query: Why is $p{\implies}q$ defined to have a truth value if $p$ is false?
positive_passages.docid: 578
positive_passages.text: In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?
negative_passages.docid: 73231
negative_passages.text: Why in implication, if premise is false then implication is true?
negative_passages.docid: 70585
negative_passages.text: Why does Assuming[x > 0, TrueQ[x > 0]] return False?
negative_passages.docid: 157625
negative_passages.text: How to prove conditional in Coq?
negative_passages.docid: 47222
negative_passages.text: Soundness of a rule of a proof system with respect to the truth tables?
negative_passages.docid: 106401
negative_passages.text: Prove: $p-mq | f(m)$ where $m$ is any integer
negative_passages.docid: 142762
negative_passages.text: how to prove that (P⊃Q)≡(¬Q⊃¬P) ( P ⊃ Q ) ≡ ( ¬ Q ⊃ ¬ P ) is disallowed in intuitionistic logic?
negative_passages.docid: 51369
negative_passages.text: Natural deduction proof of $p \lor (p\implies q)$ with propositional calculus
negative_passages.docid: 119888
negative_passages.text: Is it possible to convert $\forall$ quantifiers to $\exists$ quantifiers without negation?
negative_passages.docid: 27550
negative_passages.text: How is (p ∧ q ∧ ¬r) ∨ (p ∧ ¬q ∧ ¬r) ≡ (p ∧ ¬r) ∨ (¬q ∧ q)? Is it really distributive property?
negative_passages.docid: 31846
negative_passages.text: Does $\neg(x > y)$ imply that $y \geq x$?
negative_passages.docid: 63952
negative_passages.text: Disproving: If $(\Gamma\vdash(\phi\,\lor\psi))$, either $(\Gamma\vdash\phi)$ or $(\Gamma\vdash\psi)$ is true
negative_passages.docid: 121361
negative_passages.text: Showing that $\lnot Q \lor (\lnot Q \land R) = \lnot Q$ without a truth table
negative_passages.docid: 29031
negative_passages.text: How does "If $P$ then $Q$" have the same meaning as "$Q$ only if $P$ "?
negative_passages.docid: 116527
negative_passages.text: Validating a proof for necessary and sufficient conditions on sets A & B such that $P(A) \cup P(B) = P(A\cup B)$
negative_passages.docid: 106799
negative_passages.text: Proof Verification: Converse of Intermediate Value Theorem
negative_passages.docid: 68607
negative_passages.text: Could an argument with false Premises and a true Conclusion be logically valid?