Formal Logic/Sentential Logic/Properties of Sentential Connectives

= 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 $$\varphi\,\!$$ and $$\psi\,\!$$, there is a related tautology $$\varphi \leftrightarrow \psi\,\!$$.

Bivalence
Every formula has exactly one of two truth values.


 * $$\vDash\ \mathrm{P} \lor \lnot \mathrm{P}\,\!$$     Law of Excluded Middle
 * $$\vDash\ \lnot (\mathrm{P} \land \lnot \mathrm{P})\,\!$$     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.


 * $$\vDash\ \mathrm{P} \rightarrow \mathrm{P}\,\!$$
 * $$\vDash\ \mathrm{P} \leftrightarrow \mathrm{P}\,\!$$

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


 * $$\mathrm{P} \land \mathrm{Q}\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$\mathrm{Q} \land \mathrm{P}\,\!$$
 * $$\mathrm{P} \lor \mathrm{Q}\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$\mathrm{Q} \lor \mathrm{P}\,\!$$
 * $$\mathrm{P} \leftrightarrow \mathrm{Q}\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$\mathrm{Q} \leftrightarrow \mathrm{P}\,\!$$

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


 * $$(\mathrm{P} \land \mathrm{Q}) \land \mathrm{R}\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$\mathrm{P} \land (\mathrm{Q} \land \mathrm{R})\,\!$$
 * $$(\mathrm{P} \lor \mathrm{Q})\lor \mathrm{R}\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$\mathrm{P} \lor (\mathrm{Q} \lor \mathrm{R})\,\!$$
 * $$(\mathrm{P} \leftrightarrow \mathrm{Q})\leftrightarrow \mathrm{R}\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$\mathrm{P} \leftrightarrow (\mathrm{Q} \leftrightarrow \mathrm{R})\,\!$$

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.


 * $$\mathrm{P} \land (\mathrm{Q} \land \mathrm{R})\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$(\mathrm{P} \land \mathrm{Q}) \land (\mathrm{P} \land \mathrm{R})\,\!$$


 * $$\mathrm{P} \land (\mathrm{Q} \lor \mathrm{R})\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$(\mathrm{P} \land \mathrm{Q}) \lor (\mathrm{P} \land \mathrm{R})\,\!$$


 * $$\mathrm{P} \lor (\mathrm{Q} \land \mathrm{R})\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$(\mathrm{P} \lor \mathrm{Q}) \land (\mathrm{P} \lor \mathrm{R})\,\!$$


 * $$\mathrm{P} \lor (\mathrm{Q} \lor \mathrm{R})\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$(\mathrm{P} \lor \mathrm{Q}) \lor (\mathrm{P} \lor \mathrm{R})\,\!$$


 * $$\mathrm{P} \lor (\mathrm{Q} \rightarrow \mathrm{R})\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$(\mathrm{P} \lor \mathrm{Q}) \rightarrow (\mathrm{P} \lor \mathrm{R})\,\!$$


 * $$\mathrm{P} \lor (\mathrm{Q} \leftrightarrow \mathrm{R})\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$(\mathrm{P} \lor \mathrm{Q}) \leftrightarrow (\mathrm{P} \lor \mathrm{R})\,\!$$


 * $$\mathrm{P} \rightarrow (\mathrm{Q} \land \mathrm{R})\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$(\mathrm{P} \rightarrow \mathrm{Q}) \land (\mathrm{P} \rightarrow \mathrm{R})\,\!$$


 * $$\mathrm{P} \rightarrow (\mathrm{Q} \lor \mathrm{R})\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$(\mathrm{P} \rightarrow \mathrm{Q}) \lor (\mathrm{P} \rightarrow \mathrm{R})\,\!$$


 * $$\mathrm{P} \rightarrow (\mathrm{Q} \rightarrow \mathrm{R})\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$(\mathrm{P} \rightarrow \mathrm{Q}) \rightarrow (\mathrm{P} \rightarrow \mathrm{R})\,\!$$


 * $$\mathrm{P} \rightarrow (\mathrm{Q} \leftrightarrow \mathrm{R})\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$(\mathrm{P} \rightarrow \mathrm{Q}) \leftrightarrow (\mathrm{P} \rightarrow \mathrm{R})\,\!$$

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


 * $$\vDash\ (\mathrm{P} \land \mathrm{Q}) \land (\mathrm{Q} \land \mathrm{R}) \rightarrow \mathrm{P} \land \mathrm{R}\,\!$$
 * $$\vDash\ (\mathrm{P} \rightarrow \mathrm{Q}) \land (\mathrm{Q} \rightarrow \mathrm{R}) \rightarrow (\mathrm{P} \rightarrow \mathrm{R})\,\!$$
 * $$\vDash\ (\mathrm{P} \leftrightarrow \mathrm{Q}) \land (\mathrm{Q} \leftrightarrow \mathrm{R}) \rightarrow (\mathrm{P} \leftrightarrow \mathrm{R})\,\!$$

Conditionals
These tautologies and equivalences are mostly about conditionals.


