Abstract
This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page Tarski consequence-definition paper is based on a monistic fixed-universe framework—like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as ‘the class of all individuals’. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework—like the Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations—as in post-WWII model theory. In the early 1960s, many logicians—mistakenly, as we show—held the ‘contrary alternative’ that Tarski had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege-Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski’s philosophy underwent many other radical changes.
Dedicated to Professor Roberto Torretti, philosopher of science, historian of mathematics, teacher, friend, collaborator—on his eightieth birthday.
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Notes
- 1.
Unless explicitly said otherwise, ‘consequence’ is used in a ‘logical’ sense so that the expression ‘logical consequence’ is redundant. Thus, we are not discussing the ‘consequences’ of events, of actions, or of inactions, etc.
- 2.
- 3.
One of the History and Philosophy of Logic referees required us to give more emphasis to this point than it had received in the submitted version. We are grateful that this needed improvement was brought to light; it helps us to highlight a major philosophical and technical change between Tarski’s pre-WWII attitude and his post-WWII thinking. After the war uninterpreted constants became legitimate and were assigned an important role. Contrast [57].
- 4.
Tarski did not dwell on the philosophical ramifications of this ‘relevance’ or ‘pertinence’ requirement, which might have been foreshadowed by Aristotle. But others have noted its importance (Corcoran [8]).
- 5.
Tarski is not as explicit as one might wish. This point is not made in any single sentence though it can be gleaned from the paragraph that begins on p. 414. Incidentally, this is the only place in [69] that uses ‘form’ in the required sense of logical form. See Corcoran’s [14] piece ‘Logical form’ in [1].
- 6.
According to Leonard Jacuzzo [36], whose 2005 dissertation research involved comparisons of dozens of introductory logic texts, most books he studied teach the affirmative answer to this question (personal communication). Our own less extensive experience confirms his sad finding.
- 7.
- 8.
Space limitations preclude discussion of how Frege arrived at PCR (or a similar monistic universal variable-range view), why it was so widely adopted, how its conflict with pluralistic views escaped notice for so long, and when its soundness came under scrutiny.
- 9.
Today the set of elements of a Boolean algebra, or of any other algebraic structure such as a group or a ring, is called its carrier. The maximal element of the Boolean algebra is usually denoted by the digit ‘1’ and is called its unity. If a given Boolean algebra is ‘formalized’ using a first-order language whose individual variables range over the carrier, then the carrier is the universe of discourse of the ‘theory of the algebra’. But this can distort Boole’s [3] viewpoint. Take Boole’s ‘universe of men (sc. humans)’. Boole used ‘1’ to denote this class and ‘0’ to denote the null class: two elements of the carrier of the corresponding Boolean algebra. In such cases, where a Boolean algebra of classes is under discussion, there are two things competing for the names ‘universe’, ‘universe of discourse’, ‘universal class’ and the like: the carrier and the carrier’s unity. The carrier is often the powerset of the carrier’s unity. In such cases, the carrier’s unity is the union of the carrier. The Boolean tradition would incline towards using such terms for the unity. The modern abstract-algebra viewpoint that abstracts from the nature of the elements of an abstract algebra would incline toward using such terms for the carrier. Tarski called the carrier or the set of elements of a given Boolean algebra its universe of discourse on the first page of the article beginning on p. 347 of [72].
- 10.
To the best of Corcoran’s knowledge, there is no chance that any unpublished writing by Tarski, either a passage in the Tarski-Corcoran correspondence (preserved at the Bancroft Library at U.C. Berkeley), or anywhere else, would suggest that Tarski [63] conceives of various models having various universes of discourse. In fact, according to Paolo Mancosu, evidence to the contrary is to be found not only in the Tarski-Corcoran correspondence (Mancosu [26 and 39, p. 451]) but also in Tarski’s unpublished 1940 lecture ‘On the completeness and categoricity of deductive systems’ also in the Bancroft Library (Mancosu [41, pp. 754–756]).
- 11.
