Strahm, Thomas

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42 REVIEWSFinally, McGinn’s inquiry into the ﬁve logical properties assumes an uncommon method-ological stance which, arguably, is dictated by the realist view of these properties. In general,since truth conditions by themselves underdetermine semantics, our conception of the se-mantic properties of terms should dictate the assignments of truth-conditions to the sentencesin which the terms occur, not the other way around. Hence McGinn’s contention that thequantiﬁer is vastly over-rated as a tool of logical and linguistic analysis. We should notbe seduced by the success of the reduction of identity, existential, and modal sentences toquantiﬁcations over properties, things, and possible worlds, respectively. Even if these re-ductions ﬁx the truth-conditions for identity, existence, and modal sentences, the appeal toquantiﬁcation distorts the semantic facts about ‘=’, ‘exists’, and ‘possibly’ because it fails toyield adequate analyses of the concepts of identity, existence, and modality. More generally,McGinn believes that there are substantive facts about the semantic functioning of the syn-tactic correlates of the ﬁve logical properties, and that these facts are epistemically accessibleto us. McGinn gets to these semantic facts by ﬁrst doing metaphysics—getting an accountof the nature of the logical properties—and then using the results to develop an account ofthe semantic functioning of the corresponding terms.Thus, semantics cannot be the royal road to metaphysics, because the correct semantics isconstrained by the correct metaphysics. Surely, on some level this is correct. We can’t argueagainst the Parmenidean by claiming that there exists a counterexample to the argument:some are eternal, so all are eternal. Since the counterexample requires the possibility ofthere being more than one thing, the required semantics of the quantiﬁers presupposesthat Parmenidean metaphysics is wrong. McGinn’s work highlights the extent to whichmetaphysics penetrates and permeates the conceptual framework of logic. Inquiry into thecentral concepts of logic cannot ignore metaphysics. Even if one does not agree with allthe conclusions of the book, one has to grant that Logical Propertiesmakes a good case forbringing, as McGinn puts it, philosophy back into philosophical logic.Matthew McKeonDepartment of Philosophy, Michigan State University, East Lansing, Michigan, 48824-1032, USA.Sergei Tupailo. Realization of analysis into explicit mathematics. The Journal of Sym-bolic Logic, vol. 66 (2001), pp. 1848–1864.The paper under review explores realizability interpretations of various intuitionistic sub-systems of second-order arithmetic into Feferman’s explicit mathematics. For the lattersystems, this therefore provides an alternative method to establish their lower proof-theoreticbounds.Explicit mathematics was introduced by Feferman (A language and axioms for explicitmathematics, JSL XLIX 308(1); Constructive theories of functions and classes, JSL LXIX308(2)) in the seventies. Beyond its original aim to provide a basis for Bishop-style con-structivism, the explicit framework has gained considerable importance in proof theory inconnection with the proof-theoretic analysis of subsystems of second-order arithmetic andset theory. The most famous example in this connection is the reduction of the classicalsubsystem of second-order arithmetic based on Δ12 comprehension and bar induction to themost prominent framework of explicit mathematics, 0, achieved by Ja¨ger (A well-orderingproof for Feferman’s theory T0, Archiv fu¨r mathematische Logik und Grundlagenforschung,vol. 23 (1983), pp. 65–77) and Ja¨ger and Pohlers (Eine beweistheoretische Untersuchung von(Δ12-CA) + (BI) und verwandter Systeme, Sitzungsberichte der Bayerischen Akademie derWissenschaften, Mathematisch-naturwissenschaftliche Klasse, 1982, pp. 1–28).The language of explicit mathematics, more precisely its operational or applicative core,allows for a very smooth and elegant formulation of abstract Kleene-style notions of realiz-ability. Moreover, even realizability for second-order languages ﬁnds its natural place, sincehttps:/www.cambridge.org/core/terms. https://doi.org/10.1017/S1079898600004741Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 15:47:47, subject to the Cambridge Core terms of use, available atREVIEWS 43collections of (ﬁrst-order) individuals, so-called classes or types, are directly named or rep-resented by objects of ﬁrst order, which in turn can act as arguments of abstract operationsin the applicative basis of explicit mathematics.The subsystems of intuitionistic second-order arithmetic that are shown in this articleto be realizable into the system 0 and its subsystems are based on schemes of arithmeticcomprehension, choice, and replacement, as well as a principle of inductive generation. Thestrongest of these systems is the theory   introduced by Griﬀor and Rathjen (The strengthof some Martin-Lo¨f type theories, Archive for Mathematical Logic, vol. 33 (1994), pp. 347–385), which has the same proof-theoretic strength as Δ12 comprehension plus bar induction,hence exhausting all of 0.The present paper explores realizability in explicit mathematics of the relatively simplelanguageof second-order arithmetic. This keeps the amountof technical details at aminimumand demonstrates the method most distinctly. The paper can be seen as an importantpreparatory work for two subsequent papers by Tupailo, dealing with a much more involvedrealizability interpretation of constructive set theory 	, possibly augmented by largeset axioms, into explicit mathematics (Realization of constructive set theory into explicitmathematics: a lower bound for impredicative Mahlo universe, Annals of Pure and AppliedLogic, to appear; On non-wellfounded constructive set theory, to appear in the proceedingsof the Ninth Annual CSLI Colloquium on Logic, Language and Computation; both papersare available at  	  ). The former of thesetwo papers provides a relative lower bound of the impredicative Mahlo universe in explicitmathematics.In summary, the author oﬀers an interesting, concisely written paper, which directly relatesstrong subsystems of intuitionistic analysis to explicit mathematics, without going throughordinal-theoretic proof theory.Thomas StrahmInstitut fu¨r Informatik und angewandte Mathematik, Universita¨t Bern, Neubru¨ckstrasse10, CH-3012 Bern, Switzerland. strahm@iam.unibe.ch.F. A.Muller. Sets, classes, and categories. British Journal for the Philosophy of Science,vol. 52 (2001), pp. 539–573.Muller skillfully expounds the challenge for set theoretic foundations for category theory:Simple categorical methods, with evidently low proof theoretic strength, get formalized byusing very large sets. The standard device, a Grothendieck universe, is a set U which itselfsatisﬁes all the ZF axioms. The category Top of all topological spaces, for example, is takenas the category of all topological spaces that exist in U . That is a set, though not in U , andthere is a set of all functors F : Top	 Set and much more. The categorical theorems used inpractice survive almost entire.As Muller says, this is “not just a bit abundant [it is] mindbogglingly, ﬂabbergastinglyabundant”(550). A universe amounts to an inaccessible cardinal. Muller oﬀers a moreelegant approach using his strengthened form of Ackermann’s theory of sets and classes.He motivates his theory and proves numerous theorems on its expressive power. It hasclasses of every ﬁnite rank over the class of all sets, and so is more agile than Go¨del-Bernays. Muller argues that his sets and classes reﬂect the true diﬀerence between logicaland combinatorial collections, and his sets are sharply delineated in Cantor’s sense while hisclasses are not (though they have extensionality). He shows his theory equiconsistent withZF.As to ZF-style foundations, the reviewer proposes a related strategy within ZF, deﬁning‘small’ sets as elements ofV (+). All usualmathematics apart fromcategory theory is donein small sets. Deﬁne the category Top as all small topological spaces and continuous mapsand similarly for similar categories. Both approaches are decidedly deﬂationary compared tohttps:/www.cambridge.org/core/terms. https://doi.org/10.1017/S1079898600004741Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 15:47:47, subject to the Cambridge Core terms of use, available at

Book Review: The Journal of Symbolic Logic: Tupailo Sergei . Realization of analysis into explicit mathematics. The Journal of Symbolic Logic, vol. 66 (2001), pp. 1848-1864

Book Review: The Journal of Symbolic Logic: Tupailo Sergei . Realization of analysis into explicit mathematics. The Journal of Symbolic Logic, vol. 66 (2001), pp. 1848-1864

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