5 Tautologies and Boolean Algebras

His conclusion:

Nevertheless, the only way to verify these results (if this were thought worth while) is for the problem to be attacked quite independently, by a differ- ent machine. This corresponds exactly to the situation in most experimental sciences.

We note that it is becoming more and more apparent that the processing, and also the storage, of large quantities of information outside the human brain leads to social problems that go far beyond questions of the reliability of math- ematical deductions.

5. In conclusion, we quote an impression concerning mechanical proofs, even ones done by hand, which is experienced by many.

After stating a proposition to the effect that “the functionTW,η₀θ˜is correctly defined,” a gifted and active young mathematician writes (Inventiones Math., vol. 3, f.3 (1967), 230):

The proof of this Proposition is a ghastly but wholly straightforward set of computations. It took me several hours to do every bit and as I was no wiser at the end—except that I knew the definition was correct—I shall omit details here.

The moral: a good proof is one that makes us wiser.

of modified logical systems, for example those of the intuitionist type, requires more careful analysis of this list.

Proof of Proposition 5.1. Let E be a finite set of formulas in L, and let P be a logical polynomial (with a fixed representation) overE. For any map v :E → {0,1}, we extendv to P using the same rules that defined the truth function| |in Section 2.5. We set

P^v=

P, ifv(P) = 1,

¬P, ifv(P) = 0.

5.2. Fundamental Lemma.Let E^v ={Q^v|Q∈ E}. Then for any v we have F ∪ E P^v (usingMP).

This lemma expresses the following idea. It is natural to prove Proposition 5.1 by induction on the length of the tautology. However, the component parts of a tautology themselves might not be tautologies. The operation of takingP to P^v forces any formula to be “v-true” and makes it possible for us to use induction.

5.3. Proof of 5.1 assuming the Fundamental Lemma. Let P be a tau- tology, so that P^v = P for all v, Set E = {P1, . . . , Pr}. By the fundamen- tal lemma, F ∪ {P₁^v, . . . , P_r^v} P using MP for any v: We show that then F ∪ {P₁^v, . . . , P_r^v₋₁} P using MP. Descending induction on r then gives the required assertion (the assumption that P is a logical polynomial inP1, . . . , Pr

is not used in the induction step).

The Deduction Lemma 4.5 shows thatF ∪{P₁^v, . . . , P_r^v₋₁} (P_r^v⇒P) using MP; to see this we only need examine the proof and notice that the deduction used only MP and the tautologies in F, since the rule of deduction Gen was not needed.

Since for any v there exists av that coincides withv onP₁, . . . , P_r−1 but takes a different value on P_r, it follows that P_r ⇒ P and ¬P_r ⇒ P are deducible from F ∪ {P₁^v, . . . , P_r^v₋₁} using MP. On the other hand, the tau- tology C4: (Pr⇒P)⇒((¬Pr ⇒P)⇒P) lies in F. Applying MP twice, we

deduceP.

5.4.Proof of the Fundamental Lemma. We use induction on the number of connectives in the representation ofP as a logical polynomial overE. If there are no connectives, that is,P ∈ E, then the assertion is obvious. Otherwise,P has the form¬QorQ1∗Q2, where ∗is one of the binary connectives.

(a)The caseP =¬Q. Ifv(Q) = 0, thenQ^v=¬Q=P=P^v. ThatQ^v=P^v is deducible fromF ∪ E^v is precisely the induction assumption.

On the other hand, if v(Q) = 1, then Q^v = Q, P^v = ¬¬Q. Here Q is deducible from F ∪ E^v by the induction assumption, and then the tautology Q⇒ ¬¬QinF along with MP gives a deduction ofP^v.

(b)The case P =Q₁∗Q₂. For the different connectives and possible values of v(Q1) and v(Q2) we first tabulate the formulas for which deductions exist by

the induction assumption and the formulas for which we must find deductions.

In the columns under ∧ and ∨ we give formulas from which (Q1∧Q2)^v and (Q1 ∨Q2)^v, respectively, are deducible using MP and the tautologies in F (tautologies Cl, C2, and C5). Hence it suffices to find deductions of each of formulas 1–16 from F and the pair of formulas in the appropriate row in the second column using MP.

Deduction of formulas1–16.

Given:

deductions of Must Find: Deduction of (Q1∗Q2)^v

v(Q1)v(Q2) Q^v1 andQ^v2 ⇒ ∧

0 0 ¬Q1,¬Q2 1. Q1⇒Q2 5.¬ ¬(Q1⇒ ¬Q2) 0 1 ¬Q1, Q2 2. Q1⇒Q2 6.¬ ¬(Q1⇒ ¬Q2) 1 0 Q1,¬Q2 3.¬(Q1⇒Q2) 7.¬ ¬(Q1⇒ ¬Q2) 1 1 Q1, Q2 4. Q1⇒Q2 8.¬(Q1⇒ ¬Q2)

