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I interpret the expression 'If P then Q' as asserting that if P is true Q is automatically true. So, we would say 'If P then Q' is true only when it indeed is the case that P being true implies Q is true. However, in logic, the truth of 'If P then Q' is determined solely on the basis of the truth values of P and Q individually and not by verifying whether Q follows from P, or is implied by P.

So I just don't get how we can decide the truth of if-then statements by just looking at the truth values of P and Q. For it to be true don't we need to prove somehow that the truth of Q follows from the truth of P?

In one of the logic books that I read, they explained conditional statements in this manner: 'If P then Q' asserts that it is not the case that P is true and Q is false. I liked this. It makes me understand the truth table of conditional statements well. However, by this explanation, I'm not able to see why would one use the words 'If then' then. How the idea that it is not the case that P is true and Q is false follows from the meaning of words 'if then' (or 'implies' for that matter).

Should I completely forget about the meaning of if-then sentences as used in ordinary language and assign them a new meaning?

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We go to Jack and Jill's party in April.

Later we can't remember whose birthday was being celebrated.

But I tell you that I think Jill has an autumn birthday.

So you say "Ah, if Jill has an autumn birthday, then Jack's birthday is in April".

That's a perfectly natural ordinary-language thing to say. But in no sense are you asserting that there is some sort of connection between their birth dates. It's just that either Jill's birthday is in April or Jack's is. So if not the first, then the second.

Plenty of ordinary-language conditionals are like this, saying no more about the world than the corresponding material conditional.

Of course often we assert if P then Q when we think that there is e.g. some causal mechanism which would make the truth of $P$ bring about the truth of $Q$. But equally, we often assert P or Q when we think that there is e.g. some causal mechanism which would ensure that one of $P$ and $Q$ is true ("either the email will successfully be delivered or you'll get an error message"). But what I actually assert when I claim P or Q is true just so long as one or other (or perhaps both) of $P$ and $Q$ is true: I don't thereby assert there is a causal relation. Likewise, perhaps for if P then Q; arguably even if we assert it on the basis of some belief about a connection or implication relation in the background, that isn't part of the literal content of the conditional assertion.

For more on this standard sort of line see Chs. 18 and 19 of An Introduction to Formal Logic downloadable from https://logicmatters.net

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  • $\begingroup$ This. If-then statements even in natural language do not in general express (nor are understoood to express) causality; they generally do express deduction, inference, implication, as they do in logic; I don't think that it's instructive to treat the natural language if-then and the mathematics if-then as being fundamentally distinct, even if the former is more nuanced. $\endgroup$
    – ryang
    Sep 5, 2023 at 5:45
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    $\begingroup$ On a separate note: I posit that the frequently-reported mental disconnect that false→false is true is due not to any expectation of causality but because, calling a conditional and its consequent both an "implication", we are frequently tempted to wrongly conflate them. $\endgroup$
    – ryang
    Sep 5, 2023 at 5:45
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Material conditionals can be interpreted as a case-by-case account of whether $Q$ holds if we have $P$. Clearly, if we already have $Q$, then we have it if we also have $P$. If you don’t have an intuition for the Principle of Explosion, it is harder to motivate $Q$ being true if $P$ is true for a false $P$. Still, it seems pretty clear that assuming contradictions can’t happen, anything follows from one.

Logical implication deals with conditionals that hold such that in all cases that $P$ is true, $Q$ is also true. It generally tracks with $\vDash$, but is often used to show meta-theorems. For example,

$\Gamma, A \vdash B \implies \Gamma \vdash (A \to B)$

is a statement of the Deduction Meta-theorem for Propositional Calculus which uses $\implies$. There are treatments of $\implies$ in modal systems such as C.I. Lewis’s where $(P \implies Q):=\Box(P\to Q)$. Intuitionistic Propositional Calculus is obtained in a similar manner.

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In my opinion, the best way to understand the arrow $\to$ is by pure definition, that is, it is a binary connective forming a complex proposition by joining two other propositions, the antecedent and consequent, such that the complex propostion is false if and only if the antecedent is true and the conequent is false.

However, I offer a semantic approach to help with your intuition...

Assume we have the following conditional statement: if Jonnny is grounded, then Johnny cannot play outside.

Is this conditional false if the antecedent is false and the consequent is true? In other words, is this conditional false if Jonny is not grounded and cannot play outside? The answer is no because it may be dinner time, or it may be time for Johnny to be asleep in bed. In other words, the condition that "Johnny is grounded" is sufficient to prevent Johnny from playing outside, but it is not necessary to prevent Johnny from playing outside.

Now, is this conditional false if the antecedent is false and the consequent is false? In other words, is this conditional false if Johnny is not grounded and can play outside? For obvious reasons, no. This interpretation does not imply the conditional is false.

The only interpretation that guarantees the conditional is false is when the antecedent is true and the consequent is false. In other words, the conditional is false if and only if Johnny is indeed grounded and Johnny can play outside.

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