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The sun being above the horizon is a necessary condition for direct sunlight; but it is not a sufficient condition, as something else may be casting a shadow, e.g., the moon in the case of an eclipse. 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".
An example: we are given the conditional fact that if it is a bear, then it can swim. Then, all 4 possibilities in the truth table are compared to that fact. If it is a bear, then it can swim — T; If it is a bear, then it can not swim — F; If it is not a bear, then it can swim — T because it doesn’t contradict our initial fact.
material conditional (material implication) implies, if P then Q, it is not the case that P and not Q propositional logic, Boolean algebra, Heyting algebra: is false when A is true and B is false but true otherwise.
For any given interpretation, a given formula is either true or false under it. [69] [79] ... If p then q; and if p then r; therefore if p is true then q and r are true
The proposition to be proved is P. We assume P to be false, i.e., we assume ¬P. It is then shown that ¬P implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, Q and ¬Q, and appealing to the law of noncontradiction. Since assuming P to be false leads to a contradiction, it is concluded that P is ...
A function symbol, with some valence greater than or equal to 0. These are often denoted by lowercase roman letters such as f, g and h. Examples: f(x) may be interpreted as "the father of x". In arithmetic, it may stand for "-x". In set theory, it may stand for "the power set of x". In arithmetic, g(x,y) may stand for "x+y".
These examples, one from mathematics and one from natural language, illustrate the concept of vacuous truths: "For any integer x, if x > 5 then x > 3." [11] – This statement is true non-vacuously (since some integers are indeed greater than 5), but some of its implications are only vacuously true: for example, when x is the integer 2, the statement implies the vacuous truth that "if 2 > 5 ...
A form of deductive reasoning that concludes one disjunct must be false if the other is true and a disjunction is given (if and not , then ). disquotationalism A theory of truth that focuses on the disquotation principle, which suggests that the function of the truth predicate is to remove quotation marks to form equivalent sentences.