# What is a logical constant, exactly?

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 🙇

• I'm most aware with constants as a symbols in language. Formally in mathematical theories like first order theories like ZFC etc. we have a language which is a set of symbols for functions, relations and constants (ZFC is theory over language with one element, $\in$). In particular models of theory particular symbols for constant will be interpreted as particular element of a model. For instance $\mathbb{R}$ over language $\{+,c\}$ where $+$ is interpreted as addition in reals and $c$ is interpreted as $0$, we get that this structure fulfill $\phi(c):=\forall x \; x+c=0$ Jun 21, 2023 at 19:25
• Are you looking for an explanation on criteria ("if I encounter something I have never seen ebfore, how could I decide if it is a logical constant or not?") ir are you looking for the definitive list of things that are considered logical constants? Jun 21, 2023 at 19:55
• @Z.A.K, Ah, I'd be much more interested in criteria, since then I'd have the tools to check if anything I'm working with is a logical constant, and second to that, sets of things whose members are logical constants. Jun 21, 2023 at 20:09
• Jun 22, 2023 at 12:06

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.

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.

• Hey Z.A.K, thank you very much for taking the time to explain :) I think I get it now. So logical constants can be a bunch of stuff to different people that doesn't really matter to my studies of logic, other than what is already virtually universally agreed upon, that being pretty much anything constituting an instance of a given propositional/argument form, namely: The propositional constants, and the connectives that bind them? The connectives in a propositional/argument form also stay as logical constants, though the variables themselves are not, since they aren't, well, constant (varied)? Jun 21, 2023 at 20:41
• > logical constants can be a bunch of stuff to different people that doesn't really matter to my studies of logic, other than what is already virtually universally agreed upon, < Yes, that's exactly right! (Not that I mean to discourage you from studying what philosophers wrote on the subject if you find that interesting, but it pretty much won't come up in mathematical logic courses.) Jun 21, 2023 at 20:52
• Oh OK thank you ^.^ May 1000 quokkas guide you in your dreams 🙇 Jun 21, 2023 at 20:59