JAAKKO HINTIKKA

QUANTUM LOGIC AS A FRAGMENT OFINDEPENDENCE-FRIENDLY LOGIC

Received 30 April 2000; received in revised version 3 August 2001

ABSTRACT. The working assumption of this paper is that noncommuting variables areirreducibly interdependent. The logic of such dependence relations is the author’s indepen-dence-friendly (IF) logic, extended by adding to it sentence-initial contradictory nega-tion ¬ over and above the dual (strong) negation ∼. Then in a Hilbert space ∼ turnsout to express orthocomplementation. This can be extended to any logical space, whichmakes it possible to define the dimension of a logical space. The received Birkhoff andvon Neumann “quantum logic” can be interpreted by taking their “disjunction” to be¬(∼A & ∼B). Their logic can thus be mapped into a Boolean structure to which anadditional operator ∼ has been added.

KEY WORDS: quantum logic, independence-friendly logic, negation, Boolean structures

As indicated in another paper (Hintikka, forthcoming), I have come tobelieve that a number of the conceptual problems in quantum theory can beunderstood on the basis of the assumption that noncommuting variables aremutually dependent. But what is here meant by dependence and how can itbe expressed logically? In a logical language, the dependence of a variableon another one is expressed by the dependence of the quantifier which theformer variable is bound to on the quantifier involving the latter variable.Furthermore, the dependence of a quantifier on another one is indicatedby its occurring within the scope of the latter. The scope of a quantifier isindicated by the pair of parentheses associated with it. Because the scopesof the quantifiers of a received Frege–Russell-type first-order languageare therefore ordered transitively and asymmetrically, mutual dependencecannot be expressed by means of the received Frege–Russell logic.

However, in the extension of the received first-order logic called inde-pendence-friendly (IF) first-order logic, all possible patterns of depen-dence and independence among first-order quantifiers are expressible, andhence so are all possible patterns of dependence and independence be-tween different variables. (For IF first-order logic, see Hintikka, 1996).This logic arises from the received first-order logic by merely allowinga quantifier (Q2y) to be independent of another one, say (Q1x), in whosesyntactical scope it occurs. This is indicated by writing it (Q2y/Q1x). Thisnovelty brings mutually dependent quantifiers (and ipso facto mutuallydependent variables) within the scope of an explicit logical treatment.

Journal of Philosophical Logic 31: 197–209, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Accordingly, I have argued, many of the conceptual problems of quan-tum theory become amenable to analysis by means of logic, viz. IF first-order logic. In this sense, IF logic is the logic of quantum theory.

But if so, what are we to say of the structures – orthomodular lattices– which are supposed to constitute the logic of quantum theory? Theselattices were discovered by John von Neumann and G. Birkhoff (1936).What is their relation to IF logic? This paper is calculated to provide ananswer. The answer is that the received quantum logic is a fragment of IFfirst-order logic, suitably extended in one respect. This respect is negation.In the game-theoretical semantics which is the foundation of IF first-orderlogic, the only negation characterizable by semantical (game) rules is astrong (dual) negation which does not obey the law of excluded middle(cf. Hintikka and Sandu, 1997). However, one can none the less extendIF logic further by admitting sentence-initial contradictory negation to thebasic IF first-order logic. The result will be called extended IF first-orderlogic. It is the extension just mentioned. It is a stronger logic than the plainIF first-order logic, yet still a first-order one in that all quantification is overindividuals.

Unlike its companion paper (Hintikka, forthcoming), the present onedepends essentially on the use of Hilbert spaces as a mode of representa-tion of quantum theoretical phenomena. It is therefore in order to relatethe logical language representation used here to Hilbert space representa-tion, indeed to any explicit mathematical representation of the same sort.Let us assume that the following is a law that governs certain physicalphenomena:

(∀t1)(∀t2) . . . (∀x1)(∀x2) . . . S[t1, t2, . . . , x1, x2, . . .].(1)

Here (1) might, for instance, be the translation of the basic equation ofsome physical theory from the language of Hilbert spaces to a logical no-tation. It might, for instance, be the Schrödinger equation. The variables t1,t2, . . . , x1, x2, . . . are the variables that normally occur in such an equation.

