Reviews : Etudes critiques–Buchbesprechungen

ReviewsEtudes critiques – Buchbesprechungen

Hintikka, J., The Principles of Mathematics Revisited, Cambridge UniversityPress, Cambridge 1996, pp. xii + 288.

Hintikka’s book presents an innovative approach to quantification in the framework ofgame-theoretic semantics and relies on such a framework for tackling the main traditionalsubjects in the philosophy of logic and in the philosophy of mathematics. The title couldsuggest that the very content of Russell’s Principles of Mathematics is to be revisited, butthat is not the case. The title is rather motivated by an analogy of style and aim. Let usremember that Russell’s aims were “the proof that all pure mathematics deals exclusivelywith concepts definable in terms of a very small number of fundamental logical concepts,and that all its propositions are deducible from a very small number of fundamental logi-cal principles” and “the explanation of the fundamental concepts which mathematicsaccepts as indefinable” (p. xv). Russell, like Frege before him, introduced new logicalnotions and, relying on them, supplied a new philosophical understanding both of languageand of mathematical concepts. Hintikka’s book starts by taking into account first orderlogic, which corresponds to part of the logical work done by Frege and Russell. Hintikkaextends such a logic by introducing a new notion of quantifier and interprets the logicalconstants from the point of view of game-theoretic semantics. Both the new notion of quan-tifier and the game-theoretic semantics were already presented by him in other writings. Inthis book Hintikka relies on the concept of an independent quantifier to propose a new wayof understanding the basic mathematical concepts. From this point of view the analogy withRussell’s Principles is justified. On the other hand it is not clear if what has been done bystarting from first order logic and going to higher logic (Russell’s way) or developing a firstorder set theory (Zermelo-Fraenkel’s way) can be reconstructed in the game-theoreticsemantics framework enlarged by adding the independent quantifiers.

The main idea of game-theoretic semantics is that, given a model M of the language,it is possible to associate to every sentence an ideal game played by two players, one ofthem aiming at truth and the other one aiming at falsity. The game is played by apply-ing rules which are meant to fix, or to conform to, the meaning of the logical operators.The application of a rule introduces a simpler game. At the end atomic sentences arereached and the game is won by the player whose claims are right. The rules for thegame G(S), where S is a sentence, are the following:

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Etudes critiques – Buchbesprechungen – Reviews 283

If S is S1 ∨ S2, the verifier chooses Si (i = 1 or i = 2) and the game goes on as G(Si).

If S is S1 ∧ S2, the falsifier chooses Si (i = 1 or i = 2) and the game goes on as G(Si).

If S is ∃ x S[x], the verifier chooses a member, say b, of the domain of M and thegame goes on as G(S[b]).

If S is Ax S[x], the falsifier chooses a member, say b, of the domain of M and thegame goes on as G(S[b]).

If S is +S, the verifier and the falsifier interchange their roles and the game goes onas G(S).

If S is an atomic sentence, then if S is true the verifier wins (and the falsifier loses)and if S is false the falsifier wins (and the verifier loses).

The truth values of atomic sentences are taken for granted. In general, given a specificmodel, a sentence S is defined as true in it when there is a winning strategy for the ini-tial verifier in the game G(S) and is defined as false when there is a winning strategyfor the initial falsifier in the game G(S).

Hintikka starts speaking of a winning strategy for a player as “a rule that determineswhich move that player should make in any possible situation that can come up in thecourse of a play of that game” (p. 27) and such that the player who follows it wins, butlater he presents a winning strategy as a set of functions. At the end of the book Hin-tikka discusses the possibility and the desirability of a limitation to recursive functions.Hintikka thinks that such a limitation is possible but not desirable. However, he exploitsit to reconstruct a constructivistic logic, which however is not intuitionistic logic proper.

The existence of a winning strategy for the initial verifier (falsifier), and not itsknowledge by the initial verifier (falsifier), is what makes a sentence true (false). Thisexplains why excluded middle holds for the language of first order logic. Even in thecase in which a winning strategy is expressed by a rule, its existence for the initial ver-ifier in the game G(S) only requires that the initial verifier possesses a potential abilityto move according to the rule defining the winning strategy. As matter of fact, he couldnot know the rule or, at least when empirical matters are involved, he could lack theinformation required to apply the rule. This should make completely clear that truth isnot understood from a constructivistic point of view, as conceived, by example, by Dum-mett.

