Formal Logic/Sentential Logic/Properties of Sentential Connectives

Sentential Logic Expressibility Substitution and Interchange Properties of Sentential Connectives Here we list some of the more famous, historically important, or otherwise useful equivalences and tautologies. They can be added to the ones listed in Interdefinability of connectives. We can go on at quite some length here, but will try to keep the list somewhat restrained. Remember that for every equivalence of and , there is a related tautology . Bivalence Every formula has exactly one of two truth values.      Law of Excluded Middle      Law of Non-Contradiction Analogues to arithmetic laws Some familiar laws from arithmetic have analogues in sentential logic. Reflexivity Conditional and biconditional (but not conjunction and disjunction) are reflexive. Commutativity Conjunction, disjunction, and biconditional (but not conditional) are commutative.    is equivalent to       is equivalent to       is equivalent to    Associativity Conjunction, disjunction, and biconditional (but not conditional) are associative.    is equivalent to       is equivalent to       is equivalent to    Distribution We list ten distribution laws. Of these, probably the most important are that conjunction and disjunction distribute over each other and that conditional distributes over itself.    is equivalent to       is equivalent to       is equivalent to       is equivalent to       is equivalent to       is equivalent to       is equivalent to       is equivalent to       is equivalent to       is equivalent to    Transitivity Conjunction, conditional, and biconditional (but not disjunction) are transitive. Other tautologies and equivalences Conditionals These tautologies and equivalences are mostly about conditionals.      Conditional addition      Conditional addition    is equivalent to         Contraposition    is equivalent to         Exportation Biconditionals These tautologies and equivalences are mostly about biconditionals.      Biconditional addition      Biconditional addition    is equivalent to       is equivalent to    Miscellaneous We repeat DeMorgan's Laws from the Interdefinability of connectives section of Expressibility and add two additional forms. We also list some additional tautologies and equivalences.      Idempotence for conjunction      Idempotence for disjunction      Disjunctive addition      Disjunctive addition    Image:EquivalenceSign.png         Demorgan's Laws    Image:EquivalenceSign.png         Demorgan's Laws    Image:EquivalenceSign.png         Demorgan's Laws    Image:EquivalenceSign.png         Demorgan's Laws    is equivalent to         Double Negation Deduction and reduction principles The following two principles will be used in constructing our derivation system on a later page. They can easily be proven, but—since they are neither tautologies nor equivalences—it takes more than a mere truth table to do so. We will not attempt the proof here. Deduction principle Let and both be formulae, and let be a set of formulae. Reduction principle Let and both be formulae, and let be a set of formulae. Sentential Logic Expressibility Substitution and Interchange