A Logical Axiom that is Not a Schema/Scheme

Question

Pretty straightforward. I am just looking for an example (or two) of a logical axiom that is not a scheme or schema. One that is in a common logical system would be nice.

Oh, and what is the difference between a scheme and a schema?

Answer 1

Even $p=p$ may be considered as a schema, insofar as $p$ may stand for any singular term (or any formula, if one takes '$=$' as a biconditional...)

What about this:

$$\forall x (x = x)$$

I don't know if it is actually used as an axiom, but it seems to me the most plausible candidate of a formula that cannot, as far as I see, be used as a schema.

UPDATE: I think Mauro is right in his comment that even this formula could be used as a schema (though the rules and axioms governing quantifiers will usually give renaming of bound variables for free).

So the answer to the original question seems to be somewhat calculus-dependent. With regard to propositional variables, there are two common ways. One is described by Mauro in his answer: The proof system offers some schemata and allows to add any instance of any of them to the derivation.

The other way is to have a substitution rule. Such proof systems will not offer axiom schemata, but really axioms. So it might allow to add $P \vee \neg P$, but not allow to add $Q \vee \neg Q$. To derive the latter formula, one would need to write two lines instead:

1.   P v ~P          Axiom XY
2.   Q v ~Q          Substitution in 1: Q for P

Here, $P \vee \neg P$ might really be considered an axiom that is not a schema. Of course, through the substitution rule, $P \vee \neg P$ will still feel a bit like a schema. If so, then it's just a characteristic of first-order logic: If one formula is valid, then it will remain valid if you uniformly replace any propositional variable by another, or any predicate sign by another, or any individual constant by another, or finally any variable by another.

Answer 2

The classical laws of thought:

• Law of identity: $p = p$
• Law of non-contradiction: $\neg (p \wedge \neg p)$
• Law of excluded middle: $p \vee \neg p$

Edit: LNC and LEM may be schemas (as pointed out by Nate in the comments)

Answer 3

The usual presentation of a logic calculus, like e.g. propositional logic, defines a language with sentence variables : $p_0, p_1, \ldots$ and conncetives : $\lnot, \lor, \to, \ldots$ and express axioms in "schematic" forms, like :

$\varphi \to (\psi \to \varphi)$.

We can instead, express the axioms directly into the object language, like :

$p_0 \to (p_1 \to p_0)$.

In this case, we have to add to modus ponens an second inference rule, the substitution rule :

from the formula $\varphi$, infer $\varphi[\psi/p]$

where $\varphi[\psi/p]$ is the formula which results by substitution of $\psi$ for each occurrence of $p$ throughout $\varphi$.

See :

• Alonzo Church, Introduction to Mathematical Logic (1956), page 72.