Things are less clear concerning the relation to the usual classical semantic approach.The axiom of choice allows us to prove that S is true in M according to the game-theo-retic semantics if and only if it is true in M according to the usual model theoretic seman-tics. For the existence of a winning strategy for the initial verifier in the game-theoreticsemantic analysis of a sentence of the form

Ax ∃ y S(x, y)

is equivalent to the truth of the following sentence:

∃ f Ax S(x, f(x))

Hintikka holds that “∃ y“ in “Ax ∃ y S(x, y)“ is a paradigmatic example of a dependentquantifier and that the axiom of choice contributes to the analysis of this notion. How-ever this notion would only be a part of the logic of quantifiers. It is also possible to

284 Etudes critiques – Buchbesprechungen – Reviews

existentially quantify independently of a previous (universal) quantification, as it hap-pens, according to Hintikka, in actual mathematical practice. The independence from aprevious quantifier is expressed by Hintikka by means of a “/” followed by the previ-ous quantifier as in

Ax (∃ y/Ax) S(x, y)

Adopting the game-theoretic semantics point of view, Hintikka explains “that the choiceby the verifier of a value of y must be made in ignorance of the value of x” (p. 51). Sucha formula is logically equivalent to the first order formula

∃ y Ax S(x, y)

Hintikka stresses that for other kinds of formulas there is no first order equivalent for-mula. His example is

Ax Az (∃ y/Az) (∃ u/Ax) S(x, y, z, u)

From this kind of formula it can be seen that in order to warrant the independence fromAx for the choice relative to (∃ u/Ax) the verifier must be conceived as a team of two play-ers, one choosing y and the other one choosing u, otherwise information on x could beattained from the information on y. In general, Hintikka assumes that every existentialquantifier is independent of any previous existential quantifier.

The above formula could be written in branching notation

Ax ∃ y

S(x, y, z, u)

Az ∃ u

but Hintikka claims that the /-notation allows a more general approach. The resultingnew logic is called by Hintikka independence-friendly (IF) first order logic. He presentsit by referring only to formulas in prenex negation normal form and only from thesemantic point of view, by emphasizing its expressive power and neglecting to developa corresponding notion of proof.

In IF logic, there are sentences which are neither true nor false in the defined senses.A simple example of such formulas is Ax (∃ y/Ax) (x=y), since for every domain of morethan one element neither the initial verifier nor the initial falsifier has a winning strat-egy in the game associated to it. So Hintikka specifies that when a sentence containingindependent quantifiers is said to be equivalent to a usual first or second order sentence,equivalence must be taken only as truth in the same models.

Each sentence of this logic can be translated into a second order Σ11 sentence, which

is equivalent to it in Hintikka’s sense. It follows that the downward Löwenheim-Skolemtheorem holds also for the formulas of the IF logic. Other metatheoretical properties ofstandard first order logic hold for IF logic. However, the logical truths of such logic, likethe logical truths of second order logic, are not axiomatizable. In spite of the non-axiomatizability and the translatability of IF formulas into equivalent second order for-mulas, Hintikka tries to assimilate IF first order logic to the standard first order logic,by presenting the latter as a special case of the first. Hintikka claims that the notion ofan independent quantifier belongs to the first order level and, as such, does not concep-tually involve second order ideas.


Hintikka credits Frege with having discovered the notion of informational depend-ence paradigmatically expressed by Ax ∃ y, but criticises him for having neglected theequally relevant notion of informational independence paradigmatically expressed byAx (∃ y/Ax). From the game-theoretic semantics perspective, Frege’s logic would be agame of perfect information, i.e. a game in which all information (given by the previ-ous moves and choices) is available to the players. Hintikka introduces informationalindependence to mean that some information is not available, i.e. some previous movesand choices are not known by the relevant player when he makes his choice. Ignoranceof a move, done by a player of the verifier team, means that its informational content,i.e. the choice made via it, cannot be attained by another player of the same team dur-ing the game, but does not exclude, when possible, that it be fixed in advance, by apply-ing a suitable winning strategy. This is the case of a sentence like ∃ x ∃ y (x = y), where∃ y is independent of ∃ x, but the choice for x and y can be fixed in advance to be iden-tical to some n. So there is no need to correct Hintikka on this point, as R. Cook andS. Shapiro have done in their review article “Hintikka’s Revolution: The Principles ofMathematics Revisited” (British Journal of Philosophy of Science, 49 (1998), pp. 309-316).