Even at that time, William Craig was a distinguished mathematical logician, former doctoral student of Quine at Harvard, Full Professor of Philosophy at Berkeley. He was a member of the UC Berkeley Group for Logic and Methodology, in which Tarski was still active. He was internationally known for what was then called the Craig Interpolation Lemma. The Craig Interpolation Theorem, as it is sometimes known today, is ‘one of the basic results of the theory of models [2], almost on a par with, say, the compactness theorem’ (Boolos et al. [4, p. 260]). An entire chapter of [4] is devoted to this theorem, which is to be distinguished from a less deep but equally famous result then called Craig’s theorem, now sometimes the Craig Axiomatizability Lemma or the Craig Reaxiomatization Lemma (Boolos et al. [4, p. 198]).
- 12.
- 13.
We never use the expression ‘formally implies’ in Russell’s sense without explicitly adding ‘in Russell’s sense’ or an equivalent. In fact, as Nabrasa pointed out, few if any logicians do either. He reminded us that the fact that a person defines an expression in a certain sense, in and of itself, is no evidence that the person uses the expression in that sense (Frango Nabrasa, personal communication).
- 14.
This is yet another example of Russell’s habit of using previously established terminology in a sense never before employed and without explaining or even alluding to the previous senses. His friend and colleague G. E. Moore criticized him for this in connection with using ‘implies’ in the sense of the truth-functional conditional. Previously no traditional logician would have said that ‘Some animal is not a dog’ logically implies ‘Every oak is a tree’.
- 15.
See p. xxii of Corcoran’s Editor’s Introduction to Tarski [72]. A no-expressible-countermodels definition of consequence defines a premise-conclusion argument expressed in a given language to be valid if no argument in the same form expressible in the same language has all true premises and false conclusion. Such a definition is featured in Quine’s Philosophy of Logic [47], where great care is taken to ensure that the language has the required ‘richness’, to use Tarski’s expression [64, p. 416]. Even today, some authors claim, of course without giving any justification, that the no-expressible-countermodels conception is the classical definition of consequence. E.g. in [4, p. 101] where ‘argument’ is elliptical for ‘expressed argument’, we find: ‘Logic teaches that the premisses […] (logically) imply or have as a (logical) consequence the conclusion […], because [sic] in any argument of the same form, if the premisses are true, then the conclusion is true’.
- 16.
Tarski should not be construed as referring to possible objects as opposed to actual objects. By ‘possible objects’ Tarski means ‘objects’; the use of the modal adjective ‘possible’ is entirely empty—it is what is sometimes called redundant rhetoric, filler, or expletive. Tarski’s usage of modal words is almost always, if not absolutely always, expletive, like putting ‘absolutely’ before ‘always’, ‘entirely’ before ‘empty’, ‘no matter how small’ after ‘every real number’ or ‘if any’ after ‘all odd perfect numbers’. See Corcoran [18] and [19, p. 266]. Incidentally, Tarski may not be speaking very strictly in saying that designations of all objects are needed.
- 17.
Henry Hiż had been a fellow Quine PhD student at Harvard with William Craig and Robert McNaughton, supervisor of Corcoran’s 1963 dissertation. McNaughton introduced Corcoran to Hiż in 1961 or 1962 and to Craig in 1963. Hiż helped with the second edition of Logic, Semantics, Metamathematics and with the Editor’s Introduction [72, pp. viii, xxvii].
- 18.
- 19.
- 20.
The following background has been generously supplied by David Hitchcock (personal communication): ‘Tarski wrote the paper in 1935. He delivered the German version at a conference in Paris in September 1935, and appears to have left a copy of the paper with the conference organizers for publication in the conference proceedings, which came out in 1936. The Polish version appeared in the first (January) issue of the 1936 volume of Przegl ąd filozoficzny (Polish for ‘Philosophical Review’, the leading Polish philosophy journal), and so must have been submitted by the end of 1935, allowing time for typesetting and correcting proofs. Thus, Tarski wrote the paper no later than 1935. It is likely that he did not write it earlier than 1935, since Carnap reports in his autobiography that Tarski visited Vienna in June 1935 and that Carnap persuaded Tarski at that time to present his ideas on semantics at the September 1935 conference. Tarski had just finished translating his truth monograph into German (at the end of the historical note which Tarski added to his German translation of his truth monograph, there appears in Latin in italics, centered two lines below the end of the text the sentence
“Nachwort” allatum est die 13. Aprilis 1935.
