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| Inference tests' results
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Double negation
| $$\frac A{{\sim}{\sim}A}$$
| Valid? Yes!
| $$\frac{{\sim}{\sim}A}A$$
| Valid? Yes!
| $$\overline{A\to{\sim}{\sim}A}$$
| Valid? Yes!
| $$\overline{{\sim}{\sim}A\to A}$$
| Valid? Yes!
| Self-negation
| $$\frac A{{\sim}A}$$
| Valid? Nope. | Here's a counter-example: A=T $$\frac{{\sim}A}A$$
| Valid? Nope. | Here's a counter-example: A=F $$\overline{A\to{\sim}A}$$
| Valid? Nope. | Here's a counter-example: A=T $$\overline{{\sim}A\to A}$$
| Valid? Nope. | Here's a counter-example: A=F Idempotency of &
| $$\frac A{A\;\&\;A}$$
| Valid? Yes!
| $$\frac{A\;\&\;A}A$$
| Valid? Yes!
| $$\overline{A\to(A\;\&\;A)}$$
| Valid? Yes!
| $$\overline{(A\;\&\;A)\to A}$$
| Valid? Yes!
| Law of noncontradiction
| $$\overline{{\sim}(A\;\&\;{\sim}A)}$$
| Valid? Nope. | Here's a counter-example: A=N $$\overline{{\sim}({\sim}A\;\&\;A)}$$
| Valid? Nope. | Here's a counter-example: A=N & intro.
| $$\frac{A\quad B}{A\;\&\;B}$$
| Valid? Yes!
| Simplification, & elim.
| $$\frac{A\;\&\;B}A$$
| Valid? Yes!
| $$\frac{A\;\&\;B}B$$
| Valid? Yes!
| $$\overline{(A\;\&\;B)\to A}$$
| Valid? Yes!
| $$\overline{(A\;\&\;B)\to B}$$
| Valid? Yes!
| Symmetry of &
| $$\frac{A\;\&\;B}{B\;\&\;A}$$
| Valid? Yes! | $$\overline{(A\;\&\;B)\to(B\;\&\;A)}$$
| Valid? Yes!
| Ex contradictione quodlibet
| $$\frac{A\quad{\sim}A}B$$
| Valid? Nope. | Here's a counter-example: A=B, B=F $$\frac{A\;\&\;{\sim}A}B$$
| Valid? Nope. | Here's a counter-example: A=B, B=F $$\frac{{\sim}A\;\&\;A}B$$
| Valid? Nope. | Here's a counter-example: A=B, B=F $$\overline{(A\;\&\;{\sim}A)\to B}$$
| Valid? Nope. | Here's a counter-example: A=B, B=F $$\overline{({\sim}A\;\&\;A)\to B}$$
| Valid? Nope. | Here's a counter-example: A=B, B=F Associativity of &
| $$\frac{(A\;\&\;B)\;\&\;C}{A\;\&\;(B\;\&\;C)}$$
| Valid? Yes! | $$\frac{A\;\&\;(B\;\&\;C)}{(A\;\&\;B)\;\&\;C}$$
| Valid? Yes!
| Idempotency of ∨
| $$\frac A{A\lor A}$$
| Valid? Yes!
| $$\frac{A\lor A}A$$
| Valid? Yes!
| $$\overline{A\to(A\lor A)}$$
| Valid? Yes!
| $$\overline{(A\lor A)\to A}$$
| Valid? Yes!
| Law of excluded middle
| $$\overline{A\lor{\sim}A}$$
| Valid? Nope. | Here's a counter-example: A=N $$\overline{{\sim}A\lor A}$$
| Valid? Nope. | Here's a counter-example: A=N $$\overline{A\to(B\lor{\sim}B)}$$
| Valid? Nope. | Here's a counter-example: A=B, B=N $$\overline{A\to({\sim}B\lor B)}$$
| Valid? Nope. | Here's a counter-example: A=B, B=N Addition, ∨ intro.
| $$\frac A{A\lor B}$$
| Valid? Yes!
