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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).
Then if is true, that rules out the first disjunct, so we have . In short, P → Q {\displaystyle P\to Q} . [ 3 ] However, if P {\displaystyle P} is false, then this entailment fails, because the first disjunct ¬ P {\displaystyle \neg P} is true, which puts no constraint on the second disjunct Q {\displaystyle Q} .
One cannot prove in K that if "p is necessary" then p is true. The axiom T remedies this defect: T, Reflexivity Axiom: p → p (If p is necessary, then p is the case.) T holds in most but not all modal logics. Zeman (1973) describes a few exceptions, such as S1 0. Other well-known elementary axioms are: 4:
A n+1-ary predicate P represents an n-ary function f if, and only if, it is the case that: ,, …, is true if, and only if, (,, …,) =. Similarly, a unary predicate P represents a set S if, and only if, it is the case that: Px is true if, and only if, x is a member of S .
If P is an n-ary predicate symbol and t 1, ..., t n are terms then P(t 1,...,t n) is a formula. Equality. If the equality symbol is considered part of logic, and t 1 and t 2 are terms, then t 1 = t 2 is a formula. Negation. If is a formula, then is a formula. Binary connectives.
In thermodynamics, the phase rule is a general principle governing multi-component, multi-phase systems in thermodynamic equilibrium.For a system without chemical reactions, it relates the number of freely varying intensive properties (F) to the number of components (C), the number of phases (P), and number of ways of performing work on the system (N): [1] [2] [3]: 123–125
Belnap's logic is designed to cope with multiple information sources such that if only true is found then true is assigned, if only false is found then false is assigned, if some sources say true and others say false then both is assigned, and if no information is given by any information source then neither is assigned.
OR := λp.λq.p p q NOT := λp.p FALSE TRUE IFTHENELSE := λp.λa.λb.p a b. We are now able to compute some logic functions, for example: AND TRUE FALSE ≡ (λp.λq.p q p) TRUE FALSE → β TRUE FALSE TRUE ≡ (λx.λy.x) FALSE TRUE → β FALSE. and we see that AND TRUE FALSE is equivalent to FALSE. A predicate is a function that returns a ...