Formal Logic/Sentential Logic/Properties of Sentential Connectives

← Expressibility ↑ Sentential Logic Substitution and Interchange →



Properties of Sentential ConnectivesEdit

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  .

BivalenceEdit

Every formula has exactly one of two truth values.

       Law of Excluded Middle
       Law of Non-Contradiction

Analogues to arithmetic lawsEdit

Some familiar laws from arithmetic have analogues in sentential logic.

ReflexivityEdit

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

 
 

CommutativityEdit

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

           
           
           

AssociativityEdit

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

           
           
           

DistributionEdit

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.

           
           


           
           
           
           


           
           
           
           

TransitivityEdit

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

 
 
 

Other tautologies and equivalencesEdit

ConditionalsEdit

These tautologies and equivalences are mostly about conditionals.

 
 
 
       Conditional addition
       Conditional addition
                 Contraposition
                 Exportation

BiconditionalsEdit

These tautologies and equivalences are mostly about biconditionals.

       Biconditional addition
       Biconditional addition
 
                     

MiscellaneousEdit

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
 
                 Demorgan's Laws
                 Demorgan's Laws
                 Demorgan's Laws
                 Demorgan's Laws
                 Double Negation

Deduction and reduction principlesEdit

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 principleEdit

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

 

Reduction principleEdit

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

 
 


← Expressibility ↑ Sentential Logic Substitution and Interchange →