query: What would happen if $p\rightarrow q$ were defiend to be false if $p$ is false?
positive_passages.docid: 15504
positive_passages.text: In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True?
negative_passages.docid: 5035
negative_passages.text: Can unprovability unprovable? Is there an $\omega$-fold unprovability?
negative_passages.docid: 1027
negative_passages.text: Why is this true? $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$
negative_passages.docid: 52322
negative_passages.text: Implication in mathematics - How can A imply B when A is False?
negative_passages.docid: 157625
negative_passages.text: How to prove conditional in Coq?
negative_passages.docid: 45323
negative_passages.text: How to prove $\vdash\neg P\to (P\to Q)$?
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: 140194
negative_passages.text: Using natural deduction to prove that $p \implies q \vdash \lnot p \lor q$
negative_passages.docid: 60617
negative_passages.text: How compute $\lim_{p\rightarrow 0} \|f\|_p$ in a probability space?
negative_passages.docid: 46939
negative_passages.text: Natural deduction, Proof $\vdash$ $P\Rightarrow(Q\Rightarrow P)$
negative_passages.docid: 65379
negative_passages.text: Deriving $\neg R$ from $\{R↔(R∨(P∧¬P)), R↔¬P, ¬P→(P↔(Q→Q)), P→Q\}$
negative_passages.docid: 54791
negative_passages.text: Is it true or false that $\lvert P([0, 1]) \setminus P((0, 1)) \rvert > \lvert P(\mathbb{N}) \rvert$
negative_passages.docid: 9282
negative_passages.text: Prove using natural deduction $((P \land Q) \rightarrow R) \vdash (P \rightarrow R) \lor (Q\rightarrow R)$
negative_passages.docid: 96458
negative_passages.text: Does $X_n \overset{L^2}{\rightarrow} X$ imply $X_n^2 \overset{L^1}{\rightarrow} X^2$?
negative_passages.docid: 148237
negative_passages.text: Is ¬¬(¬¬P → P) provable in intuitionistic logic?
negative_passages.docid: 6119
negative_passages.text: If both $P$ and $Q$ are true , how can I tell that $P$ implies $Q$?
negative_passages.docid: 56325
negative_passages.text: If we accept a false statement, can we prove anything?