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The truth table of $p\implies q$ is as follows:

$$\begin{array}{c|c|c} \boldsymbol{p} & \boldsymbol{q} & \boldsymbol{p\implies q} \\ \hline \color{green}{\text{T}} & \color{green}{\text{T}} & \color{green}{\text{T}} \\ \hline \color{green}{\text{T}} & \color{red}{\text{F}} & \color{red}{\text{F}} \\ \hline \color{red}{\text{F}} & \color{green}{\text{T}} & \color{green}{\text{T}} \\ \hline \color{red}{\text{F}} & \color{red}{\text{F}} & \color{green}{\text{T}} \end{array}$$

I know it's just a definition, but I'd like to know the motivation behind it. Here's what I think:

Strictly speaking, the meaning of the proposition $p\implies q$ can be interpreted as if $\boldsymbol p$ is true, then $\boldsymbol q$ must be also true. This means that there's no case in which $\boldsymbol p$ is true and $\boldsymbol q$ is false. So if $p$ is true and $q$ is false, then $p\implies q$ becomes false and in other cases, it is true.

Am I right?

  • Yes, that is how I think of this truth table too. When you state an implication, you're only lying if the conclusion is false while the premise is true. – Sambo Mar 03 '24 at 16:37
  • In classical propositional logic, when we say, "If it is raining, then it is cloudy," we don't mean rain causes cloudiness, or that it is always cloudy when it is raining (e.g. sunshowers). We mean only that, at present, it is not both raining and not cloudy. $R \implies C ~~\equiv ~~ \neg (R \land \neg C)$. If you are familiar with the basic methods of proof, you can find a formal justification of this "definition" starting from what might be called "first principles" at https://dcproof.wordpress.com/2017/12/28/if-pigs-could-fly/https://dcproof.wordpress.com/2017/12/28/if-pigs-could-fly/ – Dan Christensen Mar 03 '24 at 18:32
  • Yes. Also, related to what you are saying- a big part of why → is important is because of Modus Ponens. Which, says that Given P and P → Q, we can deduce Q – Michael Carey Mar 03 '24 at 20:19
  • @MichaelCarey I call it Detachment. It is one of those "first principles" from which the above "definition" can be derived. See https://www.dcproof.com/DeriveImplies.html Each of the rules of inference used there can be thought of as a kind of first principle. – Dan Christensen Mar 03 '24 at 22:32

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