Two important related subjects discussed by Hintikka are negation and truth. Hin-tikka is able to show that arithmetical truth is definable, in the sense that the existenceof a winning strategy for an arithmetic sentence is expressible by an IF formula of thearithmetical language. There is such a possibility because negation of truth, which isweaker than falsity, cannot be expressed in the IF logic and therefore T-biconditionalsdo not hold in general. Addition of contradictory negation increases the expressivepower of IF logic, making infinite, well-ordering, mathematical induction and othernotions expressible, but also making truth for the so extended arithmetical language notdefinable in it. Hintikka underlines that his definition of truth is not given in a compo-sitional way and tries to motivate why compositionality should not be a desirablerequirement.

In general Hintikka thinks that IF first order logic allows to avoid both higher orderentities and set-theoretical principles in mathematical theorizing. Although it is not clearthat it is so, the reasons and ideas which underlie his claim are worth considering. In thisconnection it should be noticed that in some not initial parts of the book Hintikka takesthe existence or the non-existence of a winning strategy as a combinatorial fact. It wouldbe interesting, and quite in line with Hintikka’s foundational aims, to expand and to clar-ify this way of understanding his notions of truth and falsity, even beyond the ideally epis-temic framework of game-theoretic semantics, since an epistemic idealization does notseem to be able to justify the great generality of the adopted notion of winning strategy.

Hintikka’s book is not carefully written. There are omissions and wrong formulas,also in the more formal appendix “IF First-Order Logic, Kripke, and 3-Valued Logic”by Gabriel Sandu. Metamathematical results are not always correctly reported and com-mented. The already mentioned review by Cook and Shapiro and, above all, the reviewby Neil Tennant in Philosophia Mathematica (6 (1998), pp. 90-115) have alreadypointed out most, and perhaps all, shortcomings. Still, the book is commendable. It isinsightful and raises deep questions.

Pierdaniele GiarettaUniversità degli Studi di Padova


Passioni e Ragioni. Un itinerario nella filosofia della psicologia, byClotilde Calabi. Milano: Edizione Angelo Guerini e Associati, 1996.Pp. 157.

Emotions are often taken to be opposed to reason. Some, like the Stoics, went so far asto claim that emotions are illnesses of the soul which one has to try and get rid of. Bycontrast, many contemporary thinkers, be they philosophers or psychologists, haveattempted to show that a proper understanding of the mind reveals intimate relationsbetween emotions and mental states such as perceptions, beliefs and judgements; insteadof being seen as waging a war against reason, emotions are thought to be part of a nor-mal cognitive set. This approach is characterised by a more positive evaluation of emo-tional phenomena. Clotilde Calabi’s book stands firmly in this tradition. The main claimwhich is argued for is that emotional experiences are objective and similar to perceptualexperiences (116).

Independently of the intrinsic interest of the questions which it raises, this neat lit-tle book recommends itself by its clarity, the careful way in which claims and argumentsare presented and the wealth of insights into our emotional life. Another remarkable fea-ture consists in the detailed and informed discussions of the thoughts of major histori-cal figures, such as Hume, Descartes, William James and Gilbert Ryle.

The book is divided into two main parts. The first of them offers an analysis of pride,one of Hume’s favourite emotions. The Humean distinction between the object of prideand its cause is discussed and clarified. The related distinction between the cause of anemotion and its reason is also examined. In agreement with most contemporary philoso-phers, Calabi claims that beliefs, and in particular evaluative beliefs, constitute an essen-tial component of pride: to be proud of possessing a beautiful house, I need to believethat I own a beautiful house. In reply to well-known difficulties with the thesis that sucha belief is both causally and conceptually related to the emotion, the author follows Don-ald Davidson’s claim that depending on the chosen descriptions of the cause and theeffect, causal relations can be ascribed by analytic propositions. Thus, the analyticity ofthe proposition that the cause of the tempest caused the tempest is perfectly compatiblewith there being a causal relation involved. I should like to add that there is room fordiscussion here. One can accept the Davidsonian claim and still have some doubts whereemotions are concerned. A characteristic of the tempest case is that there is a descrip-tion under which the cause can be specified independently of its effect. Thus, the ques-tion is whether there is a similarly independent description of pride. If one takes seri-ously the claim that it is an essential feature of pride that it is related to certain beliefs,the answer seems to be that there cannot be such a description.

