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Suppose I have a statement like "2+2=4". I believe the truth of that statement causes the truth of a statement like "2+2=4 OR 3=5". Have any philosophers or mathematicians developed the notion of a causal theory of truth? Such a theory should ideally require two statements to be relevant for there to be a causal connection between them.

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  • I think the answer given so far is exactly what you are looking to go beyond? If so then see SEP on Lewis's counterfactuals for a quick-but-dense intro.
    – Canyon
    Aug 20 '17 at 23:18
  • What you're likely looking for in mathematics is the notion of "forcing". Nothing causal, since mathematical entities are typically assumed to be abstract if they're countenanced, but certain statements "force" via the forcing proof method other statements to be true. For example, the axiom of constructibility (V=L) forces the continuum hypothesis (among other independent axioms) when added to ZF set theory.
    – Dennis
    Aug 21 '17 at 2:35
  • Isn't "theory of truth" is simply, logic? Aug 22 '17 at 5:56
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Causal set theory, https://sites.google.com/site/lisaglaserphysics/research/causal-set-theory or https://en.wikipedia.org/wiki/Causal_sets or just google "causal sets", imposes a poset ordering on events corresponding to "causal connection" (your words). The logical analogy is just implication, and I'd think that's all there is to it. Although maybe substructural logical connectives better fit your intuition, which you don't really elaborate in sufficient detail for a sharp answer. For example https://arxiv.org/abs/gr-qc/0109053 which involves resource-aware linear implication in this case.

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Yes, Whitehead and Russell did it. What you call "a causal theory of truth" is called logic in philosophy. Whitehead and Russell showed that logic and mathematics are one and the same.

That example given in your question is called the "principle of addition" in W&R's Principia Mathematica:

If q is true, then 'p or q' is true.

But I'm curious what lead you to this idea.

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  • Well, in mathematics, there are vacuous implications. Such as "The angles of a triangle add up to 180 degrees" implies the four color theorem vacuously. Intuitively, the truth of the first has nothing to do with the truth of the second. Has anyone made this intuitive notion rigorous?
    – user107952
    Aug 20 '17 at 20:20
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    Yes, they are proved in 2.02 and 2.21 in Principia Mathematica: A true proposition is implied by any proposition; a false proposition implies any proposition. Aug 20 '17 at 20:44

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