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

"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.)

• 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. Feb 8, 2013 at 20:31
• You just asked a million dollar question. All logic wants is a T or an F. Lucksawitz (sic) had a 1.5. between them. He was frowned upon by some. Bottom line, GIGO, garbage in, garbage out. The purists seem to set up a black and white world to me. Aug 16, 2017 at 2:12
• Lukasiewicz. Well there are many logics to choose from now is my understanding. Aug 16, 2017 at 2:15
• @Gordon I share your view. The problem seems to the assumption that the world accords with our formal truth-values. I would see this as THE central error made by most metaphysicians. There is a failure to take note of the small print of Aristotle's logical rules, and maybe the OP would do well to read it. C.W.A Whittaker's book on 'De Interpretatione' would be my recommendation.
– user20253
Aug 16, 2017 at 16:15
• @PeterJ sorry, I forgot to put the @ x. Aug 16, 2017 at 17:27

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.

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.

• That's an excellent answer covering Symbolic and Mathematical logic.
– SF.
Feb 16, 2013 at 10:44

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.

• 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. Feb 9, 2013 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! Feb 9, 2013 at 23:20

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.

• 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. Feb 15, 2013 at 4:49

## 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.

• How do you define "that which 'is'"? Do you do it without recourse to what is true? Also, why think that what is true is necessarily so (i.e., "must be")? Maybe logical truths, but truths modeled in a formal language that aren't tautologies? Aug 16, 2017 at 0:20
• But "necessary" means "necessarily true", and "that which is" means "those propositions (or whatever you're assigning truth values to) which are true". The worry is that you're tacitly assuming the concept you're trying to define -- truth. You might be right that truth can be defined for various restricted domains, but that doesn't seem to be what OP is asking. Also, logic doesn't typically have anything to say about causality (though you might study the logic of causation) and does not define truth in terms of it. Aug 16, 2017 at 0:38
• @Dennis, I deleted the last comment because I am poorly trying to express something in a small space. I understand - I think - what your objection might be. IF it is believed that "What is" is synonymous with "Truth" - then my argument might be perceived to be "begging the question". However, I believe these two terms are wrongfully conflated. If it addresses your objection - I could show how "What Is" and "Truth" are necessarily separate concepts. Am I understanding the objection correctly? Aug 16, 2017 at 0:41
• yes, that would address my worry/objection. I have doubts that the result would remain within the realm of logic, even if successful. The OP's question is really about a very formal/mathematical conception of logic, while what you have to say seems interesting I worry that it strays from that conception of logic. One reason to worry is Tarski's theorem on the undefinability of truth. See also, the page on a truth predicate. Aug 16, 2017 at 0:47
• @Dennis - I started trying to address your objection, but, I realized that I could just avoid the whole matter if I just added `"entailed"` to - "which is [entailed]" and "must be [entailed]". I hope this clarifies and removes what seems like circular reasoning. Again, I am hoping to avoid the Red Herring about whether "Existence" and "Conclusions" are of the same "Form/Nature" existing in the same "Domain of Reality". Hopefully this is sufficient. This might be better suited for another question, or side-discussion. Let me know! Aug 16, 2017 at 3:20

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.

• 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. Feb 26, 2013 at 2:33