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For example, specifying the set {1, 3, 5} as the universe and assigning 0 to the variable y, for example, is an interpretation for the wff ∀x Q(x, y), where Q(x, y) means x is greater than y. ∀x Q(x, y) with that interpretation reads, for example, "Every number in the set {1, 3, 5} is greater than 0".
As can be seen from the above example, a wff becomes a proposition when it is given an interpretation.
There are, however, wffs which are always true or always false under any interpretation. Those and related concepts are discussed below.
A wff is said to be satisfiable if there exists an interpretation that makes it true, that is if there are a universe, specific predicates assigned to the predicate variables, and an assignment of values to the free variables that make the wff true.
For example, ∀x N(x), where N(x) means that x is non-negative, is satisfiable. For if the universe is the set of natural numbers, the assertion ∀x N(x) is true, because all natural numbers are non-negative. Similarly ∃x N(x) is also satisfiable.
However, ∀x [N(x) ⋀¬N(x)] is not satisfiable because it can never be true. A wff is called invalid or unsatisfiable, if there is no interpretation that makes it true.
A wff is valid if it is true for every interpretation*. For example, the wff ∀x P(x) ⋁∃x ¬P(x) is valid for any predicate name P , because ∃x ¬P(x) is the negation of ∀x P(x). However, the wff ∀x N(x) is satisfiable but not valid.
Note that a wff is not valid iff it is unsatisfiable for a valid wff is equivalent to true. Hence its negation is false.
Two wffs W1 and W2 are equivalent if and only if W1 ↔W2 is valid, that is if and only if W1 ↔W2 is true for all interpretations.
For example ∀x P(x) and ¬∃x ¬P(x) are equivalent for any predicate name P. So are ∀x [ P(x) ⋀Q(x) ] and [ ∀x P(x) ⋀∀x Q(x) ]for any predicate names P and Q .
English sentences appearing in logical reasoning can be expressed as a wff. This makes the expressions compact and precise. It thus eliminates possibilities of misinterpretation of sentences. The use of symbolic logic also makes reasoning formal and mechanical, contributing to the simplification of the reasoning and making it less prone to errors.
Transcribing English sentences into wffs is sometimes a non-trivial task. In this course we are concerned with the transcription using given predicate symbols and the universe. To transcribe a proposition stated in English using a given set of predicate symbols, first restate in English the proposition using the predicates, connectives, and quantifiers. Then replace the English phrases with the corresponding symbols.
Example: Given the sentence "Not every integer is even", the predicate "E(x)" meaning x is even, and that the universe is the set of integers, first restate it as "It is not the case that every integer is even" or "It is not the case that for every object x in the universe, x is even."
Then "it is not the case" can be represented by the connective "¬", "every object x in the universe" by "∀ x", and "x is even" by E(x).
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