| $$\frac B{A\lor B}$$
| Valid? Yes!
| $$\overline{A\to(A\lor B)}$$
| Valid? Yes! | $$\overline{B\to(A\lor B)}$$
| Valid? Yes!
| Exclusive ∨ intro.
| $$\frac{A\quad{\sim}B}{A\lor B}$$
| Valid? Yes! | $$\frac{{\sim}A\quad B}{A\lor B}$$
| Valid? Yes!
| Affirming a disjunct, exclusive ∨ elim.
| $$\frac{A\lor B\quad A}{{\sim}B}$$
| Valid? Nope. | Here's a counter-example: A=B, B=N $$\frac{A\lor B\quad B}{{\sim}A}$$
| Valid? Nope. | Here's a counter-example: A=N, B=B Dual nd ∨ elim.
| $$\frac{A\lor B}{A\quad B}$$
| Valid? Yes!
| Symmetry of ∨
| $$\frac{A\lor B}{B\lor A}$$
| Valid? Yes! | $$\overline{(A\lor B)\to(B\lor A)}$$
| Valid? Yes!
| Disjunctive syllogism
| $$\frac{A\lor B\quad{\sim}A}B$$
| Valid? Nope. | Here's a counter-example: A=B, B=F $$\frac{A\lor B\quad{\sim}B}A$$
| Valid? Nope. | Here's a counter-example: A=F, B=B Disjunctive syllogism (standard implication form)
| $$\frac{{\sim}A\lor B\quad A}B$$
| Valid? Nope. | Here's a counter-example: A=B, B=F $$\frac{A\lor{\sim}B\quad B}A$$
| Valid? Nope. | Here's a counter-example: A=F, B=B ∨ elim. (with conditionals)
| $$\frac{A\lor B\quad A\to C\quad B\to C}C$$
| Valid? Yes!
| Associativity of ∨
| $$\frac{(A\lor B)\lor C}{A\lor(B\lor C)}$$
| Valid? Yes! | $$\frac{A\lor(B\lor C)}{(A\lor B)\lor C}$$
| Valid? Yes!
| Constructive dilemma
| $$\frac{A\lor B\quad A\to C\quad B\to D}{C\lor D}$$
| Valid? Yes! | $$\frac{A\lor B\quad A\to C\quad B\to D}{D\lor C}$$
| Valid? Yes!
| Destructive dilemma
| $$\frac{A\to C\quad B\to D\quad{\sim}C\lor{\sim}D}{{\sim}A\lor{\sim}B}$$
| Valid? Nope. | Here's a counter-example: A=N, B=N, C=F, D=F $$\frac{A\to C\quad B\to D\quad{\sim}C\lor{\sim}D}{{\sim}B\lor{\sim}A}$$
| Valid? Nope. | Here's a counter-example: A=N, B=N, C=F, D=F De Morgan's laws
| $$\frac{{\sim}(A\;\&\;B)}{{\sim}A\lor{\sim}B}$$
| Valid? Yes! | $$\frac{{\sim}A\lor{\sim}B}{{\sim}(A\;\&\;B)}$$
| Valid? Yes! | $$\frac{{\sim}(A\lor B)}{{\sim}A\;\&\;{\sim}B}$$
| Valid? Yes! | $$\frac{{\sim}A\;\&\;{\sim}B}{{\sim}(A\lor B)}$$
| Valid? Yes!
| Distributivity
| $$\frac{A\;\&\;(B\lor C)}{(A\;\&\;B)\lor(A\;\&\;C)}$$
| Valid? Yes! | $$\frac{A\lor(B\;\&\;C)}{(A\lor B)\;\&\;(A\lor C)}$$
| Valid? Yes! | $$\frac{(B\lor C)\;\&\;A}{(B\;\&\;A)\lor(C\;\&\;A)}$$
| Valid? Yes! | $$\frac{(B\;\&\;C)\lor A}{(B\lor A)\;\&\;(C\lor A)}$$
| Valid? Yes! | $$\frac{(A\;\&\;B)\lor(A\;\&\;C)}{A\;\&\;(B\lor C)}$$
| Valid? Yes! | $$\frac{(A\lor B)\;\&\;(A\lor C)}{A\lor(B\;\&\;C)}$$
| Valid? Yes! | $$\frac{(B\;\&\;A)\lor(C\;\&\;A)}{(B\lor C)\;\&\;A}$$
| Valid? Yes! | $$\frac{(B\lor A)\;\&\;(C\lor A)}{(B\;\&\;C)\lor A}$$
| Valid? Yes!
