Is there such a thing as a 'necessary truth'?

Question

Wikipedia (note the redirect) defines 'necessary truth' as statements which "could not be untrue", and I assume that this is how the term is usually used. A search through the SEP shows that while there is no article dedicated to the topic, this phrase is used in over 100 articles.

This question is in two parts:

1. What is a necessary truth?
   Are these three definitions equally correct:

   • (A) A statement that cannot be untrue
   • (B) A statement which is true in all possible worlds
   • (C) A statement whose negation implies a contradiction

   The only way I can imagine (A) or (B) to be applicable would be if (C) is also applicable. Is there any other way to determine if something is a necessary truth other than by showing (C)?

2. Do necessary truths exist?
   Based on qualification (C) above, I assume that the answer hinges on whether or not logical contradictions are impossible (and is therefore closely related to a few questions already seen on this site, such as this one and this one). However, it appears to me that Quine (in "Two Dogmas of Empiricism") disagrees. Would someone care to shed light on the issue?

Answer 1

I do not think you can find a brief answer to this debated issue.

See at least Modal Logic, Varieties of Modality, The Epistemology of Modality and Possible Worlds.

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Def (A) of necessity

A statement that cannot be untrue

is quite useless; "cannot be" means "impossible". Thus, necessity is simply the negation of the possibility of the negation, which is quite "standard" in modal logic, but only moves the problem one step back.

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Def (C)

A statement whose negation implies a contradiction

reduces necessity to (logical) entailment. But the relation of entailment is basically the formalization of "logical necessity" : B necessarily follows from A iff A entails B (i.e. iff B is logical consequence of A).

And also a contradiction is "defined" as a logical impossibility.

So again, we have a sort of "circularity".

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Def (B)

A statement which is true in all possible worlds

is the base of the modern semantics for the languages of modal logic [see SEP's entry] :

The power and appeal of basic possible world semantics is undeniable. In addition to providing a clear, extensional formal semantics for a formerly somewhat opaque, intensional notion, cashing possibility as truth in some possible world and necessity as truth in every such world seems to tap into very deep intuitions about the nature of modality and the meaning of our modal discourse.

Unfortunately, the semantics leaves the most interesting — and difficult — philosophical questions largely unanswered [emphasis added].

Answer 2

According to Wittgenstein, necessary truths are rules for the governance of language. Their role is therefore normative, and to be contrasted with empirical or contingent truths, which are descriptive.

This is a simple and insightful solution to a problem which has confounded, and continues to confound, philosophers.

For more, see this answer: How fundamental is logic?

Answer 3

I'd say that in the Western philosophical tradition, the notion of truth as a stable category has been a constant since at least Aristotle.

In Buddhism, in particular the philosophy of Nagarjuna he negates the truthfulness of truth; in broad terms this follows from the analysis of impermenance; hence Nagarjuna would deny that there are neccessary truths; it part of the doctrine of sunyata.

What is a necessary truth? Are these three definitions equally correct:

This can be certainly discussed. But another direction one can take is that we have three different but workable definitions of truth that are useful, for example (B) is how the semantics of intuitionistic logic is discussed in mathematical logic - in this example (C) does not hold, at least in the form of the law of the excluded middle.