and ξ differ only on variables that do not occur freely in Q, and on x. The
proposition is proved.
The following almost obvious fact is the basis for many phenomena that attest to the inadequacy of formal languages for completely describing intuitive concepts (see “Skolem’s paradox” below):
2.12. Proposition. The cardinality of the class of φ-definable sets does not exceed
card(alphabet ofL) +ℵ0.
Here and below, by “card(alphabet of L)” we mean the cardinality of the al- phabet of L without the set of variables.
Proof. If the language hasℵ0variables, then there are at most card(alphabet ofL) +ℵ0 formulas.
If, on the other hand, it has an uncountable set of variables, then we note that every definable set can be defined by a formula whose variables belong to a fixed countable subset of the variables that is chosen once and for all.
2.13. Corollary. If M is infinite and card(alphabet of L) < 2cardM, then
“almost all” sets are undefinable.
Thus, the only way to define all subsets of M is to include a tremendous number of names in the language. For languages that are to describe actual mathematical reasoning this is an unrealistic program. Essentially, any finitely describable collection of modes of expression allows us to define only a countable number of sets. However, it is often technically useful to include in the alphabet, for example, names for all the elements ofM.
In the following sections we proceed to study systematically sets of true formulas.
3.3The setTφLis closed under the rules of deductionMP (modus ponens)and Gen (generalization). By definition, this means that ifP andP⇒Qlie inTφL, thenQalso lies inTφL, and that ifP lies inTφL, then∀xP lies inTφLfor any variable x. The verification is immediate: if |P|φ= 1 and |P ⇒Q|φ = 1, then we must have|Q|φ = 1; if|P|φ(ξ) = 1 for allξ, then also|∀xP|φ(ξ) = 1. The formula Qis called a direct consequence of the formulas P and P ⇒ Qusing the rule of deductionMP. The formula∀xP is called adirect consequence of the formula P using the rule of deductionGen.
The intuitive meaning of these rules of deduction is as follows. The rule MP corresponds to the following type of argument: “If P is true, and if the truth of P implies the truth of Q, thenQ is true.” Thus, one might say that the semantics of the expression “if . . . then” in natural languages is divided between the semantics of the connective ⇒ and the semantics of the rule of deduction MP in languages ofL1. Neglecting this point of view often leads to confusion when one attempts to explain the rules for assigning truth values to the formulaP ⇒Q.
The rule Gen corresponds to the practice in mathematics of writing “identi- ties” or universally true assertions. When we write (a+b)2=a2+2ab+b2or “in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides,” the quantifiers ∀a ∀b and ∀ triangles are omitted.
Putting the quantifiers back in does not change the truth values, and has the advantage of freeing the notation for later use.
3.4.The setTφLcontains all tautologies. To define what a tautology is, we first introduce the notion of alogical polynomial over a set of formulasE. This is an element in the minimal set of formulas that containsEand is closed with respect to constructing formulas from shorter formulas using logical connectives.
A sequence of formulasP1, . . . , Pn and representations of eachPi, either in the form Q, where Q ∈ E, or in the form ¬Q or Q1∗Q2, where Q, Q1, Q2
lie in {Pi, . . . , Pi−1}, is called a representation of Pn as a logical polynomial over E. The representation of Pn is not necessarily unique: for example, if E ={P, Q, P ⇒Q}, thenP ⇒Qhas two representations.
Let : E → {0,1} be any map. If we are given a representationr of the formulaPn as a logical polynomial overE, then we can use the formulas in 2.5 to determine|Pn|r recursively.
A formula P is called a tautology if there exist a set of formulas E and a representation r of P as a logical polynomial over E such that|P|r= 1 for all maps :E → {0,1}. The property of being a tautology is effectively decidable, since; by syntactically analyzing P we can enumerate all representations of P as a logical polynomial. All tautologies obviously belong toTφL.
Here are our first examples of tautologies:
A0. P⇒P; A1. P⇒(Q⇒P);
A2. (P⇒(Q⇒R))⇒((P ⇒Q)⇒(P ⇒R));
A3. (¬Q⇒ ¬P)⇒((¬Q⇒P)⇒Q);
B1. ¬¬P⇒P, P ⇒ ¬¬P; B2. ¬P⇒(P ⇒Q).
HereP, Q, andRare arbitrary formulas inL; the form in which these tautologies are written makes it clear what representation as a logical polynomial over {P, Q, R}is intended.
Thus, tautologies are formulas that are true regardless of the truth or falsity of the component parts (if the notion of component is suitably chosen). Bl is the law of the excluded middle: a double negation is equivalent to the original assertion. B2 is the mechanism by which a contradiction in a set of formulasE in Lleads to the deducibility of any formula, and thereby destroys the entire system. (See Proposition 4.2 below.)
Example of how a tautology is verified. We give three versions of how to verify that the simple formula Al is a tautology.
Version(a). By the formulas in 2.5, we have
|P ⇒(Q⇒P)|= 1− |P|+|P| |Q⇒P|
= 1− |P|+|P|(1− |Q|+|P| |Q|) = 1, since|P|2=|P|.
Version(b). We tabulate|P ⇒(Q⇒P)|as a function of|P|and|Q|:
|P| |Q| |Q⇒P| |P ⇒(Q⇒P)|
0 0 1 1
0 1 0 1
1 0 1 1
1 1 1 1
This is an example of a “truth table.”
Version(c). The basic property of the connective⇒is thatP ⇒Qis false only if P is true andQis false. IfP ⇒(Q⇒P) were false, then P would be true and Q⇒P would be false; then, in turn, Qwould be true and P would be false, a contradiction.
The reader would do well to verify that the more complicated axioms, for example A2, are tautologies, and to decide which of the three versions he prefers.
