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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".
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)).
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 ...
If x is true, then the result of expression x → y is taken to be that of y (e.g. if x is true and y is false, then x → y is also false). But if x is false, then the value of y can be ignored; however, the operation must return some Boolean value and there are only two choices. So by definition, x → y is true when x is false (relevance ...
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.
The ratio scale then segments by orders of magnitude used in science and technology, expressed in various units of measurement. An early expression of ratio scale was articulated geometrically by Eudoxus : "it was ... in geometrical language that the general theory of proportion of Eudoxus was developed, which is equivalent to a theory of ...
Implication alone is not functionally complete as a logical operator because one cannot form all other two-valued truth functions from it.. For example, the two-place truth function that always returns false is not definable from → and arbitrary propositional variables: any formula constructed from → and propositional variables must receive the value true when all of its variables are ...
Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true. The truth table of p EQ q (also written as p = q, p ↔ q, Epq, p ≡ q, or p == q) is as follows: The Venn diagram of A EQ B (red part is true)