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Jun 28, 2011 · 4) Likewise, when p is false and q is false, (p⇒q) then checks as true, meaning that it's a valid statement. q is at still at least as true as p; it's not less true. We still haven't introduced a false conclusion starting with true premises. (P =>Q) may be considered true when all we know is that P is false or Q is true.

Oct 8, 2011 · P implies Q means that Q is true when ever P is; it does not mean in addition that Q is false when ever P is...otherwise as a net result Q will equate to P..... No. The sentiment here is causation and in that : P is a sufficient condition for Q and there may be other as well.

Aug 7, 2021 · The p implies q statement is often described in various ways including: (1) if p then q (i.e. whenever p is true, q is true) (2) p only if q (i.e. whenever q is false, p is false) I see the truth t...

Jan 14, 2016 · Start with the given statement, $$ [p \land (p \rightarrow q)] \rightarrow q.$$ As you noticed, from the first logical equivalence in Table 7, you can replace the part in the round brackets to get the equivalent statement

Jan 19, 2020 · Given the premises p→q and ¬p→¬q, prove that p is logically equivalent to q. I understand why this works, but I do not know how to construct a complete formal proof. So far, I have this: premises: p→q ¬p→¬q (p→q) ∧ (¬p→¬q) ⇔ T apply contrapositive law (p→q) ∧ (q→p) ⇔ T ∴ (p↔q) ⇔ T

Mar 7, 2016 · To finish proving the equivalency $ P \to Q \equiv \neg P \lor Q ~$ we also need to show $ (\neg P \lor Q) \vdash (\neg P \lor Q) $. I don't see an obvious inference rule at this point, but we could show it by contradiction.

So, (P→Q) holds true. Suppose Q true. Then Q holds true. So, we can infer (Q→Q) holds true. Since (P∨Q), (P→Q), and (Q→Q) hold true, it follows that Q holds true. Since [¬P∧(P∨Q)] comes as the only assumption still in place, we may infer {[¬P∧(P∨Q)]→Q} as true as desired.

$\begingroup$ Basically, I feel like the truth value of an if-then statement is partially independent of the truth values of P and Q. They cannot determine the truth value of if P then Q on their own, except on row two, because if P is true and Q is false, of course P cannot imply Q.

Dec 7, 2019 · M: The truth table of p→q says that if p is false and then p→q is true. The author of the books goes on to clarify the doubt people have about the above statement, he uses the following scenario to explain why M is true, "Perhaps you are bothered by the fact that P ⇒ Q is true in the last two lines of this table. Here’s an example that ...

The meaning of something like (p->q) isn't so much that p implies q in the sense of some sort of connection between p and q, but rather that it is not the case that p stands as true and q stands as false simultaneously. Truth tables don't list probabilities. They tell you what logical value of the output IS on the basis of the input(s).