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We write 'If A, then B' to mean that if A is true, then B must be true because B is a logical consequence of A i.e. it is impossible for A to be true but B to be false.

Let us consider one such statement:

'If S1 , then S2' is true, where S1, S2 are expressions.

Let it also be the case that : 'If Si , then S2' is true, and let there be many such Si. For all such Si and S1, assume they do not contradict each other, and indeed there is a finite deduction from all Si and S1 to S2.

Now for a given S2, I want to generate a finite conjunction of expressions which, so to speak, encodes necessary properties of the premises so that the conclusion (S2 in this case) is the logical consequence of the premise (all such Si). Ideally, this expression must take in some input, and yield some Si, and with appropriate set of constraints, we should be able to generate all Si. (For sufficiency, we may append some disjunction expressions as well if need be).

Is there a method to generate such an expression?

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  • Not very clear... Every conjunction with S2$ will have S2 as consequence. Sep 2, 2020 at 13:56
  • @MauroALLEGRANZA I am thinking on lines of reverse mathematics. Philosophically, premise is contained in the consequence...so I am looking to encode this contained thing so that I get all premises. I hope it clears something...
    – Ajax
    Sep 2, 2020 at 14:01
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    "Philosophically, premise is contained in the consequence" is false, and this is why your project can not succeed. It is true, to some extent, in Aristotelian logic, and this is why Kant believed it once, when such restricted logic was all there was. And even then it was only true when there are no extra-logical axioms that also contribute to the consequence. In general, S2 can be implied by premises with concepts not even alluded to in S2, and so nothing finite can "encode" them. Reverse mathematics only works when the form of premises is severely constrained by pragmatic context.
    – Conifold
    Sep 2, 2020 at 19:15
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    @Ajax "are you denying" Yes and it is obvious it cannot possibly work. If you could derive all possible premises P₁ ∧ P₁ ∧ … ∧ Pₙ from a conclusion C, then the premises would follow from the conclusion, which would makes the implication an equivalence, and the conclusion therefore equivalent to the conjunction of premises, and then this would no longer be the general case but a trivial case of P₁ ∧ P₁ ∧ … ∧ Pₙ⇔.C. Sep 2, 2020 at 19:34
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    @Ajax Adding to the conclusion to infer the premise will move it ever closer to equivalence with the premises. The idea cannot work. Sep 2, 2020 at 19:55

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