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  2. Propositional calculus - Wikipedia

    en.wikipedia.org/wiki/Propositional_calculus

    The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic, [1] in which formulas are interpreted as having precisely one of two possible truth values, the truth value of true or the truth value of false. [19]

  3. Propositional function - Wikipedia

    en.wikipedia.org/wiki/Propositional_function

    However, when a value is assigned to x, such as lava, the function then has the value true; while one assigns to x a value like ice, the function then has the value false. Propositional functions are useful in set theory for the formation of sets. For example, in 1903 Bertrand Russell wrote in The Principles of Mathematics (page 106):

  4. Necessity and sufficiency - Wikipedia

    en.wikipedia.org/wiki/Necessity_and_sufficiency

    The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". [9] [1] By contraposition, this is the same thing as "whenever P is true, so is Q". The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q).

  5. List of logic symbols - Wikipedia

    en.wikipedia.org/wiki/List_of_logic_symbols

    is false when A is true and B is false but true otherwise. → {\displaystyle \rightarrow } may mean the same as ⇒ {\displaystyle \Rightarrow } (the symbol may also indicate the domain and codomain of a function ; see table of mathematical symbols ).

  6. Implicational propositional calculus - Wikipedia

    en.wikipedia.org/wiki/Implicational...

    However, if one adds a nullary connective ⊥ for falsity, then one can define all other truth functions. Formulas over the resulting set of connectives {→, ⊥} are called f-implicational. [1] If P and Q are propositions, then: ¬P is equivalent to P → ⊥; P ∧ Q is equivalent to (P → (Q → ⊥)) → ⊥; P ∨ Q is equivalent to (P ...

  7. Material implication (rule of inference) - Wikipedia

    en.wikipedia.org/wiki/Material_implication_(rule...

    Then if is true, that rules out the first disjunct, so we have . In short, P → Q {\displaystyle P\to Q} . [ 3 ] However, if P {\displaystyle P} is false, then this entailment fails, because the first disjunct ¬ P {\displaystyle \neg P} is true, which puts no constraint on the second disjunct Q {\displaystyle Q} .

  8. Truth table - Wikipedia

    en.wikipedia.org/wiki/Truth_table

    Thus, the function f itself can be listed as: f = {((0, 0), f 0), ((0, 1), f 1), ((1, 0), f 2), ((1, 1), f 3)}, where f 0, f 1, f 2, and f 3 are each Boolean, 0 or 1, values as members of the codomain {0, 1}, as the outputs corresponding to the member of the domain, respectively. Rather than a list (set) given above, the truth table then ...

  9. Truth function - Wikipedia

    en.wikipedia.org/wiki/Truth_function

    In logic, a truth function [1] is a function that accepts truth values as input and produces a unique truth value as output. In other words: the input and output of a truth function are all truth values; a truth function will always output exactly one truth value, and inputting the same truth value(s) will always output the same truth value.