# What context do quantifiers make valid expressions?

Say I have a formal language such that x is an individual constant and symbolically has a particular value (say 2) a proposition such as x+1=3 already has the value of true, and I cannot define a binding operation such as a quantifier on to x like ∀xP(x) as this is a badly formed sentence and has no truth value.

Say I have a free variable x in my formal language, I can in the context where x can change to values in the domain of reals define ∀xP(x) as either being true or false.

If I were to have x as a varying value and wanted to look at (substitute) x=2 for example would it still be wrong for a quantified expression to have a truth value when x=2? Would it still be wrongly formed sentence (in the case of trying to quantify over a constant)? as x is still free to vary from values other than 2 more generally so could it have a truth value when x is two but still able to vary, or does it act as if x is a constant and I can only talk about quantification in the more free case where x has no particular value of interest e.g., when I haven't defined 'when x=....'.

• Not very clear... In formula x+1=3 you can "instantiate" free variable x with value 2 and the result is a true arithmetical sentence. In formula ∀x(x+1=3) there is no free variable: the expression is a sentence that has a definite truth value in a suitable interpretation: for example, in the domain of natural numbers it is false. Mar 23, 2022 at 11:56
• Having said that, what do you mean with "valid expressions"? A syntactically correct expression? This must be specified by the syntax of the language. Mar 23, 2022 at 12:02
• Can you continue the example? What would you write down when you "want to look at x=2"? Mar 23, 2022 at 12:33
• For more on this topic, you can see textbooks: van Dalen, Simpson, Chiswell & Hodges. Mar 23, 2022 at 12:36
• There is language made of symbols and there is a world made of objects: words and sentences (expressions=strings of symbols that are syntactically correct=well-formed) speaks of objects and facts of the world. The same for a formal language: 2+1=3 is a sentence of the formal language of arithmetic that expresses a true fact about the "world of number" while ∀x(x+1=3) is a false sentence (about the same arithmetical world. We cannot substitute variable x into ∀x(x+1=3) with the constant 2 (the name for the number two) because x is not free in the formula. Mar 23, 2022 at 16:06

Say I have a formal language such that x is an individual constant and symbolically has a particular value (say 2) a proposition such as x+1=3 already has the value of true, and I cannot define a binding operation such as a quantifier on to x like ∀xP(x) as this is a badly formed sentence and has no truth value.

Then the symbols "x" and "2" are synonymous. "x" is just another way of writing "2", and the statement "x = 2" is a tautology (like "2 = 2" or "x = x"). This is not the usual convention in mathematics, but if you want to define your syntax in that way, you can do so.

Say I have a free variable x in my formal language, I can in the context where x can change to values in the domain of reals define ∀xP(x) as either being true or false.

Earlier, you told us that "x" means "2" at the level of syntax. Now you are telling us that "x" is a variable. This means that "x" has two different meanings, and the sentence "x + 1 = 3" is syntactically ambiguous - you could mean "x, the variable, plus one is three" or you could mean "two, the number, plus one is three". Normally, we try to avoid this in formal languages, because the whole point of using formal languages is to eliminate syntactic ambiguity. As a result, it is going to be very difficult to engage in formal reasoning with such an ambiguous language.

This is entirely different from the semantic notion of writing something like "let x = 2." In that context, "x" is still syntactically a variable, but we are describing the interpretation of the sentence in which the variable x takes on the value two. The variable x is a different object from the symbol "x," because the former is a semantic object, and the latter a syntactic object. This distinction between syntax and semantics is, I believe, the point of confusion in your question. Substitution is a syntactic transformation which converts a semantic interpretation back into "pure" syntax. Importantly, substitution preserves the semantic meaning of a sentence, under a given interpretation, but does not preserve its syntax, nor its semantic meaning under other interpretations.

Similarly, we must distinguish between the "∀" symbol (a syntactic object) and the universal quantifier (a semantic object). The quantifier symbol "∀" binds to a variable symbol such as "x," and the (semantic) universal quantifier binds to a (semantic) variable such as x. Although the "∀" symbol can be described in purely syntactic terms (by appealing to substitution), at a semantic level, it must be understood in terms of semantic interpretations. Specifically, it states that the formula evaluates to true under every interpretation of x (possibly subject to other requirements, such as the axioms of the surrounding theory).

• Thank you very much for this response, If I am doing normal Mathematics, and I have what would be a constant 'a' in the context of my problem and have it with a value of '2', would that be similar to having x be another name for two as in a formal language? Mar 30, 2022 at 9:36
• I guess my issue, is differing from the 'variable at a value in our model' vs a 'constant'. Mar 30, 2022 at 9:36
• @user1007028: Mathematicians do that sort of thing from time to time, when they introduce symbols like π or e, and physicists when they introduce symbols like c. Ultimately, it's not really that significant whether you think of it as a syntactic constant or a semantic constant, but you do need to be consistent about this. If it's a syntactic constant, then it can't appear bound to a quantifier, because it's just another way of writing a number. If it's a semantic constant, then quantifying over it is not ungrammatical, but it might not make logical sense. Mar 30, 2022 at 15:14
• What would be an example of a syntactic constant and semantic constant? For example the speed of light in a vacuum (c) or perhaps I have a function f(x)=ax and I have decided a=2 from now on, implying ax=2x? Obviously as Mathematicians we will use a symbol 'a' that has no fixed value and call it a 'constant' I would understand that this usage of the word is not correct in Logic? Mar 31, 2022 at 9:48