The second part of the book is more general in its scope. It considers some of themain classical theories of emotions – that is, those of Descartes, James, Watson and Ryleamong others – and traces the relations between these theories and the conceptions ofthe mind in which they are embedded. Dismissing one theory after another, the authormakes her way to the account she favours. This account, which shares features with theso-called adverbial theory of perception, underlines the analogy between emotions andperceptions. We are told that an emotion is “an experience that asserts the existence ofdeterminate objects or states of affairs which possess particular properties (there are joy-

Dialectica Vol. 55, Nº 3 (2001), pp. 286-288


ful events, nice or odious persons, etc.)” (116). One way to interpret this claim is to readit as rather implausibly claiming that emotions are or involve evaluative judgements.However, this is not what the author means. In fact, Calabi denies that emotions can bedecomposed into evaluative judgements, on the one hand, and feelings, on the otherhand. Following C.D. Broad, she makes the following claim:

But it does not happen that I see the dog, judge that it is growling, evaluate that it is dan-gerous and, as a consequence, feel fear. I directly perceive the dangerousness of the dogand my perception is an experience of fear, that is, a perception characterised by a par-ticular affective tone. (120)

The idea is that emotions are non-evaluative cognitive states which possess certain phe-nomenal qualities. Thus, paraphrasing Broad, one could say that according to this view,being afraid of a dog is seeing the dog fearingly – hence the analogy with the adverbialtheory of perception, according to which ‘I see a square’, for instance, is taken to mean‘I see squarely’, a proposition which does not seem to refer to sense-data. Seeing thedog in this way means that certain features of what we perceive emerge as salient:

To say that the dog is dangerous involves seeing as salient some of its characteristics,such as its dimensions, its sharp teeth, its barking, its frontal position, as opposed to thecolour of its fur or to its race (...). (124)

The question which arises is how to understand the relation between the phenomenalproperties of the experience and the patterns of salience (to use a phrase coined byRonald de Sousa, The Rationality of Emotion, Cambridge, Mass., MIT Press, 1987). Weare told that the affective tonality corresponds to an order of salience (125). However, ifthis is taken to mean that there is nothing more to an affective tonality that an order ofsalience, some doubts are bound to arise. It is not clear that having one’s attention takenby the size, the teeth, etc. of a dog amounts to having an experience of fear. One cansurely close one’s eyes while still fearing the dog. And it seems one could focus one’sattention on the relevant features from mere curiosity. This suggests that the perceptionof the dog and its salient features has to be distinguished from the fear of the dog. Thisis not to deny, of course, that the fear in question is based on the perception of the dogand also that emotions are liable to determine a certain order of salience. Moreover, thethesis that perceiving the dog differs from fearing it is also compatible with the claimthat the perception of the dog and the emotion of fear do not have to follow each othertemporally: the perception of the dog can coincide with the arousal of fear. The point issimply that the same perception could have failed to trigger fear. To take this point intoaccount, it can be claimed that in addition to a perception with a certain order of salience,fear also requires proprioception of a particular kind, that is, an awareness of the bodilychanges which go with fear. This seems to be the author’s claim when she says that “theaffective tonality corresponds to an order of salience on the one hand and is closelyrelated to proprioception on the other hand” (125). If so, it would be possible to perceivea dog with the order of salience which is characteristic of fear while not feeling any emo-tion of fear.

The book closes with a discussion of the fascinating question whether machinescould someday feel emotions. This depends on whether emotions can be defined inpurely functional terms. The answer seems to be negative in so far as the emotions wefeel depend on our physiology – emotions involve events in viscera, skeletal muscles


and endocrine glands. The author ends with an analogy between the question whethermachines can feel emotions and that, discussed by Casanova, whether the true pleasureof smoking a pipe is one of the soul or of the senses. Casanova’s answer is simply thatwhat seems a pleasure is pleasure. In the same way, the question whether machines canfeel emotions depends, we are told, on whether it can seem to a machine that it has anemotion. Such a seeming would be sufficient and cannot be ruled out. It can be added– and Clotilde Calabi would probably agree – that as long as the structure of the machinediffers from our physiology, its states could hardly be claimed to be of the same kind ashuman emotions. So, whether the true pleasure of smoking a pipe is one of the sensesor one of the soul, it does not seem that machines are likely to ever experience it.

Christine TappoletUniversité de Montréal

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