tonk, plonk and plink - nuel d. belnap

Upload: kherian-gracher

Post on 07-Apr-2018

253 views

Category:

Documents


0 download

TRANSCRIPT

  • 8/6/2019 Tonk, Plonk and Plink - Nuel D. Belnap

    1/7

    Tonk, Plonk and Plink

    Nuel D. Belnap

    Analysis, Vol. 22, No. 6. (Jun., 1962), pp. 130-134.

    Stable URL:

    http://links.jstor.org/sici?sici=0003-2638%28196206%2922%3A6%3C130%3ATPAP%3E2.0.CO%3B2-G

    Analysis is currently published by The Analysis Committee.

    Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.

    Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/journals/anacom.html.

    Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.

    The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academicjournals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers,and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community takeadvantage of advances in technology. For more information regarding JSTOR, please contact [email protected].

    http://www.jstor.orgSun Dec 30 10:48:38 2007

    http://links.jstor.org/sici?sici=0003-2638%28196206%2922%3A6%3C130%3ATPAP%3E2.0.CO%3B2-Ghttp://www.jstor.org/about/terms.htmlhttp://www.jstor.org/journals/anacom.htmlhttp://www.jstor.org/journals/anacom.htmlhttp://www.jstor.org/about/terms.htmlhttp://links.jstor.org/sici?sici=0003-2638%28196206%2922%3A6%3C130%3ATPAP%3E2.0.CO%3B2-G
  • 8/6/2019 Tonk, Plonk and Plink - Nuel D. Belnap

    2/7

    A N A L Y S I S

    TONK, PLONK AND PLINK1

    A N. PRIOR has recently discussed2 the connective tonk, where.tonk is defined by specifying the role it plays in inference. Priorcharacterizes the role of tonk in inference by describing how it behavesas conclusion, and as premiss: (1) A I- A-tonk-B, and (2) A-tonk-B t B(where we have used the sign ' k ' for deducibility). We are then led bythe transitivity of deducibility to the validity of A I- B, " which promisesto banish falscbe Spitzjndigkeit from Logic for ever."A possible moral to be drawn is that connectives cannot be definedin terms of deducibility at all; that, for instance, it is illegitimate todefine and as that connective such that (1) A-and-B t A, (2) A-and-B I- B,and (3) A, B I- A-and-B. We must first, so the moral goes, have a notionof what and means, independently of the role it plays as premiss and asconclusion. Truth-tables are one way of specifying this antecedentmeaning; this seems to be the moral drawn by J. T. Stevenson.3 Thereare good reasons, however, for defending the legitimacy of definingconnections in terms of the roles they play in deductions.It seems plain that throughout the whole texture of philosophy onecan distinguish two modes of explanation: the analytic mode, whichtends to explain wholes in terms of parts, and the synthetic mode, whichexplains parts in terms of the wholes or contexts in which they occur.4In logic, the analytic mode would be represented by Aristotle, whocommences with terms as the ultimate atoms, explains propositions orjudgments by means of these, syllogisms by means of the propositionswhich go to make them up, and finally ends with the notion of a scienceas a tissue of syllogisms. The analytic mode is also represented by thecontemporary logician who first explains the meaning of complexsentences, by means of truth-tables, as a function of their parts, and thenproceeds to give an account of correct inference in terms of the sentencesoccurring therein. The loczls classicuf of the application of the syntheticmode is, I suppose, Plato's treatment of justice in the Rept/blic, where hedefines the just man by reference to the larger context of the community.Among formal logicians, use of the synthetic mode in logic is illustratedby Kneale and Popper (cited by Prior), as well as by Jaskowski, Gentzen,Fitch, and Curry, all of these treating the meaning of connectives as

    1 This research was supported in part by the Office of Naval Research, Group PsychologyBranch, Contract No. SAR/Nonr-609(16).2 ' , ANALYSIShe Runabout Inference-ticket 21.2, December 1960.3 ' 21.6, June 1961. Cf. p. 127:oundabout the Runabout Inference-ticket ', ANALYSIS"The important difference between the theory of analytic validity [Prior's phrase for what ishere called the synthetic view] as it should be stated and as Prior stated it lies in the factthat he gives the meaning of connectives in terms of permissive rules, whereas they should

