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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.

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That's an excellent answer covering Symbolic and Mathematical logic. –  SF. Feb 16 '13 at 10:44
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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.

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I think SF. was asking a more precise question, namely how different logics define true and false respectively. If there is no difference between them, you should point that out, otherwise your answer isn't detailed or exhaustive enough to be considered useful. –  iphigenie Feb 9 '13 at 13:52
    
See Whitehead's account of propositions in PR and AI. By confiscating meaning and value under the auspices of truth and falsity, he complains we miss much of the novelty offered to the world. Propositional feelings comprise the entire creative advance of the universe, not just a component of judging subjects. Using this criteria, one could just as easily claim that false, erroneous, or "non-conformal" propositions contribute more to the advancement of the world. Thanks for posting this discussion and the helpful dialogue, you folks have raised some good points! –  Myron Moses Jackson Feb 9 '13 at 23:20
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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.

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While I think you are right to point out that calling "True" and "False" "axiomatic expressions" is a bit confused, I have my doubts about the accuracy of your description. The "(potential) results" of a "compatible logical grammar" doesn't seem to accurately describe "truth" and "falsity", at least it doesn't do so with any level of clarity. More commonly they are referred to as "truth-values" or "semantic values". Additionally, your final paragraph seems to be baseless speculation about the reasons behind OP's question. I think an edit is needed to clarify your answer. –  Dennis Feb 15 '13 at 4:49
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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.

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This answer seems to be missing the distinction between propositions or statements being true and agents "telling the truth". The latter, while interesting in its own right as a matter of ethics, is not what Truth in logic is used to talk about. –  Paul Ross Feb 26 '13 at 2:33
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