But suppose now that the law that governs the relevant phenomenais like (1) except that the variables x1, x2, . . . are interdependent, whilet1, t2 . . . are not. Such a law is not expressed by any sentence of an ordinaryfirst-order language nor in a customary notation of mathematical physics.However, such a law can be formulated by means of IF logic as follows:

(∀t1)(∀t2) . . . (∀x1)(∀x2) . . . (∃z1/∀x1)(∃z2/∀x2) . . .

((x1 = z1) & (x2 = z2) & · · · & S[t1, t2, . . . , x1, x2, . . .]).(2)

Here (2) is equivalent with

QUANTUM LOGIC AS A FRAGMENT OF IF LOGIC 199

(∃f1)(∃f2) . . . (∀t1)(∀t2) . . . (∀x1)(∀x2) . . .(3)

(x1 = f1(t1, t2, . . . , x2, x3, . . .)) &

(x2 = f2(t1, t2, . . . , x1, x3, . . .)) & · · ·& S[t1, t2, . . . , x1, x2, . . .]).

The leading idea of my approach to quantum logic can be expressed bysaying that while quantum theorists express their fundamental law in aprima facie form like (1), what they in practice do is to assume (2). Thetacit reason for their proceeding in this way is that in the usual mathemat-ical notation there is no standard way of expressing (2), that is, no generalcounterpart to the slash notation.

In (3), the functions f1, f2, . . . are known as Skolem functions. In gen-eral, a sentence F is true if and only if a full set of its Skolem functions〈f1, f2, . . .〉 exists. From a game-theoretical vantage point, such a fullordered set of Skolem functions defines a winning strategy for the ver-ifier in a semantical game G(F) connected with the sentence F whosetruth-value is to be evaluated. In this perspective, the existence of sucha winning strategy is what the truth of S means. From the same vantagepoint, we can also characterize explicitly the notions of dependence andindependence that are relied on here. Dependence and independence meansimply informational dependence and independence in the sense of gametheory.

In one sense, the slash notation used in (2) nevertheless is not a per-fectly self-explanatory one. This sense can be seen by pointing out thatthe existential quantifiers and identities in (2) are merely a way of ex-pressing the dependence and independence relations between the variablest1, t2, . . . , x1, x2, . . . . As such they must be considered in logical rules asparts of the quantifier prefix and not as parts of the expression in the scopeof the quantifiers.

Sequences of variables 〈x1, x2, . . .〉 in (1) can be said to representSkolem vectors or state vectors or simply vectors. Like any other simi-lar proposition, proposition (1) has the effect of restricting the admissiblestate vectors to a subspace of the state space. The propositional relationsbetween propositions correspond to certain relations between the corre-sponding subspaces. The crucial question is what the correspondence inquestion is.

For many purposes, it suffices to consider the simple case

(∀t)(∀x)(∀y)(∃z/∀x)(∃u/∀y)(4)

((x = z) & (y = u) & S[t, x, y]).

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Its Skolem function form is

(∃f )(∃g)(∀t)(∀x)(∀y)(5)

((x = f (t, y)) & (y = g(t, x)) & S[t, x, y]).In (4), t may be thought of as time and x, y as a pair of noncommutingvariables.

In the same way as the sequence of variables x1, x2, . . . can be thoughtof as a vector in a state space of all such vectors, in (3), the sequence ofSkolem functions 〈f1, f2, . . .〉 can be thought of as an operator in the usualsense, that is, as mapping of the state space into itself. It will be called aSkolem operator.

