Search results
Results from the WOW.Com Content Network
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).
The symbol for material implication signifies the proposition as a hypothetical, or the "if–then" form, e.g. "if P, then Q". The biconditional statement of the rule of transposition (↔) refers to the relation between hypothetical (→) propositions , with each proposition including an antecedent and consequential term.
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.
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 ...
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 ...
In predicate logic, a predicate P over some domain is called decidable if for every x in the domain, either P(x) holds, or the negation of P(x) holds. This is not trivially true constructively. Markov's principle then states: For a decidable predicate P over the natural numbers, if P cannot be false for all natural numbers n, then it is true ...
Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows. The objects are the elements of G, and for any two elements x and y of G, there exists a unique morphism from x to y if and only if.