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Let's consider the following sentence: $$ A \implies B $$

If $A$ has no a construction / proof, does it mean that sentence is true?

  • "If A has no construction / proof" is a bit vague. Are you assuming $\lnot A$? It seems you are asking if it is possible to establish $\lnot A \vdash A \Rightarrow B$, which is provable if you have the axiom-schema $\vdash \bot \to X$ (it is written in different ways, but it always boils down to that). In general, every statement that preserves constructability is provable in natural deduction, and $\lnot A \vdash A \Rightarrow B$ preserves constructability because both the left and right side of the $\vdash$ are not constructible. – DanielV Nov 02 '16 at 22:19
  • What does it mean $\vdash$? Does it differ with $\Vdash$? –  Nov 02 '16 at 22:26
  • $\vdash$ means provable in a given axiom system.It is what follows when you apply theorems and transformations from a given proof system. $\Vdash$ means the statement is true with respect to semantic definition of logic. – IamThat Nov 04 '16 at 12:51

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See Brouwer–Heyting–Kolmogorov interpretation :

A proof of $A \to B$ is a function [or procedure] that converts a proof of $A$ into a proof of $B$.

If there is no proof of $A$, we will never have a proof of $B$, but the said function still count as a proof of $A \to B$.

Consider the formula :

$\bot \to P$.

Since there is no proof of $\bot$, any mapping may count as a proof of $\bot \to P$, since it has to be applied to an empty domain.

Thus, the above formula is provable.

Mauro ALLEGRANZA
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  • I have to admin that I don't understand: "but if we have the said function, we may still assert $A \to B$ ". –  Nov 04 '16 at 11:46