Now we can already ascertain a most interesting thing concerning thelogic of mutually dependent quantifiers. From (3) it can be seen that theadmissible state vectors are eigenvectors of Skolem operators, in analogywith the mathematics of the Hilbert space treatment of quantum theory.This is what it means for them, say for 〈x1, x2, . . .〉 to satisfy the condi-tions xi = fi(x1, x2, . . . , xi−1, xi+1, . . .) where 〈f1, f2, . . .〉 is a Skolemoperator.

We can also see what the problem of quantum logic amounts to. Thequestion is what logical relations correspond to the natural relations be-tween subspaces of a Hilbert space. In quantum theory, what is normallyconsidered are closed linear subspaces. However, counterparts to logicaloperators have to be defined for all propositions, corresponding to all arbi-trary subspaces.

One question here is: What relations between subspaces correspond tothe two kinds of negation? Contradictory negation obviously correspondsto complementation. But what about the strong negation ∼? The candidatefamiliar from the received quantum logic for the role of a counterpart to∼ is the orthocomplement of a closed subspace. It can now be seen thatorthocomplementation corresponds to the strong (dual) negation used inIF logic. Since we are considering only the state space of the mutually de-pendent variables, for instance the state space at a fixed time, we can lookaway from the quantifier (∀t) in (4) and the quantifiers (∀t1), (∀t2), . . . in(2)–(3). This is just like considering the Schrödinger equation as governingthe state of a quantum-theoretical system at some fixed time, looking awayfrom its diachronic applications. Thus instead of (4) we can consider

(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x = z) & (y = u) & S[x, y]).(6)

Now the strong negation of any sentence S means that there exists awinning strategy for the falsifier in the game G(S). For instance what thismeans in the case of the game correlated with (6) is that the falsifier has a

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way of choosing values of x and y in such a manner that she wins no matterwhat values of z and u are chosen by the verifier independently of x and y,respectively. This can naturally be expressed by an exchange of universaland existential quantifiers of the quantifier prefix plus a strong negationprefixed to the expression in their scope. Applied to (6) while keeping inmind that the existential quantifiers and identities belong to the quantifierprefix, this procedure yields

(∃x)(∃y)(∀z/∃x)(∀u/∃y)((x = z) & (y = u) & ∼S[x, y])(7)

But now it is obvious that the quantifier combinations

(∃x)(∀z/∃x) and (∃y)(∀u/∃y)(8)

are equivalent to

(∀z)(∃x/∀z) and (∀u)(∃y/∀u),(9)

respectively. (The equivalence of (8) and (9) offers an interesting exam-ple of a logical law characteristic of IF logic having no counterpart intraditional first-order logic.) Hence (7) is equivalent with

(∀z)(∀u)(∃x/∀z)(∃y/∀u)((x = z) & (u = y) & ∼S[z, u]).(10)

In the case of Hilbert spaces, this means that 〈x, y〉 of (7) and 〈z, u〉 of(8) are different eigenvectors of the same operator. But it is known fromthe theory of Hilbert spaces that two eigenvectors of the same operatorare always orthogonal. Hence, the vectors 〈x, y〉, 〈z, u〉 are in the case ofa Hilbert space orthogonal to each other (see, e.g., Bub, 1997, p. 251).This is enough to show that the two subspaces, viz. those of vectors 〈x, y〉satisfying S[x, y] and those satisfying ∼S[x, y] are orthogonal to eachother. In brief, strong negation corresponds to orthogonality in the case ofa Hilbert space.

Hence we can conclude that the vectors corresponding to the strongnegation ∼ are all orthogonal to the vectors in the given subset of theHilbert space. The subspace corresponding to ∼ is therefore included inthe orthogonal complement of the given one. It is also easily seen that theconverse inclusion holds. Thus the strong negation ∼ corresponds in thecase of Hilbert spaces to orthocomplementation.

This is an interesting result in that it can be thought of as suggestingto us a way of defining a generalized notion of orthogonality for an arbi-trary formula expressed in an IF first-order language, simply by taking thedual negation ∼ as giving the counterpart of the orthogonal complement.Moreover, we can extend the same correlation between strong negation and

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orthogonal complementation to arbitrary subsets of the state space. Whatis meant is that if S corresponds to any subset σ , then ∼S corresponds tothe orthogonal complement of the linear closure of σ .

