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Let us assume a logical system with only 3 axioms (laws of thought):

Axiom 1: A statement that is true will remain true till a change is made in the system.
Axiom 2: A statement may not be true and not true.
Axiom 3: A statement must either be true or not true.

Basic rules are that statements refer to the truth of other statements and can use boolean operators AND, OR and NOT to tell if-then statements.

Then, we state a statement, G:

G can not be proven as true.

Now, let us fill in this table (with ticks and crosses):

G is       Provable true     Provable false     Not provable true/false
-----------------------------------------------------------------------
True            ?1                 ?2                  ?3
False           ?4                 ?5                  ?6

Obviously, blank 2 and 4 cannot be, and can be crossed.

If G is true, then it states that it can not be proven true. Therefore, blank 1 is crossed out.

If G is false, then it can be proven, either as true or as false. Under our current assumption (of it being false), we can only prove it as being false.

Is this enough to conclude that G is false?

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  • what about slot 3? Slot 6 would be viable except that you've assumed it away in the second to last sentence.
    – Dave
    Mar 12, 2015 at 14:40
  • You have to take care in "handling" both provable and true in the same context ... See Gödel's Incompleteness Theorems and Tarski's undefinability theorem.
    – Mauro ALLEGRANZA
    Mar 12, 2015 at 15:33
  • Is statement G self-referential? If so, you need to give us rules for what happens (does the internal G evaluate first). Also in undefined cases, does the logic default to true or false? (e.g., the material conditional defaults to True in sentential logic in order to handle untested conditionals).
    – virmaior
    Mar 13, 2015 at 0:45

1 Answer 1

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Unless I'm missing the point, none of your axioms offer any definition of proof or state that all true statements in your system must be provable. Given that, there's no reason a statement might not be both true and not provably true --or, for that matter, false, but not provably false.

In my opinion, your axiomatic system would need a formal definition of proof, as well as a definition of what kinds of statements belong in its domain of discourse, in order for this question to be meaningful.

You're basically retreading ground that was covered at the dawn of modern symbolic logic. You may want to look into the work of Tarski, Russell and Godel to see how these questions originally arose, and how they were handled.

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  • This topic does seem a lot more complicated than I imagined; sorry for bothering you.
    – ghosts_in_the_code
    Mar 12, 2015 at 15:57
  • 2
    It's not a bother, that's what the site is here for. I edited my answer to provide a little more context.
    – Chris Sunami
    Mar 12, 2015 at 16:38

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