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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).
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. [1]
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)).
Some of these connectives may be defined in terms of others: for instance, implication, p → q, may be defined in terms of disjunction and negation, as ¬p ∨ q; [75] and disjunction may be defined in terms of negation and conjunction, as ¬(¬p ∧ ¬q). [51]
For example, in ∀x ∀y (P(x) → Q(x,f(x),z)), x and y occur only bound, [19] z occurs only free, and w is neither because it does not occur in the formula. Free and bound variables of a formula need not be disjoint sets: in the formula P ( x ) → ∀ x Q ( x ) , the first occurrence of x , as argument of P , is free while the second one ...
It may be defined either by appending one of the two equivalent axioms (¬q → p) → (((p → q) → p) → p) or equivalently p∨(¬q)∨(p → q) to the axioms of intuitionistic logic, or by explicit truth tables for its operations. In particular, conjunction and disjunction are the same as for Kleene's and Łukasiewicz's logic, while the ...
It is therefore often useful to consider these two square root functions as a single function that has two values for positive x, one value for 0 and no value for negative x. In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general.
The form of a modus ponens argument is a mixed hypothetical syllogism, with two premises and a conclusion: If P, then Q. P. Therefore, Q. The first premise is a conditional ("if–then") claim, namely that P implies Q. The second premise is an assertion that P, the antecedent of the conditional claim, is the case.