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If we have a statement of the form $P \implies Q$ and this statement is true, then we say that $P$ is a sufficient condition for Q.

Suppose the following statement:

$$2 \, \text{is odd} \implies I \, \text{is invertible}, \qquad \text{where $I$ is the identity matrix}$$

The above material conditional is True just because $P$ is False or $Q$ is True. But there isn't anything that allows to guarantee (sufficient) $Q$ from $P$. Why in such cases we call $P$ a sufficient condition for $Q$?

user599310
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  • Why closing the question when in the proposed answer the word sufficient doesn't appear? ... – user599310 Sep 21 '22 at 13:45
  • The thing is, despite the name, material implication doesn't actually express "causation", as one would think when considering it outside of the mathematical standpoint. $P\implies Q$ simply means, whenever P is true, Q is also true. As to why the statement is to be considered true when P is false; it doesn't actually have to be. We only define it to be true because it happens to be helpful when moving on to first-order logic, rather than propositional logic. You can read my answer to a similiar question (this gets asked a lot) here: https://math.stackexchange.com/a/4534223/858891 – Dark Rebellion Sep 21 '22 at 20:59
  • @DarkRebellion My question was about "sufficient" condition. Why to call them this way when the only thing that matters is the truth values of $P$ and $Q$? The term "sufficient" doesn't appear on this answer also. – user599310 Sep 21 '22 at 23:14
  • The "why" of material implication can only be truly understood with some familiarity with the basic methods of proof. Examples and analogies will only get you so far. Students must be able to derive (1) $A \land B \to(A \to B)$, (2) $A \land \neg B \to \neg ( A \to B)$, (3) $\neg A \land B \to (A\to B)$, (4)$ \neg A \land \neg B \to (A \to B)$. The logic is simple ("first principles"). The formal proofs are blessedly short. Each of these results correspond to a line in the truth table for $A\to B$. – Dan Christensen Sep 22 '22 at 01:24

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