In (modal) logic can you only assign values to variables if they are free and not in a quantifier?

Question

Here is the skeleton of the question, not enough to get a full answer, but enough to help clarify my question:

Let N be the structure (N,I) with domain {0,1,…} the natural numbers and where I(≺)is the set of pairs (x, y) where x is strictly less than y

The question then provides lots of different variable assignments for x, y, z grouped as A1, A2... and asks if various statements are true.

One of which is: N⊨A1∀x∃y(x≺y)

Here A1 is the variable assignment.

In this situation, if x and y are not free, like above, are they still assigned to? Are they variables and if not what are they?

Another brief question I have is, are the natural numbers implicitly defined within first order logic or must they always be defined?

I.e. in some first order language, can I write the sentence 1>3 given that ">" is defined and 1 and 3 are not? Is this sentence then unambiguous in meaning?

E.g. 2 Take a language with constants C = {0,1}, predicates P = {=2} and functions F = {+2,−1,×2}, written prefix.

Would +(3,0) be a term in the language?

"in some first order language, can I write the sentence 1>3?" In the FOL of arithemtic, provided that you have defined the $<$ relation, because $1$ is defined as $s(0)$ (the successor of $0$) and $2=s(1)$ and so on. — Mauro ALLEGRANZA, Nov 25 '22 at 13:25
Regarding assignment, the basic result is that if a formula has no free variable (i.e. is a sentence), then for every structure $\mathfrak A$, either the formula is satisfied by every assignment (and it is true in $\mathfrak A$) or it is not satisfied at all (and it is false in $\mathfrak A$). — Mauro ALLEGRANZA, Nov 25 '22 at 13:28
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. — Community, Nov 25 '22 at 13:30
"Would +(3,0) be a term in the language?" No, because $3$ is not. — Mauro ALLEGRANZA, Nov 25 '22 at 13:31
Re te title: assignment functions assign values of all variables of the language. Thus, for e.g. interpretation in $\mathbb N$ an example of assignment $v: \text {Var} \to \mathbb N$ will be: $v(x_1)=0, v(x_2)=1, \ldots$. — Mauro ALLEGRANZA, Nov 25 '22 at 13:33
Re: Regarding assignment: So is that to say that formulae with no free variables will evaluate to truth values completely disregarding variable assignments? (In other words, variables do not get assigned values even if the interpretation says that x = 1 y = 2...)? — Michael Fan, Nov 25 '22 at 13:34
See the post Connection between interpretation, variable assignment and truth valuation — Mauro ALLEGRANZA, Nov 25 '22 at 13:37

In (modal) logic can you only assign values to variables if they are free and not in a quantifier?

0 Answers0