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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).
In ordinary language terms, if both p and q are true, then the conjunction p ∧ q is true. For all other assignments of logical values to p and to q the conjunction p ∧ q is false. It can also be said that if p, then p ∧ q is q, otherwise p ∧ q is p.
The predicate calculus goes a step further than the propositional calculus to an "analysis of the inner structure of propositions" [4] It breaks a simple sentence down into two parts (i) its subject (the object (singular or plural) of discourse) and (ii) a predicate (a verb or possibly verb-clause that asserts a quality or attribute of the object(s)).
Let p = Pr(X > Y), and then test the null hypothesis H 0: p = 0.50. In other words, the null hypothesis states that given a random pair of measurements (x i, y i), then x i and y i are equally likely to be larger than the other. To test the null hypothesis, independent pairs of sample data are collected from the populations {(x 1, y 1), (x 2, y ...
from (if p and q are true then r is true) we can prove (if q is true then r is true, if p is true) Importation (()) (()) [38] If p then (if q then r) is equivalent to if p and q then r: Idempotence of disjunction () [107]
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 ...
Equivalently, if P is true or Q is true and P is false, then Q is true. The name "disjunctive syllogism" derives from its being a syllogism, a three-step argument, and the use of a logical disjunction (any "or" statement.) For example, "P or Q" is a disjunction, where P and Q are called the statement's disjuncts.
[7] [8] [9] affirming the consequent A logical fallacy in which a conditional statement is incorrectly used to infer its converse. For example, from "If P then Q" and "Q", concluding "P". alethic modal logic A type of modal logic that deals with modalities of truth, such as necessity and possibility. ambiguity