| Reflexivity of →
| $$\overline{A\to A}$$
| Valid? Yes!
| Irreflexivity of →
| $$\overline{{\sim}(A\to A)}$$
| Valid? Nope. | Here's a counter-example: A=T Aristotle's theses
| $$\overline{{\sim}(A\to{\sim}A)}$$
| Valid? Yes!
| $$\overline{{\sim}({\sim}A\to A)}$$
| Valid? Yes!
| Modus ponens, → elim.
| $$\frac{A\to B\quad A}B$$
| Valid? Yes!
| Modus tollens
| $$\frac{A\to B\quad{\sim}B}{{\sim}A}$$
| Valid? Nope. | Here's a counter-example: A=N, B=F Positive paradox
| $$\overline{B\to(A\to B)}$$
| Valid? Yes!
| Vacuous truth
| $$\overline{{\sim}A\to(A\to B)}$$
| Valid? Nope. | Here's a counter-example: A=B, B=F Contrapositive
| $$\frac{{\sim}B\to{\sim}A}{A\to B}$$
| Valid? Nope. | Here's a counter-example: A=B, B=F $$\frac{A\to B}{{\sim}B\to{\sim}A}$$
| Valid? Nope. | Here's a counter-example: A=N, B=F Symmetry of →
| $$\frac{A\to B}{B\to A}$$
| Valid? Nope. | Here's a counter-example: A=F, B=B $$\overline{(A\to B)\to(B\to A)}$$
| Valid? Nope. | Here's a counter-example: A=F, B=B Conjunctive conditional
| $$\frac{A\quad B}{A\to B}$$
| Valid? Yes! | $$\frac{A\to B}A$$
| Valid? Nope. | Here's a counter-example: A=F, B=F $$\frac{A\to B}B$$
| Valid? Nope. | Here's a counter-example: A=F, B=F Standard conditional
| $$\frac{A\to B}{{\sim}A\lor B}$$
| Valid? Nope. | Here's a counter-example: A=N, B=F $$\frac{A\to B}{B\lor{\sim}A}$$
| Valid? Nope. | Here's a counter-example: A=N, B=F $$\frac{{\sim}A\lor B}{A\to B}$$
| Valid? Nope. | Here's a counter-example: A=B, B=F $$\frac{B\lor{\sim}A}{A\to B}$$
| Valid? Nope. | Here's a counter-example: A=B, B=F $$\frac{A\to B}{{\sim}(A\;\&\;{\sim}B)}$$
| Valid? Nope. | Here's a counter-example: A=N, B=F $$\frac{A\to B}{{\sim}({\sim}B\;\&\;A)}$$
| Valid? Nope. | Here's a counter-example: A=N, B=F $$\frac{{\sim}(A\;\&\;{\sim}B)}{A\to B}$$
| Valid? Nope. | Here's a counter-example: A=B, B=F $$\frac{{\sim}({\sim}B\;\&\;A)}{A\to B}$$
| Valid? Nope. | Here's a counter-example: A=B, B=F Standard conditional negation
| $$\frac{{\sim}(A\to B)}A$$
| Valid? Nope. | Here's a counter-example: A=F, B=F $$\frac{{\sim}(A\to B)}{{\sim}B}$$
| Valid? Nope. | Here's a counter-example: A=F, B=N $$\frac{{\sim}(A\to B)}{A\;\&\;{\sim}B}$$
| Valid? Nope. | Here's a counter-example: A=F, B=F $$\frac{A\;\&\;{\sim}B}{{\sim}(A\to B)}$$
| Valid? Yes! | $$\frac{A\quad{\sim}B}{{\sim}(A\to B)}$$
| Valid? Yes!