3.5. The set TφL contains the “logical quantifier axioms,” that is, the formulas
(a) ∀x(P ⇒Q)⇒(P ⇒ ∀xQ), if all the occurrences of xin P are bound.
(b) ∀x¬P ⇔ ¬∃xP.
(c) ∀xP(x)⇒P(t), iftis free forxinP (axiom of specialization). Here we use the notationP(t) for the result of substituting t for each free occurrence of xin P. In all other respectsP andQare arbitrary formulas.
In 3.7 we verify that the formulas in 3.5 areφ-true. The intuitive meaning of these formulas is more or less clear. For example, the axiom of specialization means that ifP(x) is true for allx, thenP(t) is also true, wheret is the name of any object. The condition thattmust be free forxis the rule of hygiene for changing notation.
The set
AxL={tautologies ofL} ∪ {quantifier axioms} is called theset of logical axioms in the language L.
A set of formulasE in L will be calledG¨odelian if it is complete, does not contain a contradiction, is closed with respect to the rules of deduction MP and Gen, and contains all the logical axioms of L. The basic conclusion of our discussion is then the following:
3.6. Proposition. The set of true formulas of L (in any interpretation) is G¨odelian.
In §6 we prove that conversely, any G¨odelian set is a set of true formulas in a suitable interpretation. Thus, the concept of a G¨odelian set is the closest approximation to the concept of truth that can be attained “without regard to meaning.”
3.7. Verification that axioms 3.5are true.
(a) Let R be the formula 3.5(a). We suppose that|R|(ξ) = 0 for some ξ∈M and show that this leads to a contradiction.
In fact, then |∀x(P ⇒ Q)|(ξ) = 1 and |P ⇒ ∀x Q|(ξ) = 0. The second equation implies that |P|(ξ) = 1 and|∀x Q|(ξ) = 0. Letξ be a variation ofξ along xfor which|Q|(ξ) = 0. Then |P|(ξ) =|P|(ξ) = 1 by Proposition 2.10, sincexdoes not occur freely inP. Hence,|P →Q|(ξ) = 0, which contradicts the relation|∀x(P ⇒Q|(ξ) = 1.
(b) For allξ∈M and for all variationsξ ofξalongx, we have
|∀x¬P|(ξ) = max
ξ |¬P|(ξ) = 1−min
ξ |P|(ξ);
|¬ ∃x P|(ξ) = 1−min
ξ |P|(ξ).
Hence, the truth values of∀x¬P and ¬ ∃xP coincide, so that∀x¬P ⇔ ¬ ∃x P is identically true.
(c) Suppose that|∀x P(x)⇒P(t)|(ξ) = 0 for some pointξ∈M. We show that this leads to a contradiction. In fact, then
|∀x P(x)|(ξ) = 1, |P(t)|(ξ) = 0.
The first equation implies that|P(x)|(ξ) = 1 for all variationsξ or ξalongx.
For ξ we take the variation such that xξ = tξ. If we prove that |P(t)|(ξ) =
|P(x)|(ξ), then we obtain the desired contradiction.
We prove this by induction on the total number of connectives and quantifiers inP.
(c1) Let P be an atomic formulap(t1, . . . , tn). Letting ¯ti denote the result of substitutingt for each occurrence ofxin ti,we successively obtain
tξ=xξ (by the definition ofξ),
¯tiξ=tξi (by induction on the length ofti),
|P(x)|(ξ) =|P(t1, . . . , tn)|(ξ) =|P(¯t1, . . . ,¯tn)|(ξ) =|P(t)|(ξ).
(c2) LetP have the form¬QorQ1 →Q2, where→is a connective. Since xdoes not bindt inP by assumption, the same is true forQ, Q1, andQ2, and the necessary induction step is automatic.
(c3) Finally, let P have the form ∃y Q or ∀y Q. We shall examine the first case; the proof for the second case is analogous.
Subcase 1. y = x. Then x is bound in P; therefore, P(x) = P(t), and
|P|(ξ) =|P|(ξ) by Proposition 2.10.
Subcase 2. y = x. The induction assumption has the form |Q(t)|(η) =
|Q(x)|(η) if η is any point in M and η is a variation of η alongx for which xη =tη. We must show that the following two truth values coincide (whereξ andξ are defined as above):
|∃y Q(x)|(ξ) =
1, if|Q(x)|(η) = 1 for some variationη of ξ alongy, 0, otherwise.
|∃y Q(t)|(ξ) =
1, if|Q(t)|(η) = 1 for some variationη of ξ alongy, 0, otherwise.
We recall thatξ is the variation ofξ alongxfor whichxξ =tξ.
We first suppose that the second truth value is 1. We chooseη∈M such that
|Q(t)|(η) = 1, and then construct the variationηofηalongxfor whichxη =tη. Then, by the induction assumption, 1 = |Q(t)|(η) = |Q(x)|(η). We show that η is a variation ofξ alongy; this will imply that the first truth value is also 1.
In fact,ηwas obtained by varyingηalongx,ηwas obtained by varyingξalong y, andξwas obtained by varyingξ alongx. Hence,η is a variation ofξalong xandy; we must show the variation alongxdid not actually take place:
xη =xξ.
But the left-hand side istη by the definition ofη; the right-hand side istξ by the definition ofξ; andη was obtained by varyingξ alongy. Sincetis free for xinP =∃y Q, it follows thaty does not occur int.
It remains to verify that if the second truth value is 0, then the first is also 0. The argument is almost the same. If the second truth value is 0, then
|Q(t)|(η) = 0 for all variationsη ofξ alongy. For each such η we constructη as in the first part of the proof. As before, we verify thatη is a variation ofξ alongyand, moreover,η runs through all such variations whenηruns through all variations of ξalongy. Hence, the first truth value is also 0.
The proposition is proved.