is valid. What has happened?!!? Well, our feeling is based on an implicit additional premise—that you can get from 6 7 to 4 7 by repeatedly subtracting 1/8 . Without this premise, we can only conclude that 6 7 is tall, since there might not be a height that is 1/8 less than 6 7 . Logically might not, that is, since we know that there is in fact such a height.5 To expedite matters, we will add not one premise but 192 premises: 6 67/8 is 1/8 less than 6 7 ; 6 66/8 is 1/8 less than 6 67/8 ; 6 65/8 is 1/8 less than 6 66/8 ; . . . ; 4 71/8 is 1/8 less than 4 72/8 ; and 4 7 is 1/8 less than 4 71/8 . Using s1 to represent 6 7 , s2 to represent 6 67/8 , . . . , down to s193 representing 4 7 , the Sorites argument augmented to state the implicit premise explicitly is

Ts1

( x)( y)((Tx Eyx) → Ty) Es2s1

Es3s2

Es4s3

. . .

Es193s192

Ts193

The augmented argument is valid in classical first-order logic. On any interpretation on which all the premises are true, so is the conclusion. Here’s why: Assuming that all of the premises are true we will be able to infer that Ts2 is true, then that Ts3 is true, and so on, until we finally infer that the conclusion Ts193 is true. We’ll just show the first step; the rest are similar. From the truth of the first and third premises it must be the case that <I(s1)> I(T) and that <I(s2), I(s1)> I(E). From the truth of the second premise, we know that whatever v(x) and v(y) may be, if Tx Eyx is satisfied by v then so is Ty. In particular, this holds when v(x) = I(s1) and v(y) = I(s2). Because Tx Eyx is satisfied by v in this case, so is Ty—that is, <I(s2)> (which is <v(y)>) is a member of I(T). This means that Ts2 must be true. Repeating this reasoning we’ll conclude that Ts193 must be true as well.

3.3 An Axiomatic Derivation System for Classical First-Order Logic

By adding two axiom schemata and one rule to the axiomatic derivation system CLA for classical propositional logic we can produce a sound and complete axiomatic system for classical first-order logic.6 We call the system CL A. We

5OK, I guess we’d better quibble here. Some view mathematics as a branch of logic, and since it is a matter of mathematics that for any positive measure of height in feet and inches there is

another measure that is 1 ” less, those people would say that what we are entertaining is not a

/

8

logical possibility. So to be more specific: it is not a matter of classical first-order logic that there

is a height that is 1 less than 6 7 .

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8

6The rule and axioms are from Stoll (1961, pp. 388–390). Stoll says that they are essentially the philosopher Bertrand Russell’s rule and axioms. Chapter 9 of Stoll proves the soundness and completeness of the resulting system.

stipulate that only closed formulas may occur as assumptions or instances of axiom schemata in derivations in CL A (as suggested in footnote 3, some axiomatic systems allow open as well as closed formulas in derivations). Our rules preserve closure, so it follows that every formula in a derivation will be closed as long as assumptions and instances of axiom schemata that occur in derivations are all closed.

We will now refer to the axiom schemata CL1–CL3 from CLA as CL 1–CL 3:

CL 1. P→ (Q → P)

CL 2. (P → (Q → R)) → ((P → Q)→ (P→ R))

CL 3. (¬P → ¬Q) → (Q → P)

while Modus Ponens retains its name:

MP. From P and P → Q, infer Q.

We’ll take the existential quantifier to be defined in terms of the universal quantifier:

( x)P =def ¬( x)¬P

so the additional axiom schemata and rules for first-order logic will mention only the universal quantifier. The first new axiom schema is

CL 4. ( x)(P → Q) → (P → ( x)Q)

where P is a formula in which x does not occur free

That is, as long as the initial quantifier isn’t quantifying over anything in the antecedent of the conditional P → Q, the quantifier may be moved to the consequent. Here’s an example of a derivation using this axiom schema:

The second axiom schema states that a universally quantified formula implies any one of its instances:

CL 5. ( x)P → P(a/x)

where a is any individual constant and the expression P(a/x) means: the result of substituting the constant a for the variable x wherever x occurs free in P

We call P(a/x) a substitution instance of P. So, for example, if P is Bby, a is c, and x is y, then P(a/x) (or Bby(c/y)) is Bbc. Using CL 5 we can continue the previous derivation to derive the formula Pa (or any other substitution instance of the formula on line 5):

Every instance of each of the two new axiom schemata is a tautology in classical first-order logic.

The new rule in CL A is

UG (Universal Generalization). From P(a/x) infer ( x)P

where x is any individual variable, provided that no assumption contains the constant a and that P itself does not contain the constant a.

The first part of the condition ensures that we can derive a universally quantified formula from one of its substitution instances only if we have not made any assumptions involving the constant in that substitution instance. It rules out, for example, inferring ( x)Rx from the assumption Ra—if we assume that Ann is Romanian it doesn’t follow that everyone is Romanian! The second part of the condition rules out generalizations that are not truly general. Without it, we could have derivations like

This derivation is not truth-preserving: from the assumption that everything stands in the relation L to itself it does not follow that everything stands in the relation L to everything! Line 4 violates the second half of the condition for correct use of the rule UG, since the formula Lxa retains an occurrence of the variable a that was generalized on. With the condition that P does not contain the constant a, however, the new rule UG is truth-preserving. A correct use of UG would produce the formula ( x)Lxx on line 4, and this is clearly acceptable since it’s the assumption we started with! Of course, we cannot then go on to infer the formula on line 5 since there’s no constant in ( x)Lxx to generalize upon.

The following derivation illustrates the combined uses of CL 5 and UG:

UG has been correctly used on line 8 because a does not occur in either of the assumptions, nor does it occur in ( x)Gx.

We can use UG to derive a quantified version of the Law of the Excluded Middle, ( x)(Ax ¬Ax), which is converted to ( x)(¬Ax → ¬Ax) when the disjunction is eliminated:

Although the constant a occurs in the formula on the first line, that formula is an instance of an axiom schema rather than an assumption, and so the use of UG on the second line is legitimate. Note that in our derivations we may use derived axioms and rules from Chapter 2 since the axioms and rules of CLA are also axioms and rules of CL A. We add to the names of the derived axioms, to make clear that we are now working within the first-order axiomatic system.

To show that ( x)Gx → ( x)Gx, which is a tautology, is a theorem we first rewrite the existential quantifier using its definition to obtain ( x)Gx → ¬( x)¬Gx. We can derive the rewritten formula with the help of derived axiom schemata and rules as follows:

Given the soundness of the axiomatic system CL A, we have now demonstrated the validity of the Sorites argument in classical first-order logic in a second way (the first was the semantic argument in Section 3.2).

A final note about first-order classical logic: unlike the case in propositional logic, the set of theorems of CL A (or of any other adequate derivation system for first-order classical logic) is undecidable. Equivalently, given the soundness and completeness of CLA, the set of tautologies of classical predicate logic is