Help with Metalogic Problem: Showing Sentence Truth Value

[gravenewworld]

Could someone please help with this problem, I am really stuck.

Show that if c is a constant and t a closed term having the same denotation, then substituting t for c in a sentence does not change the truth value of the sentence. I.E. Whether a sentence F(t) is true depends only on the domain, the denotations of the nonlogical symbols in F(x), and the denotation of the closed term t.

 

[honestrosewater]

What does denotation and having the same denotation mean? Does it mean they are equal? That c is the value of t?

 

[gravenewworld]

See that is one of the concepts that is fuzzy to me. Given a language L and an interpretation M, |M| is the domain of the interpretation--everything the interpretation talks about. If c is a constant in the language L then c^M is said to be the denotation of c in M. c^M is just an element of |M|. If R is a predicate in L then R^M is the denotation of R in M, i.e. it is just the relation that is specified by R on the domain.

 

[gravenewworld]

Technically t can and c can both have the same denotation but don't have to be equal. I was able to think of an example, for instance say Jack is the brother of jill and Jack is also Jill's roomate (or something else stupid like that). If one says in english "the brother of jill" or "jill's roomate" they both refer to the same person (denotation) Jack, but the relation "the brother of jill" is not the same as the constant "jill's roomate". "the brother of jill" could be substituted for "jill's roomate" into every sentence that contains "jill's roomate" and the truth value of the sentence will never change. THus t can replace every c or vice versa, without t and c being equal, and the truth value of the sentence won't change as long as t and c have the same denotation.

 

[honestrosewater]

Sorry, I had tests yesterday. I thought I could maybe skip ahead a bit in my book and figure out the answer, but FOL is too much more complicated than propositional logic- at least, if you need a step-by-step proof. I was thinking of a similar example, where, for instance, t was (2 + 2) and c was 4. If two terms having different arities (or lengths or heights, etc.) means they can't be equal, then they obviously can't be equal. However, I don't know how that could be the case since terms don't denote themselves. The differences between c and t, such as arity, length, etc. are all differences outside of the object language L, in the metalanguage. In L, c and t have the same denotation, so they should be members of all the same predicate sets. For instance, let Px mean "x is a 0-ary function". c is a 0-ary function, but there's no way to say so in L (unless c happens to denote a 0-ary function, but then t would also denote a 0-ary function). For if c denotes "Jill's roommate", then Pc is false; Jill's roommate is not a 0-ary function. So the fact that terms don't denote themselves should make (c = t) true, if my thinking is correct. Also, c should still be the value of t, and we normally say that f(x) = y (and (2 + 2 = 4)). Of course, I could be wrong. I wish I could be of more help.