Furthermore, the conjunction of two propositions corresponds to the in-tersection of the corresponding subspaces. Hence we must have in our vonNeumann–Birkhoff style “quantum logic” all those IF laws that depend on∼ and & only.

But when we begin to consider disjunction, the picture changes. If theclosed linear subspaces σ1, σ2 correspond to the propositions S1, S2, then,as von Neumann and Birkhoff (1936, reprint, p. 6) already noted,

∼(∼S1&∼S2)(11)

corresponds to the closed linear union of σ1 and σ2. But if σ1 and σ2 arearbitrary sets, represented by S1 and S2, then the quantum-logical counter-part to their quasi-disjunction (in other words, their closed linear union) isnot (11), for (11) equals (S1 ∨S2), that is, (11) represents the set-theoreticalsum of σ1 and σ2. The question then becomes: How can we represent theclosed linear union of σ1 and σ2 by means of IF logic in terms of S1 andS2? A moment’s thought shows that the way to do it is to take the set-theoretical complement of the intersection of the orthogonal complementsof σ1 and σ2. The result can be represented in a logical language by

¬(∼S1&∼S2),(12)

where ¬, is the contradictory negation and hence takes us to the extendedIF logic.

In this way, we can see what the von Neumann–Birkhoff quantum logicis. It operates with the primitives indicated, viz., negation (correspond-ing to joint ortho-complementation), conjunction (corresponding to set-intersection) and pseudo-disjunction (corresponding to linear closure). Interms of IF first-order logic, these amount to ∼, & and (¬ ∼S1 ∨ ¬ ∼S2)

(instead of (S1 ∨ S2)). The received quantum logic is thus interpretablein extended IF first-order logic, and can be considered a fragment of thislogic.

To put the same point differently (and to expand it), the received quan-tum logic turns out to be, not an alternative to the usual logic, but a frag-ment of a richer logic that has a simpler structure and is in reality the trueunrestricted logic of quantifiers.

It is in a sense predictable even on the purely formal level what theinterpretation of the received quantum logic in extended IF logic must be.In order to compare a logic like IF logic with a lattice structure, as vonNeumann and Birkhoff were doing, you must have a maximal and a min-imal element in your structure. And maximality and minimality require,

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normally interpreted, that the laws of contradiction and excluded middlehold. Now only the former holds in IF logic, whereas the law of tertiumnon datur fails. However, it is eminently natural to try to enforce also thelatter law by interpreting disjunction of S1 and S2 as a lattice function by

(¬∼S1 ∨ ¬∼S2).(13)

This restores formally the law of excluded middle and hence yields a latticestructure. Then many of the usual laws of logic hold, as most of themdo in IF logic, including the laws definitory of lattices. However, on thisinterpretation the lattice-theoretical quasi-disjunction (13) and the normalconjunction are no longer mirror images (duals) of each other, whereforesome of the distributivity laws relating them to each other fail. The residueof distributivity that remains in force is precisely orthomodularity.

The adequacy of this interpretation of orthomodular logic within IFpropositional logic is easily seen. For the purpose, it can be noted thatboth ∼ and ¬ (considered separately from one another) behave vis-à-vis& and ∨ in the usual way. This includes of course the usual distributivitylaws.

To return to our main subject, it is also predictable on the formal levelthat the interpretation of von Neumann–Birkhoff logic should turn outto be an orthomodular lattice. What happens is that the von Neumann–Birkhoff “disjunction” (12) does not have the same dual relation to con-junction as & and v have. Hence it is only to be expected that some dis-tributivity laws should fail.

