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Should truth be defined in terms of satisfiability? Also, should satisfiability be treated as a primitive notion?

AUTIST INC
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    What do you mean "should"? – abracadabra Mar 29 '24 at 14:18
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    By Tarski's undefinability theorem, truth cannot be defined in terms of satisfiability within strong enough encoding capable formal theories such as those of current math foundations ZFC or MLTT or HoTT. Therefore it hints philosophically Tarskian truth is unsatisfying to pin down or pick out truth in an adequate sense without infinite homunculus regress. And it's possible to treat satisfiability as primitive as discussed here whose role is only to prove paradox not truth or falsity... – Double Knot Mar 29 '24 at 17:39
  • Can you motivate the question? What led you to wonder this? Thanks. – Julius Hamilton Mar 30 '24 at 15:54
  • Satisfiability is defined in terms of truth, i.e. there exist an interpretation where the sentence is true – Poscat Mar 31 '24 at 04:06
  • @Poscat, I disagree with that. For why should truth be a primitive term and not satisfiability? For we could treat satisfiability as a primitive term and define truth in terms of satisfiability. – AUTIST INC Mar 31 '24 at 05:15
  • I'm not stating an opinion, I'm stating a fact. Maybe you should clarify how you define satisfiability without truth? – Poscat Mar 31 '24 at 06:18
  • @Poscat, let satisfiability be a primitive notion and then define truth as that which is satisfiable. – AUTIST INC Mar 31 '24 at 06:23
  • Again, how do you define satisfiability? – Poscat Mar 31 '24 at 07:29
  • It’s a primitive term. I don’t have to define it. – AUTIST INC Mar 31 '24 at 13:05
  • @Poscat, for your reference: https://en.wikipedia.org/wiki/Primitive_notion?wprov=sfti1# – AUTIST INC Mar 31 '24 at 13:10
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    In general, when we do logic, we use a mathematical metatheory (such as ZFC) where there's no such thing as primitive satisfiability. Am I understanding you correctly that you want a metatheory where satisfiability (whatever that means) is baked in? – Poscat Mar 31 '24 at 15:25
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    Also note that when we talk about truth, we are talking about truth in specific structures, i.e. it does not make sense to talk about truth without specifying models, where satisfiability has nothing to do any specific model so defining truth in terms of satisfiability won't work anyways. – Poscat Mar 31 '24 at 15:31

1 Answers1

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Should truth be defined in terms of satisfiability?

What is it that we want semantics (for some formal theory, presumably) to do? Some people think of it in terms of "saying what it is our syntax and proofs are ('really') talking about", but one might ignore this issue and think in purely 'pragmatic' terms, as in, "how can semantics help me and my formal calculus?", to which a reasonable answer might be "by enabling one to talk of non-provability": formal syntax is great for saying stuff like "φ follows from S", but not so great at denying such, barring cases such as when one already had that ¬φ, of couse

So, if one has a syntactical notion S ⊢ φ, it's interesting to have a companion notion S ⊨ φ that's easier to work with in terms of expressing S ⊭ φ, and also that's why one cares for soundness and completeness: if S ⊬ φ, completeness assures us that there's some witness for S ⊭ φ, and soundness then assures that S ⊢ φ cannot be

The usual tarskian approach to 'truth (in structures)' as satisfiability is a very convenient one for such purposes

[edit] Also, something along these lines is arguably how the (early) historical development took place: after many people for a long time tried to prove Euclid's parallel postulate from the other axioms, its non-provability was finally shown through the construction of 'models' (and suitable interpretations of 'point', 'line', etc., in them) satisfying the remaining axioms and falsifying the parallel axiom

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