What is the difference between a logical operator and a logical symbol?

I was writing about my recent experience learning first-order logic, and I started to write the following sentence: "I knew about the existence of truth tables and logical...".

I wanted to say that when I started learning about FOL, I already knew about truth tables and operations like bitwise OR, AND, etc in computer science, and also of the symbols &, |, etc that can be used more generally.

So perhaps I could say "I knew of about the existence of truth tables, logical operators, and logical symbols". But I am not sure if logical operators and logical symbols are two different things. Are they?

• Your logical operators are called connectives and is a part of the logical symbols of FOL, but there're also non-logical symbols, see WP source here... Jan 28, 2022 at 7:03
• How about "I knew about the existence of truth tables, logical connectives and Boolean operations they represent". Jan 28, 2022 at 7:06
• Nothing; maybe we call the the symbol for logical operator is the "name" of the object. Jan 28, 2022 at 7:38
• Operators carry a functional meaning; symbols carry a semantic meaning. Jan 28, 2022 at 11:07

So many semiotic and ordinary speech issues hang on the distinction suggested. We usually decide on what is a symbol and what it symbolises relatively and pass from one to another fluently in a discourse. Actually, on the overall view, what is represented by something on one side takes on the role of representing something else on another side.

As far as logic is concerned, if we are to indicate the distinction briefly, we may say, in a rough formulaic fashion, that a symbol falls into the lot of syntax and is "inscribed", in contrast, what it symbolises is "described" or interpreted.

But there are some cases in which we have to be very precise. Consider the following statements:

1. Heraclitus is an ancient philosopher.
2. Heraclitus is the name of an ancient philosopher.
3. Heraclitus designates the name of an ancient philosopher.

As the standard account goes, in (1), we use the word 'Heraclitus' to talk about him himself; in (2), we mention the word 'Heraclitus' without using it to talk about the name itself, not the person himself. In (3), we ascend one level up and use the string of symbols, 'H', 'e', 'r', 'a', 'c', 'l', 'i', 't', 'u', 's', to talk about the "inscription" of the name. The usual syntactic device to express these semantic distinctions is single quotation marks:

1. Heraclitus is an ancient philosopher.
2. 'Heraclitus' is the name of an ancient philosopher.
3. ''Heraclitus'' designates the name of an ancient philosopher.

However, this device does not give proper results when the symbols denote variables, not the instances themselves. Let us consider the following clause from a recursive definition of a formal language whose symbol set consists of Latin letters, Arabic numerals, the standard propositional connectives and round brackets:

• If ϕ and ψ are formulas of , then (ϕψ) is a formula of .

ϕ and ψ are not formulas of themselves; they are metavariables denoting formulas composed in accordance with the syntactic rules of . In order to avoid clumsy notation, the usual practice is to leave such formal niceties to the reader's comprehension. Nevertheless, in some contexts, it is extremely important to be aware of and explicitly express the distinctions.

If we would employ the single-quote notation as above

'(ϕψ)'

what we would have done would be to denote just a string of symbols, the Greek letters 'ϕ, ψ', the right arrow symbol and the brackets, not to denote a formula itself. Hence, we need a different operation than simple quotation. One choice is concatenation, a basic operation over strings often denoted by '⏜'. Thus, write

• If ϕ and ψ are formulas of , then '('⏜ϕ⏜'→'⏜ψ⏜')' is a formula of .

Equivalently and almost uniformly in philosophical discourse, we employ quasi-quotation denoted by corner quotes:

Then, we substitute, for example, '(P ∧ Q)' in place of the symbol ϕ and 'R' in place of the symbol ψ, and obtain the string of symbols

((P ∧ Q) → R)

which is a well-formed formula of .

In a manner quite akin to the one discussed, we conventionally write for the Gödel code of a formula ϕ as

for we want to express that we refer to a specific natural number when we substitute a formula in place of the symbol ϕ, while neither the symbol ϕ nor the formula intended is in the language.

In ordinary usage in FOL, logical operator and logical symbol are synonymous, or perhaps we might say that the symbol represents the operator. This is because the symbols in FOL are operators in the sense that they transform from a boolean space to {T,F}, and they also transform from the space of well-formed formulae in the formal language (of e.g. FOL) to well-formed formulae. However, there are contexts, including in metalogic when constructing formulae as strings, in which an inscribed symbol may be distinguished from the operator it represents. So it would be perfectly acceptable to say that you were familiar with logical operators and logical symbols, as well as to say one or the other.