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Can there exist a "self-evident" statement? That is, can there exist a statement that offers sufficient substantiation for itself?

For example, a statement can be constructed

This statement is true.

Can such a statement be categorized as self-evident since it attempts to argue its own validity? Moreover, can another example be constructed where the proposition and its justification are expressed in one statement?

A is true because B, which is a logical justification for A, is true.

Will this statement constitute as an example of self-evident statement?

If such a "self-evident" statement can exist, is it rationally justified to doubt the statement?

Lastly, what are some examples of "self-evident" truths? Many people argue that our existence is self-evident. According to the definitions that you may provide above, will such an example constitute as self-evident?

  • Technically, "this statement is true" isnt valid logic, because it references itself.Your 2nd example involves two statements that reference each other, ultimately the same thing.If you accept Set Theory, the only "self-evident" statements are ones we accept as axioms (although, as Godel noted, they may end up contradicting each other). You could argue statements like A or (not A) (or other tautologies) are self-evident (for any statement A), but that's arguably just because of how we define logic. And, of course, things like the Declaration's preamble are only metaphorically self-evident. – barrycarter Aug 14 '18 at 19:28
  • Addendum: just because ("A" or "not A") is true doesn't mean either A or not A is provable (truth and provability aren't identical, alas). – barrycarter Aug 14 '18 at 19:30
  • I made an edit hopefully to clarify the question. You may roll it back as I assume you are aware. – Frank Hubeny Aug 14 '18 at 21:46
  • "This statement is true" is known as the Truth teller sentence, and is usual seen not as self-evident but as undecidable, see Ross's answer. In typical use Self-evidence refers not to self-justification but to some kind of intuitive support, as in "lines intersect at a point" or "I think therefore I am" – Conifold Aug 14 '18 at 21:51
  • If you accept ZF set theory is consistent, any theorem proven by it is technically "self-evident" since it's mathematically equivalent to the set of axioms. My latest guess at what self-evident "really" means is provability. You are looking for P such that "P is provably true, because P is true". There are definitely statements in ZF that don't have that property (ie, they are true but unprovable), but I think ZF requires "P is true" come from "P is provably true", not vice versa. – barrycarter Aug 18 '18 at 16:08
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One can find many self-evident statements. Let "P" be any statement that one can assign a truth-value to, that is, that one can assign either "true" or "false" in a truth-functional logic.

Now "P" is clearly not self-evident, but "P v ¬P", that is, "P" or not "P" is self-evident in that truth-functional logic.

Since "P" was arbitrary, this generates many self-evident statements.

For a reference see forall x: Calgary Remis, section 15.6 on "Disjunction", pp 112-116.

Now consider the questions:

Will this statement constitute as an example of self-evident statement?

Given an appropriate "P", "P v ¬P" should be self-evident in a truth-functional logic.

If such a "self-evident" statement can exist, is it rationally justified to doubt the statement?

One can rationalize most anything. So it is possible to doubt this self-evident statement. One could move from a truth-functional logic to some other kind of logic.

Lastly, what are some examples of "self-evident" truths? Many people argue that our existence is self-evident. According to the definitions that you may provide above, will such an example constitute as self-evident?

Let "P" be "I am alive". Then "P v ¬P" would be "I am alive or it is not the case that I am alive". I think that would be self-evident in a truth-functional logic.


Reference

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

  • Can you answer, if the statement P: " I exist" is self-evident? – mathnoob123 Aug 14 '18 at 21:44
  • That statement would not be based on the pattern I provided, however, it might be considered self-evident using some reasoning like Descartes' "I think therefore I am". However, I hear there are people who doubt even that. – Frank Hubeny Aug 14 '18 at 21:48
  • If to understand a statement one needs to learn what a "truth-functional logic" is it can hardly be called "self-evident". And if we take "P or not P" in the intuitive sense then I am not so sure either since even some mathematicians (intuitionists) consider it generally false. Maybe "P is P" would work better. – Conifold Aug 14 '18 at 22:02
  • @Conifold "P is P" might work although I am sure someone will find a way to doubt it as well. A truth-functional logic is a classic logic with the law of excluded middle where sentences take on the value of either "true" or "false" with nothing inbetween. It is the logic that works with truth tables. – Frank Hubeny Aug 14 '18 at 22:55
  • Aristotle, who first explicitly stated P or not P, did not know "classical logic", neither did Kant. It is a 19th century creation after much effort, not exactly self-evident. Aristotle also rejected P or not P with respect to the future contingents, as in the tomorrow's sea battle example. That's a general problem, "self-evident" is either false or needs too many caveats to be evident, P is P was denied by Hegel. – Conifold Aug 14 '18 at 23:11

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