—i.e. The ‘afterword’ was produced on the 13 April 1935.). The other paper that Tarski presented at the Paris conference is a kind of summary of the ideas in semantics in that work. Both papers in fact presuppose the concepts of the truth monograph.’ For further information see Hitchcock-Stroińska [33, especially, pp. 155–158].
- 21.
The De Morgan and Boole work discovered the concept of the universe of a discourse in the conceptual framework of the mathematics and science of their time. The role of universes of discourse persisted into the conceptual framework of Tarski’s time. Tarski’s work shows this role but it does not show awareness of that role nor does it show any appreciation of the De Morgan-Boole achievement. The fact that blood circulated in Plato’s veins is no reason to credit him with discovering or knowing of blood circulation.
- 22.
The wisdom of Boole’s choice of the word ‘discourse’ for a certain sort of extended exposition or discussion may be questioned and it is open to doubt whether the word had ever been used in his precise sense before. Others may have observed how much Boole enriched the English language by coining the phrase ‘universe of discourse’, but no one seems to have suggested that his use of ‘discourse’ may have been equally creative and meritorious. By the way, the word ‘discourse’ is more often used for a stretch of speech involving typically more than one sentence-like expression, e.g., a paragraph or an argumentation. It was used in this sense in Zellig Harris’s Discourse Analysis Project, which is discussed in Corcoran’s paper ‘Discourse grammars and the structure of mathematical reasoning, Part I: Mathematical reasoning and the stratification of language’ [7]. See also Corcoran [12].
- 23.
By an indexical we mean a word such as ‘I’, ‘you’, ‘here’, ‘this’, and ‘now’ whose denotation changes according to the context of the speech act it is used in. See Corcoran [20, pp. 159–160] for a discussion of Boole’s innovations involving the symbol ‘1’.
- 24.
- 25.
The idea that logicians down through the history of logic were expressing a Tarskian no-countermodels concept by the phrase ‘is a consequence of’ is implausible to say the least. Tarski’s stated goal was not to characterize the traditional concept but merely to define ‘a new concept which coincided in extent with the common one’ [63, p. 409].
- 26.
- 27.
Tarski’s oral evaluation contrasts with the evaluation published by Haskell Curry in Mathematical Reviews [24].
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Acknowledgements
We call in others to aid us in deliberating on important questions—distrusting ourselves as not being equal to deciding. —Aristotle.
We started planning this paper in 2003 on a trip the two of us made to Lisbon from Santiago de Compostela for Ricardo Santos’s Tarski Symposium. Earlier versions have been circulated (Mancosu [40, p. 576]). For bringing errors and omissions to our attention, for useful suggestions, and for other help, it is a pleasure to acknowledge the following scholars: O. Chateaubriand (Brazil), D. Hitchcock (Canada), R. Torretti (Chile), I. Grattan-Guinness (England), S. Nambiar (India), H. Masoud (Iran), R. Santos (Portugal), S. Read (Scotland), F. Rodriguez-Consuegra (Spain), J. Gasser (Switzerland), and, from USA, M. Brown, W. Goldfarb, R. Grandy, I. Hamid, L. Jacuzzo, C. Jongsma, E. Keenan, M. La Vine, R. Maddux, P. Mancosu, D. Merrill, J. Miller, M. Scanlan, S. Shapiro, J. Smith, J. Tarski, and G. Weaver. During the entire period in which this article was being written we have been in almost constant contact with our long-time friend Frango Nabrasa, whose incisive sarcasm and disarming skepticism dampened any excessive enthusiasm we might have had. David Hitchcock has been especially helpful: not only has he shared insights and given valuable criticisms, his published scholarship on this topic makes crucial, actually essential, contributions. We are also grateful to Volker Peckhaus and to two anonymous History and Philosophy of Logic referees for their useful suggestions and perceptive corrections.
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Corcoran, J., Sagüillo, J.M. (2018). The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Definition Paper. In: Garrido, Á., Wybraniec-Skardowska, U. (eds) The Lvov-Warsaw School. Past and Present. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-65430-0_30
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