| Associativity of →
| $$\frac{(A\to B)\to C}{A\to(B\to C)}$$
| Valid? Yes! | $$\frac{A\to(B\to C)}{(A\to B)\to C}$$
| Valid? Nope. | Here's a counter-example: A=F, B=F, C=F Boethius's theses
| $$\overline{(A\to B)\to{\sim}(A\to{\sim}B)}$$
| Valid? Yes!
| $$\overline{(A\to{\sim}B)\to{\sim}(A\to B)}$$
| Valid? Yes!
| Boethius's theses (rule form)
| $$\frac{A\to B}{{\sim}(A\to{\sim}B)}$$
| Valid? Yes!
| $$\frac{A\to{\sim}B}{{\sim}(A\to B)}$$
| Valid? Yes!
| Reciprocal Boethius's theses
| $$\overline{{\sim}(A\to{\sim}B)\to(A\to B)}$$
| Valid? Yes!
| $$\overline{{\sim}(A\to B)\to(A\to{\sim}B)}$$
| Valid? Yes!
| Reciprocal Boethius's theses (rule form)
| $$\frac{{\sim}(A\to{\sim}B)}{A\to B}$$
| Valid? Yes!
| $$\frac{{\sim}(A\to B)}{A\to{\sim}B}$$
| Valid? Yes!
| Peirce's law
| $$\overline{((A\to B)\to A)\to A}$$
| Valid? Yes!
| Axiom of relativity
| $$\overline{((A\to B)\to B)\to A}$$
| Valid? Nope. | Here's a counter-example: A=F, B=B Absorption
| $$\frac{A\to B}{A\to(A\;\&\;B)}$$
| Valid? Yes! | $$\frac{A\to B}{A\to(B\;\&\;A)}$$
| Valid? Yes!
| Abelard's theses
| $$\overline{{\sim}((A\to B)\;\&\;({\sim}A\to B))}$$
| Valid? Nope. | Here's a counter-example: A=B, B=N $$\overline{{\sim}(({\sim}A\to B)\;\&\;(A\to B))}$$
| Valid? Nope. | Here's a counter-example: A=B, B=N $$\overline{{\sim}((A\to B)\;\&\;(A\to{\sim}B))}$$
| Valid? Nope. | Here's a counter-example: A=B, B=N $$\overline{{\sim}((A\to{\sim}B)\;\&\;(A\to B))}$$
| Valid? Nope. | Here's a counter-example: A=B, B=N Prefixing
| $$\frac{A\to B}{(C\to A)\to (C\to B)}$$
| Valid? Yes! | $$\overline{(A\to B)\to((C\to A)\to (C\to B))}$$
| Valid? Yes!
| Suffixing
| $$\frac{A\to B}{(B\to C)\to (A\to C)}$$
| Valid? Yes! | $$\overline{(A\to B)\to((B\to C)\to (A\to C))}$$
| Valid? Yes! | Exportation/importation (currying)
| $$\frac{A\to(B\to C)}{(A\;\&\;B)\to C}$$
| Valid? Yes! | $$\frac{(A\;\&\;B)\to C}{A\to(B\to C)}$$
| Valid? Yes! | $$\overline{(A\to(B\to C))\to((A\;\&\;B)\to C)}$$
| Valid? Yes! | $$\overline{((A\;\&\;B)\to C)\to(A\to(B\to C))}$$
| Valid? Yes!
| Hypothetical syllogism
| $$\frac{A\to B\quad B\to C}{A\to C}$$
| Valid? Yes!
| Affirming the consequent
| $$\frac{A\to B\quad B}A$$
| Valid? Nope. | Here's a counter-example: A=F, B=B Negating the antecedent
| $$\frac{A\to B\quad{\sim}A}{{\sim}B}$$
| Valid? Nope. | Here's a counter-example: A=F, B=N | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||