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Trying to wrap my head around conditional statements/implication and the respective truthtable in propositional logic. Read a number of the related posts on here. I understand that there is no causal relationship between $A$ and $B$, so from a false proposition anything can follow.

This made sense to me especially with the "promise" analogy: $A \Rightarrow B$ is a promise, that can only be broken when after a true condition $A$, $B$ is false (second row of the truth table). If the condition isn't met then the promise can't possibly be broken, no matter $B$. With most real life examples this makes perfect sense to me: "If you write an A in an exam, then you get a Dollar", "if you finish supper, then you get dessert" etc.

However one particular math example confuses it again for me:

$A:$ $x$ is an even number

$B:$ $x$ is divisible by two

How can $A \Rightarrow B$ be true when $A$ is false? An odd number is never going to be divisible by two. It's like saying an odd number is even. What am I missing here? What's my misconception? Am I understanding it generally wrong?

Also on a related note, what's the proper terminology for the "if-part" and the "then-part" of a conditional statement?

LinusDieLinse
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  • This is what is called "vacously true". An "If $A$ , then $B$" statement does not tell us what we have if $A$ is false, so the implication is true whatever $B$ is (can be false or even utter nonsense). In programming languages, an "if"-statement is ignored if the condition after "if" is not satisfied. – Peter Dec 31 '20 at 09:10
  • The if part is the hypothesis, the then part is the conclusion. – Gerry Myerson Dec 31 '20 at 09:34
  • "How can something true follow from a false proposition?" Not true: everything follows from a contradiction. – Mauro ALLEGRANZA Dec 31 '20 at 09:54
  • 'If' part is called antecedent, 'then' part is called consequent. – Bram28 Dec 31 '20 at 15:30
  • See https://www.quora.com/What-is-the-difference-between-material-and-logical-implication/answer/Dan-Christensen-8 – Dan Christensen Dec 31 '20 at 16:29
  • Use the principle of vacuous truth: $\neg A \implies (A \implies B)$. It may be counterintuitive to most people (non-mathematicians), but, in addition to following directly from the truth table for $A \implies B$, it can also be derived from first principles using a form of natural deduction. – Dan Christensen Dec 31 '20 at 16:46

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Your particular example confuses you because it's an "if and only if" statement. In fact, it's $x \text{ is an even number} \Leftrightarrow x \text{ is divisible by two}$, meaning that if $x$ it's not even, it's not going to be divisible by $2$; if $x$ was odd and divisible by $2$, due to the other implication, it'd be even, and that's a contradiction. This is different from, let's say, "if you finish supper, then you get dessert", because this is $\text{you finish supper} \Rightarrow \text{ you get dessert}$, and in fact you may still get dessert even if you haven't finished your supper.

Move78
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  • I know it is an "if and only if" statement, but to my understanding this just means that the "if...then..." statements $A \Rightarrow B$ and $B \Rightarrow A$ are equivalent ($A$ and $B$ being "x even", "x divisible by two"). Shouldn't the the conditional statements the "if and only if"-statement is composed of still make sense on their own? – LinusDieLinse Dec 31 '20 at 10:57
  • @LinesDieLinse they still make sense on their own. "How can $A\Rightarrow B$ be true when $A$ is false?" Because "an odd number is never going to be divisible by two" since $B \Rightarrow A$, but not because $A \Rightarrow B$. For a second, let a number be odd if it's not even, and that's all. Can you deduce that an odd number is not divisible by two only knowing that an even number is? We only know that an odd number is a number that's not even, but we can't say anything else if we don't put $B \Rightarrow A$, and only then we know that if a number is divisible by two, it's not odd. – Move78 Dec 31 '20 at 11:57