It is indeed easy to show that the laws characterizing orthomodularlattices are valid on the interpretation just presented. The telltale ortho-modular law says, in the notation used in the usual expositions (cf., e.g.,Rédei, 1998, pp. 35–36), that if A ≤ B, then B = A ∨ (∼A ∧ B). In ournotation the premise amounts to the statement

A = (¬∼A ∨ ¬∼B)(14)

and the conclusion to

B = (¬∼A ∨ ¬∼(∼A & B)).(15)

Here the right-hand side of (15) equals

(¬∼A ∨ ¬∼∼A) & (¬∼A ∨ ¬∼B).(16)

Here the first conjunct is logically true and the second is the premise, whichproves the orthomodular law.

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The modular law does not hold, either. In the terms used here, it saysthat if A � B, then

A ∨ (B & C) = (A ∨ B) & C,(17)

where ∨ is the Birkhoff–von Neumann disjunction examplified by (12).Here (17) can be rewritten in the standard IF logic notation as

¬∼A ∨ ¬∼(B & C) = (¬∼A ∨ ¬∼B) & C.(18)

If we substitute A for B and ∼A for C, we obtain as a special case of (18)

¬∼A ∨ ¬∼(A& ∼A) = (¬∼A ∨ ¬∼A)& ∼A.(19)

The left-hand side of (19) reduces to ¬ ∼A, while the right-hand side iscontradictory. Hence (18) is not valid. Thus it is shown that we are dealingwith an orthomodular structure here.

In more general terms, we can now see that the Birkhoff–von Neumannquantum logic can be mapped into extended IF logic. In this logic, notonly is orthocomplementation interpretable, but everything else as well.Moreover, this extended IF logic has an interesting structure. In terms of¬, & and ∨ it forms a Boolean structure. But over and above this Booleanstructure, it has also the additional structure induced by ∼. This new struc-ture remains to be investigated. If the new operator is taken to be ¬ ∼,the resulting structure is a Boolean algebra with an operator in Jonsson’sand Tarski’s (1951) technical sense. For Tarski requires of the additionaloperator that it be distributive with respect to set-theoretical addition, andwe do indeed have ¬∼(S1 ∨ S2) = (¬∼S1 ∨ ¬∼S2).

The holding of all Boolean laws is interesting, for h means that we canapply probabilistic concepts in quantum logic without changing the usuallaws of probability calculus. The main novelty is that while the probabili-ties of A and ¬A always add up to one, the probabilities of A and ∼A donot. For the issues involved here, see Rédei (1998, 1999).

The question as to whether the Birkhoff–von Neumann logic admitsof an interpretation in the sole terms of suitable Boolean operations, inother words, in terms of ordinary first-order logic, is closely related to thequestion of the possibility of a hidden variable interpretation (cf. here Bub,1997, Chapters 2–3). The basic idea has been to show that Birkhoff–vonNeumann logic cannot be interpreted in a Boolean structure, i.e. mappedinto it. The viewpoint we have reached helps to show what is essential here.We have seen that each Birkhoff–von Neumann structure can in a sensebe mapped into a Boolean structure of a kind viz. one has the additionalstructure imposed on it by the strong (dual) negation ∼. However, in this

QUANTUM LOGIC AS A FRAGMENT OF IF LOGIC 205

mapping the orthomodal sum (12) is not mapped on disjunction nor ∼ onthe Boolean (contradictory) negation ¬.

Hence, this kind of mapping does not amount to a reduction of the re-ceived quantum logic to classical (Boolean) logic. What makes theBirkhoff–von Neumann logic irreducible to ordinary first-order logic is thepresence of uneliminably IF ingredients. The use of the dual negation ∼does not at first show up in the logical behavior of the different constants,for ∼ behaves in the same way with respect to ∨ and & as does ¬. For thisreason, in the Birkhoff–von Neumann logic the nonstandard behavior be-comes overt only through the tacit re-interpretation of disjunction as (12).But in fact any ingredient that introduces irreducibly IF sentences makesit impossible to interpret the resulting logic in the usual two-valued first-order logic. The reason is that it can be shown that a sentence of extendedIF logic obeys the law of excluded middle only if it belongs to the ordinaryfirst-order logic. Hence an irreducibly IF sentence cannot be equivalent toan ordinary first-order sentence. This means that an irreducibly IF struc-ture cannot be mapped into a Boolean structure and have its relations ofdependence and independence preserved.

