Section 04: Rules for quantifiers
For proofs in QL, we use all of the basic rules of SL plus four new basic rules: both introduction and elimination rules for each of the quantifiers.
Since all of the derived rules of SL are derived from the basic rules, they will also hold in QL. We will add another derived rule, a replacement rule called quantifier negation.
Substitution instances
In order to concisely state the rules for the quantifiers, we need a way to mark the relation between quantified sentences and their instances. For example, the sentence \(Pa\) is a particular instance of the general claim ∀\(xPx\).
For a wff \(\mathcal{A}\), a constant \(\mathcal{c}\), and a variable \(\mathcal{x}\), define a substitution instance of ∀\(\mathcal{xA}\) or ∃\(\mathcal{xA}\) is the wff that we get by replacing every occurrence of \(\mathcal{x}\) in \(\mathcal{A}\) with \(\mathcal{c}\). We call \(\mathcal{c}\) the instantiating constant .
To underscore the fact that the variable \(\mathcal{x}\) is replaced by the instantiating constant \(\mathcal{c}\), we will write the original quantified expressions as∀\(\mathcal{x}\)\(\mathcal{Ax}\) and∃\(\mathcal{x}\)\(\mathcal{Ax}\). And we will write the substitution instance \(\mathcal{Ac}\).
Note that \(\mathcal{A}\), \(\mathcal{x}\), and \(\mathcal{c}\) are all meta-variables. That is, they are stand-ins for any wff, variable, and constant whatsoever. And when we write \(\mathcal{Ac}\), the constant \(\mathcal{c}\) may occur multiple times in the wff \(\mathcal{A}\).
For example:
~ \(Aa\) → \(Ba\), \(Af\) → \(Bf\), and \(Ak\) → \(Bk\) are all substitution instances of ∀\(x\)(\(Ax\) → \(Bx\)); the instantiating constants are \(a\), \(f\), and \(k\), respectively.
~ \(Raj\), \(Rdj\), and \(Rjj\) are substitution instances of ∃\(zRzj\); the instantiating constants are \(a\), \(d\), and \(j\), respectively.
Universal elimination
If you have ∀\(xAx\), it is legitimate to infer that anything is an \(A\). You can infer \(A\) a , \(A\) b , \(A\) z , \(Ad\) 3 . You can infer any substitution instance, \(A\(\mathcal{c}\) for any constant \(\mathcal{c}\).
This is the general form of the universal elimination rule (∀E):
When using the ∀E rule, you write the substituted sentence with the constant \(\mathcal{c}\) replacing all occurrences of the variable \(\mathcal{x}\) in \(\mathcal{A}\). For example:
Existential introduction
It is legitimate to infer ∃\(xPx\) if you know that something is a \(P\). It might be any particular thing at all. For example, if you have \(Pa\) available in the proof, then ∃\(xPx\) follows.
This is the existential introduction rule (∃I):
It is important to notice that the variable \(\mathcal{x}\) does not need to replace all occurrences of the constant \(\mathcal{c}\). You can decide which occurrences to replace and which to leave in place. For example:
Universal introduction
A universal claim like ∀\(xPx\) would be proven if every substitution instance of it had been proven. That is, if every sentence \(Pa\), \(Pb\), ... were available in a proof, then you would certainly be entitled to claim ∀\(xPx\). Alas, there is no hope of proving every substitution instance. That would require proving \(Pa\), \(Pb\), ..., \(P\)j 2 , ..., \(Ps\) 7 , ..., and so on to infinity. There are infinitely many constants in QL, and so this process would never come to an end.
Consider instead a simple argument: ∀\(xMx\), .˙. ∀\(yMy\)
It makes no difference to the meaning of the sentence whether we use the variable \(x\) or the variable \(y\), so this argument is obviously valid. Suppose we begin in this way:
We have derived \(Ma\). Nothing stops us from using the same justification to derive \(Mb\), ..., \(Mj\) 2 , ..., \(Ms\) 7 , ..., and so on until we run out of space or patience. We have effectively shown the way to prove \(M\)\(\mathcal{c}\) for any constant \(\mathcal{c}\). From this, ∀\(yMy\) follows.
