`if's in formal logic

Question

$(\alpha \to \beta) = \neg \alpha \vee \beta $

The truth table is negative when $\alpha = 1$ and $\beta = 0$. It is troubling me that $(\alpha \to \beta) \vee (\beta \to \alpha)$ is tautological.

P, Q be $x = 1$, $y = 0$. Then either

If $x = 1$ then $y = 0$
If $y = 0$ then $x = 1$

Why are they having to end up related? Why is this example wrong? I am studying the book ``Introduction to Formal Logic - Peter Smith" and it says :

``(if $= \to$) `$\to$' stands to some ‘if’s as $‘\wedge’$ stands to ‘and’ and $‘\vee’$ stands to ‘or’; for the material conditional does at least capture the core logical role of those ‘if’s which feature in singular indicative conditionals"

What does he mean by some `if's?

Either $x = 1$ or $x \ne 1$; either $y = 0$ or $y \ne 0$. In all four cases, one of 1. and 2. holds. — Trebor, Aug 30 '23 at 06:16
See e.g. this post and related ones: the truth-functional def of the conditional has this "unpleasant feature" that is TRUE when the antecedent is FALSE. — Mauro ALLEGRANZA, Aug 30 '23 at 06:37
With "some ‘if’s" the author alludes to the fact that natural language has several usage of "if"; see The Logic of Conditionals as well as Indicative Conditionals and Counterfactuals Conditionals. — Mauro ALLEGRANZA, Aug 30 '23 at 06:39
The usage of so-called "material conditional" (the formal translation of "if" above) in mathematical logic is due to its link with Modus Ponens that is the inference rule mostly used in mathematics; see the post regarding Why do we stick with the material implication? — Mauro ALLEGRANZA, Aug 30 '23 at 06:44

`if's in formal logic

0 Answers0