The nonequivalence just relied on can be shown as follows: Assumethat S is an IF sentence that obeys the law of excluded middle, so that(S ∨∼S) is logically true. Now the separation theorem holds in IF first-order logic in a strengthened form, in that the separation formula canalways be chosen to be an ordinary first-order one. Then by applying theseparation theorem to {S} and {∼S} it can be seen that S and ∼S areequivalent to ordinary first-order sentences.

Such observations also show that the Birkhoff–von Neumann strategyof reinterpreting disjunction in order to obtain a lattice structure is notthe most clarifying one. All we obtain in this way is a substructure of thesimpler and much more fundamental structure of extended IF first-orderlogic.

In conclusion, we can now see more clearly the relation of the vonNeumann–Birkhoff quantum logic and the “classical” Frege–Russell first-order logic. The von Neumann–Birkhoff lattice structure is determined bythe structure of the Hilbert space representation of quantum theory. Butit is so determined partly because the Hilbert space structure instantiatesrelations that cannot be captured by means of Frege–Russell logic but canbe captured by IF first-order logic. Hence we can accept the claim that inorder to deal with quantum phenomena we need a new logic different fromthe classical Frege–Russell one. But that new logic is IF logic, that is, alogic that is not motivated by the Hilbert space structure of quantum theorybut by reasons that are far more general than any question in physics.

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The von Neumann–Birkhoff quantum logic is merely a fragment of thistruly general first-order logic. It is IF logic, not the von Neumann–Birkhoffquantum logic, that is a genuine alternative to – and improvement on – theFrege–Russell logic. And in this sense the von Neumann–Birkhoff logic isnot determined by the mathematical apparatus of quantum theory, but re-flects its logical structure. We are nevertheless here dealing with questionsthat are connected with important issues in logical theory. In order to havea glimpse of them, consider the von Neumann–Birkhoff counterpart

(A ∨ ¬A)(20)

to the classical tautology (A ∨ ¬A). When unpacked in terms of extendedIF logic, (A ∨ ¬A) becomes

(¬∼A ∨ ¬∼¬A).(21)

This is not well-formed, but in propositional context, it can be assumed tohave the force of

(¬∼A ∨ ¬A).(22)

This says in effect that either A or ∼A is false. In game-theoretical terms, itsays that there is a winning strategy for one of the two players in the seman-tical game connected with the contradictory negation ¬Aof any sentence A.

It is quite possible that this should be false. Indeed, since the law ofexcluded middle fails for all nonreducibly IF sentences, this will typicallyhappen in IF languages. Moreover, it is possible that this failure to beeither true or false should be due to conceptual reasons expressible in thesame language. Something like this happens in languages in which theirown truth is expressible in a Gödel-type sentence asserting that it is itselfnot true. Then (20) would be logically false even though its “classical”counterpart is logically true.

This line of thought is closely related to the use of the Kochen andSpecker theorem for the purpose of ruling out hidden variables (see Bub,1997, Chapter 3). They, too, show that the quantum-theoretical lattice can-not be imbedded in a Boolean structure (with ∨ interpreted as a Booleandisjunction). They, too, construct a sentence which is a classical tautologywhen disjunction is taken to be ∨ but a contradiction when it is taken tobe ∨ .

The approach used here can be used to do more than to interpret thevon Neumann–Birkhoff lattice logic in extended IF logic. Even though theexamination of those further possibilities needs a separate study, it is inorder to indicate some of them so as to illustrate the naturalness of theapproach used here.

