What is the difference between value and constant literal in first order logic syntax?

Question

In FOL, there are two values true and false;

In FOL, there are two constant literals : T and F;

Is there any difference between the both (values and constant literals)?

Can I say that true value is same as constant literal T and false value as F?

Answer 1

The constant literals $\text T$ and $\text F$ are symbols of the language (also used : $\top$ and $\bot$).

The truth-values TRUE and FALSE are "objects" : they are e.g. the only two objects used in the domain of interpretation (the "boolean world" of the classical propositional calculus.

Obviously, the constant literals are constant; i.e. interpretations will always assign to the syntactical objects $\text T, \text F$ the truth-values TRUE and FALSE respectively.

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Usually, the two symbols above are intrduced in propositional calculus.

$\bot$ can be used to define negation as : $A \to \bot$ and obviously $\top$ can be defined as $\bot \to \bot$.

But we can develop FOL also with propositional symbols, i.e. $0$-ary predicate symbols, and the propositional constant $\bot$.

See e.g. Dirk van Dalen, Logic and Structure, page 57.

In this case (see page 66), an interpretation of the sentences $\varphi$ of the language in a structure $\mathfrak A$ is a mapping to $\{ 0, 1 \}$, satisfying (inter alia) the condition $[⊥]^{\mathfrak A} = 0$.

Answer 2

No, there are no such "constant literals" in FOL. There may be such symbols used when we talk about FOL, but they are certainly not part of the language of FOL. (The interpretation of a constant literal in a model of a theory in FOL is an element of the "universe" of the model; that makes no sense for $T$ and $F$.)

In case that's not clear, an example. Consider $\mathcal T$, the first-order theory of groups. We can do that without a constant symbol for the identity, or if we wish we can include a constant literal $e$, so we could have an axiom $$\forall x(ex=xe=x)$$in place of the constant-free $$\exists y\forall x(yx=xy=x).$$

Suppose we do include a constant symbol $e$. Now a model of $\mathcal T$ is exactly a group, and if $G$ is a group the interpretation of $e$ is just the identity element of $G$. If there is a constant literal $T$, what element of $G$ would be the interpretation of $T$?

Or consider the first-order theory of ordered fields, with, among other things, constant symbols $0$ and $1$ and a binary predicate $\ge$. If $c$ is a constant literal the sentence $\forall x(xx\ge c)$ makes sense; in fact it's a theorem if $c$ is $0$ and an anti-theorem if $c$ is $1$. What does that sentence "mean" is $c$ is $T$?

(One could have constant literals T and F in propositional logic...)