v(Q1)v(Q2) Q^v1 andQ^v2 ∨ ⇔

0 0 ¬Q1,¬Q2 9.¬(¬Q1⇒Q2) 13. Q1⇔Q2

0 1 ¬Q1, Q2 10.¬Q1⇒Q2 14.¬(Q1⇔Q2) 1 0 Q1,¬Q2 11.¬Q1⇒Q2 15.¬(Q1⇔Q2) 1 1 Q1, Q2 12.¬Q1⇒Q2 16. Q1⇔Q2

Note that if P is deducible then for any Q the formula Q ⇒ P is also deducible (tautology A1 and MP) and if ¬P is deducible then for any Qthe formula P ⇒Qis deducible (tautology B2 and MP). This immediately yields deductions of 1, 2, 4, 10, and 12. If we remove the double negations in the∧col- umn using tautology B1 and MP, we obtain deductions of 5, 6, and 7. And 11 is deducible since by B1 the second column yields a deduction of ¬ ¬Q1. In the first and last rows the deductions of 1 and 4 yield deductions of Q2⇒Q1 by symmetry; tautology C6 and MP twice give a deduction of 13 and 16 from Q1⇒Q2 andQ2⇒Q1.

3 is deduced from C3:Q1 ⇒ (¬Q2 ⇒ ¬(Q1 ⇒ Q2)) and the second column using MP twice.

8 is deduced from C3:Q1⇒(¬¬Q2⇒ ¬(Q1⇒ ¬Q2)) and the second column using MP, applying B1 toQ2, and again using MP.

9 is deduced from C3:¬Q1⇒(¬Q2⇒ ¬(¬Q1⇒Q2)) using MP twice.

15 is deduced from 3 by C7 and C5 and MP twice.

Finally, the deduction of 3 from Q₁ and¬Q₂ yields by symmetry a deduction of¬(Q₂⇒Q₁)from¬Q₂ andQ₂. Hence on the second row the deduction of 14 is analogous to that of 15.

Proposition 5.1 is proved.

5.5. Tautologies and probability. Tautologies are statements that are true independently of the truth or falsity of their “component parts.” This assertion still holds even if the components of a tautology are assigned proba- bilistic truth values P in the algebra of measurable sets in some probability space.

An example: the tautologyR∨S∨ ¬R∨ ¬S—“either it will rain, or it will snow, or it won’t rain, or it won’t snow”¹—is a reliable weather forecast despite the great complexity of the meteorological probability space.

For a precise result, it is convenient to use the terminology of Boolean algebras.

5.6. Boolean algebras. A Boolean algebraB is a set with an operation of rank one, with two operations ∨ and ∧ of rank two, and with two distinguished elements 0 and 1, such that the following axioms hold:

(a) (A) =Afor allA∈B;

(b) ∧and∨are each associative and commutative;

(c) ∧and∨are distributive with respect to one another;

(d) (a∨b) =a∧b,(a∧b) =a∨b; (e) a∨a=a∧a=a;

(f) 1∧a=a; 0∨a=a.

Examples.

(a) B is the set of all subsets of a set M, is complement,∧is intersection,∨ is union, 0 is the empty subset, and 1 is all ofM.

(b) B is the set of open-and-closed subsets of a topological space M with the same operations.

(c) B is the algebra of measurable subsets (modulo measure-zero subsets) of a probability space M with the same operations.

In all of these cases B can be identified with the space of characteristic functions of the corresponding subsets ofM (taking the value 1 on the subset and 0 on the complement).

5.7.Boolean truth functions. LetB be a Boolean algebra, and letE be a set of formulas in a languageL. Let :E →B be any map. We extend this map to the logical polynomials over E (more precisely, to their representations) by means of the recursive formulas

P ⇔Q= (P ∧ Q)∨(P∧ Q), P ⇒Q=P∨ Q,

P∨Q=P ∨ Q, P∧Q=P ∧ Q,

¬P=P.

1 A Russian proverb (translator’s note).

In the caseB ={0,1}, these formulas coincide with the definitions in 2.5.

We note that∨and∧have different meanings in the left- and right-hand sides.

5.8.Proposition.Let the logical polynomial P be a tautology overE. Then for any map :E →B to any Boolean algebra B we have P= 1.

Proof. An example of a natural map can be obtained as follows: if we are given an interpretation ofLin a setM, then the truth functions|P|(ξ) can be considered as the characteristic functions of the definable subsets of the inter- pretation classM (compare§2). Hence, our usual truth functions are essentially Boolean-valued. They are embedded in the Boolean algebra of all subsets ofM, which decomposes as a direct product of two-point Boolean algebras {0,1}. Hence the proposition follows trivially in this case.

In the general case one could use Stone’s structure theorem for Boolean algebras. However, instead of this we shall indicate how to reduce the problem to some simple computations using Proposition 5.1. Because of Proposition 5.1, it suffices to verify that the basis tautologies are -true and that -truth is preserved when we use MP. For example, ifP= 1 andP ⇒Q= 1, then P = 0 while P ∨ Q = 1, so that Q = 1 by 5.6(f); this answers the question about MP. The truth values of the basis tautologies are computed in

a similar manner using the axioms in 5.6.

Boolean truth functions will be the basic tool in the presentation of Cohen forcing in Chapter III.