 * $$\vDash\ \mathrm{P} \land \mathrm{Q} \rightarrow \mathrm{P} \qquad \qquad \vDash\ \mathrm{P} \land \mathrm{Q} \rightarrow \mathrm{Q}\,\!$$
 * $$\vDash\ \mathrm{P} \rightarrow \mathrm{P} \lor \mathrm{Q} \qquad \qquad \vDash\ \mathrm{Q} \rightarrow \mathrm{P} \lor \mathrm{Q}\,\!$$
 * $$\vDash\ (\mathrm{P} \rightarrow \mathrm{Q}) \lor (\mathrm{Q} \rightarrow \mathrm{R})\,\!$$
 * $$\vDash\ \lnot \mathrm{P} \rightarrow (\mathrm{P} \rightarrow \mathrm{Q})\,\!$$     Conditional addition
 * $$\vDash\ \mathrm{Q} \rightarrow (\mathrm{P} \rightarrow \mathrm{Q})\,\!$$     Conditional addition
 * $$\mathrm{P} \rightarrow \mathrm{Q}\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$\lnot\mathrm{Q} \rightarrow \lnot\mathrm{P}\,\!$$      Contraposition
 * $$\mathrm{P} \land \mathrm{Q} \rightarrow \mathrm{R}\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$\mathrm{P} \rightarrow (\mathrm{Q} \rightarrow \mathrm{R})\,\!$$      Exportation

Biconditionals
These tautologies and equivalences are mostly about biconditionals.


 * $$\vDash\ \mathrm{P} \land \mathrm{Q} \rightarrow (\mathrm{P} \leftrightarrow \mathrm{Q})\,\!$$     Biconditional addition
 * $$\vDash\ \lnot \mathrm{P} \land \lnot \mathrm{Q} \rightarrow (\mathrm{P} \leftrightarrow \mathrm{Q})\,\!$$     Biconditional addition
 * $$\vDash (\mathrm{P} \leftrightarrow \mathrm{Q}) \lor (\mathrm{P} \leftrightarrow \lnot\mathrm{Q})\,\!$$
 * $$\lnot (\mathrm{P} \leftrightarrow \mathrm{Q})\,\!$$   [[Image:EquivalenceSign.png| is equivalent to]]    $$\lnot\mathrm{P} \leftrightarrow \mathrm{Q}\,\!$$    [[Image:EquivalenceSign.png| is equivalent to]]    $$\mathrm{P} \leftrightarrow \lnot\mathrm{Q}\,\!$$

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.


 * $$\vDash\ \mathrm{P} \leftrightarrow \mathrm{P} \land \mathrm{P}\,\!$$     Idempotence for conjunction
 * $$\vDash\ \mathrm{P} \leftrightarrow \mathrm{P} \lor \mathrm{P}\,\!$$     Idempotence for disjunction
 * $$\vDash\ \mathrm{P} \rightarrow \mathrm{P} \lor \mathrm{Q}\,\!$$     Disjunctive addition
 * $$\vDash\ \mathrm{Q} \rightarrow \mathrm{P} \lor \mathrm{Q}\,\!$$     Disjunctive addition
 * $$\vDash\ \mathrm{P} \land \lnot \mathrm{P} \rightarrow \mathrm{Q}\,\!$$
 * $$\mathrm{P} \land \mathrm{Q}\,\!$$   [[Image:EquivalenceSign.png]]    $$\lnot (\lnot \mathrm{P} \lor \lnot \mathrm{Q})\,\!$$      Demorgan's Laws
 * $$\mathrm{P} \lor \mathrm{Q}\,\!$$   [[Image:EquivalenceSign.png]]    $$\lnot (\lnot \mathrm{P} \land \lnot \mathrm{Q})\,\!$$      Demorgan's Laws
 * $$\lnot(\mathrm{P} \land \mathrm{Q})\,\!$$   [[Image:EquivalenceSign.png]]    $$\lnot \mathrm{P} \lor \lnot \mathrm{Q}\,\!$$      Demorgan's Laws
 * $$\lnot(\mathrm{P} \lor \mathrm{Q})\,\!$$   [[Image:EquivalenceSign.png]]    $$\lnot \mathrm{P} \land \lnot \mathrm{Q}\,\!$$      Demorgan's Laws
 * $$\mathrm{P}\,\!$$   [[Image:EquivalenceSign.png|is equivalent to]]    $$\lnot\lnot\mathrm{P}\,\!$$      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&mdash;since they are neither tautologies nor equivalences&mdash;it takes more than a mere truth table to do so. We will not attempt the proof here.

Deduction principle
Let $$\varphi\,\!$$ and $$\psi\,\!$$ both be formulae, and let $$\Gamma\,\!$$ be a set of formulae.


 * $$\mathrm{If}\ \ \Gamma \cup \{\varphi\} \vDash \psi \mathrm{,\ then}\ \ \Gamma \vDash (\varphi \rightarrow \psi)\,\!$$

Reduction principle
Let $$\varphi\,\!$$ and $$\psi\,\!$$ both be formulae, and let $$\Gamma\,\!$$ be a set of formulae.


 * $$\mathrm{If}\ \ \Gamma \cup \{\varphi\} \vDash \psi\ \ \mathrm{and}\ \ \Gamma \cup \{\varphi\} \vDash \lnot\psi \mathrm{,\ then}\ \ \Gamma \vDash \lnot\varphi,\!$$
 * $$\mathrm{If}\ \ \Gamma \cup \{\lnot\varphi\} \vDash \psi\ \ \mathrm{and}\ \ \Gamma \cup \{\lnot\varphi\} \vDash \lnot\psi \mathrm{,\ then}\ \ \Gamma \vDash \varphi\,\!$$