(Note, this use of disjunction is supposed to resemble the use of the English word 'or'. Using the example above, the disjunction expresses that it is either raining outside or there is a cold front over Kansas. Thus, in the cases listed above, the disjunction of P with Q is true in all cases except case 4. We write it P ∨ Q, and it is read ' P or Q'. Disjunction resembles conjunction in that it forms a proposition out of two simpler propositions.If it is not raining outside, then P ∧ Q is false and if there is no cold-front over Kansas, then P ∧ Q is false. Where P is the proposition that it is raining outside and Q is the proposition that a cold-front is over Kansas, P ∧ Q is true when it is raining outside and there is a cold-front over Kansas. The conjunction of P and Q is true in case 1 and is false otherwise. For any two propositions, there are four possible assignments of truth values: The conjunction of P and Q is written P ∧ Q, and expresses that each are true. Conjunction is a truth-functional connective which forms a proposition out of two simpler propositions, for example, P and Q.¬¬ P always has the same truth-value as P. In the example above, ¬ P expresses that it is not raining outside, or by a more standard reading: 'It is not the case that it is raining outside.' When P is true, ¬ P is false and when P is false, ¬ P is true. ¬ P represents the negation of P, which can be thought of as the denial of P. We then define truth-functional operators, beginning with negation.This will be true ( P) if it is raining outside and false otherwise ( ¬ P). For example, let P be the proposition that it is raining outside.

All propositions require exactly one of two truth-values: true or false. Any given proposition may be represented with a letter called a 'propositional constant', analogous to representing a number by a letter in mathematics, for instance, a = 5.

Propositional logic calculator