QUANTUM LOGIC AS A FRAGMENT OF IF LOGIC 207

First, it has been found that orthogonal complementation can be ex-pressed in the language of IF logic. In other words, it has been found thatthe logical counterpart to orthogonality relations is expressible in the ex-tended IF logic. For instance, the notion of dimension can be so expressed.A model of an extended IF language has the dimension d if there can be atmost d propositions B1, B2, . . . , Bd which are pairwise orthogonal, that is,satisfy the conditions (Bi ⊃∼Bj) or, equivalently (∼B1 ∨ ∼B2) as soonas i �= j . Then for each proposition A the following must be true:

¬((∼A∨ ∼B1) & (∼A∨ ∼B2) & · · · & (∼A∨ ∼Bd))(23)

or equivalently

¬∼((A & B1) ∨ (A & B2) ∨ · · · ∨ (A & Bd)).

By substituting for A a logically true proposition we obtain

¬∼(B1 ∨ B2 ∨ · · · ∨ Bd)(24)

or, equivalently

¬∼B1 ∨ ¬∼B2 ∨ · · · ∨ ¬∼Bd.

Hence for each proposition A it is true that A if

(A & ¬∼B1) ∨ (A & ¬∼B2) ∨ · · · ∨ (A & ¬∼Bd).(25)

This shows that a kind of component representation is always obtainablein a d-dimensional model.

All this is in analogy with the geometrical situation.Moreover, this representation must have a nondeterministic element, in

the following sense. If for some A all the disjuncts (A & ¬∼Bi) shouldreduce to (A &Bi), then the representation (23) would be “classical”, thatis to say, representable in ordinary first-order logic. But this is impossibleif A is irreducibly IF. Hence, for any A there must be at least one i suchthat Bi and ¬∼Bi are not equivalent.

Thus much of the theory of orthogonal relations can be dealt with bymeans of extended IF logic.

An even more general possibility concerns the probabilistic aspects ofquantum theory. As the received first-order logic can be imbedded in asuitable probability logic (cf. Scott and Krauss, 1966), so can the extendedIF logic be imbedded in a kind of probability logic. This subject is toolarge, however, to be dealt with in a single paper.

The main idea is nevertheless worth indicating here. The difficulty aboutextending probability logic to IF logic is that quantified sentences can-not any longer be considered as long disjunctions and conjunctions. This

208 JAAKKO HINTIKKA

difficulty can be overcome by replacing quantification over dependent vari-ables by quantification over Skolem operators. In a simple case not involv-ing mutual dependence, the probability measure of

(∀x)(∃y)S[x, y](26)

is the measure of the set∧

x

∨

y

{〈x, y〉: S[x, y]},(27)

where x, y ∈ D, 〈x, y〉 ∈ D2. But it can also be expressed as the measureof

∨

f∈F

∧

x

{〈x, y〉: y = f (x) & S[x, y]},(28)

where F is the class of all Skolem functions of (26). (In the general case,the indexes will of course range of all Skolem operators rather than Skolemfunctions.) The point is that (28) unlike (27) can be generalized to propo-sitions with mutually dependent quantifiers.

Of the many other implications of what has been found in this paper,only one of the most general ones will be mentioned here. W. V. Quine hasargued that there is in the last analysis no sharp distinction between con-ceptual truths, such as logical ones, and empirical truths. In principle, evenour laws of logic could be modified in the interest of our overall theoryof the world. Hilary Putnam (1979) has suggested that the replacement of“classical logic” by quantum logic in quantum theory can be considered acase in point. Even though I am proposing to change the logic that is beingused in trying to understand quantum theory, what is argued in the presentpaper is radically different from such views. Putnam proposes changinglogic for physics’ sake; my starting point is a change in (or, rather, an ex-tension of) our basic logic for logic’s sake, or more explicitly, for the sakeof the expressive power of a logical language. The rest of the argument iscalculated to show that once this extension has been carried out, we do notneed any separate new quantum logic. The branch of IF logic dealing withmutually dependent quantifiers is the real quantum logic.

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Department of PhilosophyBoston University, USAe-mail: hintikka@bu.edu