What is a logical constant, exactly?

Question

I've done some background research, reading up on several sources, and each seem to give examples of what logical constants are, but none seem to give a comprehensive list and/or set of criteria for what a logical constant is (except for Wikipedia, but I don't possess the necessary PhD in philosophy to understand what it's saying).

From what I've gathered thus far, I've come to understand that everything here is a logical constant:

~(p * q) v r

~(A * B) v C

Where the first line represents an instance of the logical form below it. What isn't a logical form, I believe, are actual propositions, as well as truth values "true" and "false".

So basically, in sum, my belief based of what I read is that any propositional constants, such as A,B, and C, variables, such as p, q, and r, and any of the connectives such as v, ->, etc, are logical constants, but everything else is not? To put another way, anything that would be used in a formula, with a formula encapsulating both propositional forms and instances of those forms?

Thank you

Answer 1

To avoid confusion, it's worth noting that logical constants are not related to the term constant symbol used in mathematical logic when discussing first-order languages, much less to terms such as constant function used in analysis.

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The term logical constant is a technical term of philosophy, and one that comes up very rarely (if ever) in non-philosophical discourse in mathematical logic.

To explain the meaning of the term, we have to talk briefly about formal logic. As the name suggests, formal logic concerns itself with arguments that are valid due to their form or shape. Consider the following syllogism:

1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.

This argument is formally valid: every argument of the same form would be valid. The following argument has the same form:

1. All quokkas are immortal.
2. Daffy Duck is a quokka.
3. Therefore, Daffy Duck is immortal.

How do we describe the form or shape of the argument? We describe it by delineating the variables (the things we're allowed to change, i.e. to vary, while retaining the same form) and the things we're not allowed to vary, which we then call the logical constants.

In the arguments above, Socrates, Daffy Duck, quokka, man, mortal, immortal are all variables: as long as we substitute them consistently (replacing each instance of one word with the same new word), the form or shape of the argument would remain unchanged. Using abstract variables, we can write down the shape of the argument above as:

1. All $X$ are $Y$.
2. $Z$ is an $X$.
3. Therefore, $Z$ is $Y$.

Here $X,Y,Z$ are variables, and all other things in the argument are logical constants: we don't allow them to vary. E.g. if you were to replace the word "all" with the word "some", we'd consider that an argument that has a different form (and indeed that argument would not be formally valid).

This is the basic idea behind logical constants. As one would expect, philosophers of logic wrote many tomes about criteria for which words or structures may be admissible as logical constants. That's the complicated stuff described on Wikipedia and in the SEP articles on logical constants. There are competing criteria: while there is consensus about the acceptability of words like "every", "all" and "some" as logical constants, opinions differ about other words, such as "necessarily". If you want to argue about things you haven't seen before, you have to study the competing characterizations offered by the various philosophers.

Fortunately, the logical constants you'd encounter in a mathematical logic course are only the sign for negation ($\neg$), conjunction ($\wedge$), disjunction ($\vee$), implication (usually $\rightarrow$), and the two first-order quantifiers (universal $\forall$ , existential $\exists$). Philosophers of logic all agree that these are logical constants.