Formal Logic/Sentential Logic/Properties of Sentential Connectives

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← Expressibility ↑ Sentential Logic Substitution and Interchange →



Properties of Sentential Connectives[edit]

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[edit]

Every formula has exactly one of two truth values.

     Law of Excluded Middle
     Law of Non-Contradiction

Analogues to arithmetic laws[edit]

Some familiar laws from arithmetic have analogues in sentential logic.

Reflexivity[edit]

Conditional and biconditional (but not conjunction and disjunction) are reflexive.

Commutativity[edit]

Conjunction, disjunction, and biconditional (but not conditional) are commutative.

   is equivalent to   
   is equivalent to   
   is equivalent to   

Associativity[edit]

Conjunction, disjunction, and biconditional (but not conditional) are associative.

   is equivalent to   
   is equivalent to   
   is equivalent to   

Distribution[edit]

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[edit]

Conjunction, conditional, and biconditional (but not disjunction) are transitive.

Other tautologies and equivalences[edit]

Conditionals[edit]

These tautologies and equivalences are mostly about conditionals.

     Conditional addition
     Conditional addition
   is equivalent to         Contraposition
   is equivalent to         Exportation

Biconditionals[edit]

These tautologies and equivalences are mostly about biconditionals.

     Biconditional addition
     Biconditional addition
   is equivalent to       is equivalent to   

Miscellaneous[edit]

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
   EquivalenceSign.png         Demorgan's Laws
   EquivalenceSign.png         Demorgan's Laws
   EquivalenceSign.png         Demorgan's Laws
   EquivalenceSign.png         Demorgan's Laws
   is equivalent to         Double Negation

Deduction and reduction principles[edit]

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[edit]

Let and both be formulae, and let be a set of formulae.

Reduction principle[edit]

Let and both be formulae, and let be a set of formulae.


← Expressibility ↑ Sentential Logic Substitution and Interchange →