It is important here that \(a\) was just some arbitrary constant. We had not made any special assumptions about it. If \(Ma\) were a premise of the argument, then this would not show anything about all \(y\). For example:
This is the schematic form of the universal introduction rule (∀I):
∗ The constant \(\mathcal{c}\) must not occur in any undischarged assumption.
Note that we can do this for any constant that does not occur in an undischarged assumption and for any variable.
Note also that the constant may not occur in any undischarged assumption, but it may occur as the assumption of a subproof that we have already closed. For example, we can prove ∀\(z\)(\(Dz\) → \(Dz\)) without any premises.
Existential elimination
A sentence with an existential quantifier tells us that there is some member of the UD that satisfies a formula. For example, ∃\(xSx\) tells us (roughly) that there is at least one \(S\). It does not tell us which member of the UD satisfies \(S\), however. We cannot immediately conclude \(Sa\), \(Sf\) 23 , or any other substitution instance of the sentence. What can we do? Suppose that we knew both ∃\(xSx\) and ∀\(x\)(\(Sx\) → \(Tx\)). We could reason in this way:
Since ∃\(xSx\), there is something that is an \(S\). We do not know which constants refer to this thing, if any do, so call this thing ‘Ishmael’. From ∀\(x\)(\(Sx\) → \(Tx\)), it follows that if Ishmael is an \(S\), then it is a \(T\). Therefore, Ishmael is a \(T\). Because Ishmael is a \(T\), we know that ∃\(xTx\).
In this paragraph, we introduced a name for the thing that is an \(S\). We gave it an arbitrary name (‘Ishmael’) so that we could reason about it and derive some consequences from there being an \(S\). Since ‘Ishmael’ is just a bogus name introduced for the purpose of the proof and not a genuine constant, we could not mention it in the conclusion. Yet we could derive a sentence that does not mention Ishmael; namely, ∃\(xTx\). This sentence does follow from the two premises.
We want the existential elimination rule to work in a similar way. Yet since English language words like ‘Ishmael’ are not symbols of QL, we cannot use them in formal proofs. Instead, we will use constants of QL which do not otherwise appear in the proof.
A constant that is used to stand in for whatever it is that satisfies an existential claim is called a proxy . Reasoning with the proxy must all occur inside a subproof, and the proxy cannot be a constant that is doing work elsewhere in the proof.
This is the schematic form of the existential elimination rule (∃E):
∗ The constant \(\mathcal{c}\) must not appear in ∃\(\mathcal{xAx}\), in \(\mathcal{B}\), or in any undischarged assumption.
Since the proxy constant is just a place holder that we use inside the subproof, it cannot be something that we know anything particular about. So it cannot appear in the original sentence ∃\(\mathcal{xAx}\) or in an undischarged assumption. Moreover, we do not learn anything about the proxy constant by using the ∃E rule. So it cannot appear in \(\mathcal{B}\), the sentence you prove using ∃E.
The easiest way to satisfy these requirements is to pick an entirely new constant when you start the subproof, and then not to use that constant anywhere else in the proof. Once you close the subproof, do not mention it again.
With this rule, we can give a formal proof that∃\(xSx\) and∀\(x\)(\(Sx\) → \(Tx\)) together entail ∃x\(Tx\).
Notice that this has effectively the same structure as the English-language argument with which we began, except that the subproof uses the proxy constant ‘\(i\)’ rather than the bogus name ‘Ishmael’.
Quantifier negation
When translating from English to QL, we noted that ¬∃\(x\)¬\(\mathcal{A}\) is logically equivalent to ∀\(x\)\(\mathcal{A}\). In QL, they are provably equivalent. We can prove one half of the equivalence with a rather gruesome proof:
In order to show that the two sentences are genuinely equivalent, we need a second proof that assumes ¬∃\(x\)¬\(\mathcal{A}\) and derives ∀\(x\)\(\mathcal{A}\). We leave that proof as an exercise for the reader.
It will often be useful to translate between quantifiers by adding or subtracting negations in this way, so we add two derived rules for this purpose. These rules are called quantifier negation (QN):
¬∀\(\mathcal{xA}\) ⇐⇒∃\(\mathcal{x}\)¬\(\mathcal{A}\)
¬∃\(\mathcal{xA}\) ⇐⇒∀\(\mathcal{x}\)¬\(\mathcal{A}\) QN
Since QN is a replacement rule, it can be used on whole sentences or on subformulae.