RESOLVED: I fundamentally just didn't understand implication. Thank you to everyone who helped!
I was going off of my linguistic understanding of "implies", which isn't the same. For example, the sentence "this [true thing] implies that [another thing] could be false" makes grammatical sense, and logically fits many scenarios, but it doesn't align with the truth table of logical implication.
For me personally, the interpretation that matches the actual truth table for implication is as follows: read X -> Y backwards, and phrase "implies" as "facilitates".
i.e. "Y facilitates X".
Y being true facilitates X to be true, but doesn't make any guarantees:
- false -> true = makes perfect sense (Y happened and X didn't; this satisfies the definition of implication; the expression is true);
- true -> true = makes perfect sense (Y happened and X did; this satisfies the definition of implication; the expression is true)).
Y being false offers no facilitation, so this prevents X being true:
- false -> false = makes perfect sense (Y didn't happen so X can't; this satisfies the definition of implication; the expression is true));
- true -> false = is illogical (Y didn't happen so X can't, but, if it still does, this violates the definition of implication, and so the original expression is false)).
I'm confused by an example of basic logic symbols on Wikipedia. The example under the arrow symbol "implies" is this:
If P and Q are logical predicates, P ⇒ Q means that if P is true, then Q is also true. Thus, P ⇒ Q is logically equivalent with Q ∨ ¬ P.
I get the general relationship being implied (in situations where P is true, Q is also true), but I'm confused at the rephrasing to Q ∨ ¬ P. As I read it, I end up with the following possibilities (a basic OR logic table):
- false OR not false [true] = true
- true OR not false [true] = true
- true OR not true [false] = true
- false OR not true [false] = false
In that case, Q can be true without P being true, and P can be true without Q being true. That would mean the original example isn't the total working of the relationship, just "incidentally, the two can be true simultaneously". Also, then, that the only thing that matters it that one is true, and thus the order of P and Q, and the presence or absence of ¬, therefore doesn't seem that important. This is all uncomfortably broad, so I've probably missed something.
Or is this just specific to this example? I.e., here, is P ⇒ Q just indicating a subset of potentialities in a wider, unexplained scenario, in such a way that (just sometimes) Q ⇒ P and P ⇔ Q can also be true, and that there is no obvious relationship (at least not at this level of understanding of the problem)?
That would seem very cosmic for a basic explanation. If that's meant to be a bit of a trick example, I'll take it as extra education to not just assume things. But, as this isn't a test – just free education – I'm much more inclined to believe I've grossly misunderstood.
Please can someone help explain this to me?
Best Answer
(Making big tables in comments is tedious...)
The logic table for $P \implies Q$ is not the same as that of $P \vee Q$ (they are not equivalent); they have the same number of trues and falses, sure, but not in the same places: $$ \begin{array}{c c | c} P & Q & P \implies Q \\ \hline \text{true} & \text{true} & \text{true} \\ \text{true} & \text{false} & \color{red}{\text{false}} \\ \text{false} & \text{true} & \text{true} \\ \text{false} & \text{false} & \text{true} \\ \end{array} \qquad \begin{array}{c c | c} P & Q & P \vee Q \\ \hline \text{true} & \text{true} & \text{true} \\ \text{true} & \text{false} & \text{true} \\ \text{false} & \text{true} & \text{true} \\ \text{false} & \text{false} & \color{red}{\text{false}} \\ \end{array}$$
The demonstrates that the presence or absence of $\neg$ in $\neg P \vee Q$ certainly matters, and so does the order of $P$ and $Q$:
$$ \begin{array}{c c | c} P & Q & Q \implies P \\ \hline \text{true} & \text{true} & \text{true} \\ \text{true} & \text{false} & \text{true} \\ \text{false} & \text{true} & \color{red}{\text{false}} \\ \text{false} & \text{false} & \text{true} \\ \end{array} $$