T7.1 2 of 5 Interpretations We need to give a more precise statement of the semantics for PL and one which will work for all sentences of the language. (Not just for the several sentences considered on the last page!) First notice that we define truth An interpretation for PL is an assignment of meaning which specifies a) a universe of discourse, b) the members of the universe of discourse to which any one-place predicates apply, c) the relationships between members of the universe of discourse to which any 2 or more-place predicates apply, d) the truth value of any 0-place predicate letters and e) the objects named by any individual constants. Our example interpretation was: universe of discourse = US presidents of the 18th century (George Washington and John Adams), names 'g' and 'j' for these two individuals, and 'M' applies to each of the two presidents, 'F' applies to just Washington, 'S' only to Adams, and 'B' relates Washington as president prior to Adams. We usually give an interpretations a little less formally, e.g., writing "Mx: x is male", just as we've given a symbolization key. But, either way, an interpretation associates objects with parts of the language. Semantics Restated To determine whether or not a sentence of PL is true, we need to look
at its structure. If it's an As usual, if we have a sentence with main logical operator '~','&','v','>', or '=', then we can just apply the old truth table definition to ascertain its truth value. For example, 'Mg>Sg' has what truth value? True... False... We have seen on the last page that the semantics
for quantifiers is almost as trivial. Still, it's stating the semantics
'(^x)P' is true if and only if WHAT??? Roughly, we want to say that it's true when Instead to think about the truth value of '(^x)P'
we need to think about substitution instances of P.
might want to say that '(^x)P'
is true if and only if each and every substitution instance P(a/x)
is true.This proposal will work just fine for the 18th century presidents example given above. Let P = 'Mx'. Then there are only two substitution instances, 'Mg' and 'Mj'. And '(^x)Mx' means just that these two are true. But there is a problem with our proposal. An interpretation may fail
to include a name for each member of the universe of discourse. For example, let's expand
our vision. Instead, we need to be prepared to In general, a quantifier's semantics—something
stated in terms of substitution instances—needs to be defined in
the presence of names for '(^x)P'
is true on an interpretation I if and only if P(a/x)
is true on I for all names a (where names
for all members of the universe of discourse are Similarly, '(%x)P'
is true on an interpretation I if and only if P(a/x)
is true on I for (Of course, when adding names, we leave the rest of the given interpretation unchanged.) Now, here's the interpretation but with universe of discourse: g: George Washington j: John Adams Mx: x is Male Fx: x is the first US president. Sx: x is the second US president. Bxy: x was president before y. Which of the following are true
given this interpretation with universe of discourse of (Hint: You may add names for all US presidents, say 'p |