# Facts and the denial of "Iff it is true that A, then it is true that it is true that A"

The SEP article on axiomatic theories of truth goes over a theory in which "the truth iteration axiom ∀A(T[TA] ↔ TA) is inconsistent." Apparently (and this sentence is an edit), it's inconsistent with the commutativity of negation, but so I was wondering about an "it is a fact that" operator in turn, and if there could be an axiomatic theory of facts such that fact-iteration was dropped in that theory. Offhand:

• It is a fact that S is P if and only if S is P.

... so I wonder how much difference there is between "it is true that" and "it is a fact that," except that the SEP article on propositions has a section about facts, true propositions, and states of affairs with evidence of the distinction between truths and facts. Can an axiomatic theory of truth incorporate an axiomatic theory of facts so as to lack truth-iteration but retain fact-iteration, or vice versa?

• Looks like it might just be a corollary of Tarski's undefinability theorem. You cannot have a transparent truth predicate without inconsistency. Commented Dec 21, 2023 at 2:00
• @KristianBerry The SEP doesn't explain and "the commutativity of negation" is no explanation... So can you explain why the truth iteration axiom ∀A(T[TA] ↔ TA) is inconsistent? Commented Dec 21, 2023 at 10:10
• @Speakpigeon the SEP does explain it: "However, the axioms ∀ A ( T [ T A ] ↔ T A ) and ∀ A ( T [ ¬ A]↔¬TA) are inconsistent over weak theories of syntax, so one of them has to be given up " Commented Dec 21, 2023 at 13:15
• @KristianBerry Ah, so, ∀A(T[TA] ↔ TA) and ∀A(T[¬A] ↔ ¬TA) actually aren't formal logic. They are expressions from some weak theories of syntax, whatever that means. As long as you understand the words you use. Commented Dec 21, 2023 at 17:01

I'm not the definition boss, but I'm the boss of me, definitions are arbitrary, and there can be no possible answer to a question about the difference between truth and facts without first defining truth and facts such that there can be a difference, so here are some definitions that mostly preserve the common use, following "the fact that p is the truth of < p >" but restricting it more narrowly.

A fact is a measurement result.

A truth is a relationship between measurement results, relationships, or abstract objects. (Thus all expressions of facts are truths, but very few truths express facts.)

In which case it's never a fact that S is P because whether S and P are measurements or relationships, "S is P" is a relationship. Then:

• "the snow looks white" is a fact.

• "given a hypothetical thing which returns measurement results X, the thing is snow" is a truth and not a fact (relates a relationship between a measurement and a relationship [identity] between categories)

• "if it's snow, then it looks white" is a truth and not a fact (relates a relationship between the above relationship and another measurement result).

• "snow is white" is a slightly ambiguous use of common language to imply "if it's snow, then it looks white." The "is" obviously does not imply identity.

• "It is a fact that the snow looks white" is a truth and not a fact ([this measurement result] is a member of set [measurement results])

• "It is a fact that if it's snow, then it looks white" is false.

• "It is a fact that it is a fact the snow looks white" is false, as are any further iterations with either truth or fact.

• "It is a truth that it is a fact that the snow looks white" is true.

• "It is a fact that it is a truth that it is a fact that the snow looks white" is false, as are any further iterations with either truth or fact.

• "It is a truth that it is a truth that it is a fact that the snow looks white" is true, as are any further iterations with truth (but not with fact).

• I like how you thought to interleave the operators. I'll give other posters a chance to offer answers but so as of now, your answer is the candidate for acceptance. Commented Dec 21, 2023 at 13:17
• Actually, I have a follow-up question that came to mind, so to avoid posting a new question without accepting an answer to a previous one... Commented Dec 21, 2023 at 13:40