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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} .
For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris ...
In the same way, applying the second default we obtain that Nixon is a pacifist, thus making the first default not applicable. This particular default theory has therefore two extensions, one in which Pacifist(Nixon) is true, and one in which Pacifist(Nixon) is false. The original semantics of default logic was based on the fixed point of a ...
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:
If f is an n-ary function symbol, and t 1, ..., t n are terms, then f(t 1,...,t n) is a term. In particular, symbols denoting individual constants are nullary function symbols, and thus are terms. Only expressions which can be obtained by finitely many applications of rules 1 and 2 are terms.
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 ...
Let one bit be assigned for each truth value: 01=T and 10=F with 00=N and 11=B. [4] Then the subset relation in the power set on {T, F} corresponds to order ab<cd iff a<c and b<d in two-bit representation. Belnap calls the lattice associated with this order the "approximation lattice".