  • 8/6/2019 Tonk, Plonk and Plink - Nuel D. Belnap

    3/7

    131O N K , P L O N K A N D P L I N Karising from the role they play in the context of formal inference. I tis equally well illustrated, I think, by aspects of Wittgenstein and thosewh o learned from him to treat the m eanings of words as arising from therole they play in the context of discourse. It seems to me nearly self-evident that employment of both modes of explanation is importantand useful. I t would therefore be truly a shame to see the syntheticmode in logic pass away as a result of a severe attack of tonktitis.Suppose, then, that we wish to hold that it is after all possible todefine connectives contextually, in terms of deducibility. H ow are we toprevent tonk titis? H ow are we to make good the claim that there is noconnective such as tonkl tho ug h there is a connective such as and (wheretonk and and are defined as above) ?

    I t seems to me tha t the key to a so lution2 ies in observing that evenon the synthetic view, we are not d e w g our connectives ab initio, butrather in terms of an antecedently given context o dedgcibility, concerningwhich we have som e definite notions. By that I mean that beforearriving at the problem of characterizing connectives, we have alreadymade some assumptions abou t the nature of deducibility. Tha t this isso can be seen immediately by observing Prior's use of the transitivityof deducibility in o rder to secure his ingenious result. But if we no tethat we already have som e assumptions about th e context of deducibilitywithin w hich we are operating, it becomes apparent tha t by a too carelessuse of definitions, it is possible to create a situation in which we areforced to say things inconsistent with those assumptions.The situation is thus exactly analogous to that, pointed out byPeano, which occurs when o ne attempts to define an operation, ' ? ', onrational numbers as follows :

    This definition is inadmissible precisely because it has consequenceswhich contradict prior assumptions; for, as can easily be shown, adrnit-2 - 3ting this definition would lead to (say) - - 5 .In short, we can distinguish between the admissibility of the defini-tion of and and the inadm issibility of tonk on the grounds of consistency-i.e., consistency with antecedent assumptions. We can give a preciseaccount of the requirement of consistency from the synthetic point ofview as follows.

    That there is no meaningful proposition expressed by A-fonk-B; that thereis no meaning-ful sentence A-tonk-B-distinctions suggested by these alternative modes of expression areirrelevant. Not myself being a victim of eidophobia, I will continue to use language whichtreats the connective-word' onk 'as standing for the putative propositional connective, tonk.It is equally irrelevant whether we take the sign k as representing a syntactic concept ofdeducibility or a semantic concept of logical consequence.

  • 8/6/2019 Tonk, Plonk and Plink - Nuel D. Belnap

    4/7

    132 A N A L Y S I S(1) We consider some characterization of deducibility, which may be

    treated as a formal system, i.e., as a set of axioms and rules involving thesign of deducibility, ' - ',where 'A,, . . . ,A, I- B ' is read ' B is deduciblefrom A,, . . . ,A,.' For definiteness, we shall choose as our characteriza-tion the structural rules of Gentzen:

    Axiom. A t AWes. Weakening: from A,, ..., A, t C to infer A,, ...,A, B t C

    Permzltation: from A,, . ,A, A,,,, ...,A, I- B to inferA,, . , Ai+l' Ai, ..., A, t B.

    Contraction: fromA,, .. ,A,, A, t B to infer A,, ..., A, I- BTransitivig: from A,, . , A, t- B and C,, ..., C,, B t- D

    to infer A,, ..., A,, C,, ..., C, C- D.In accordance with the opinions of experts (or even perhaps on moresubstantial grounds) we may take this little system as expressing all andonly the universally valid statements and rules expressible in the givennotation: it completely determines the context.

    (2) We may consider the proposed definition of some connective, sayplonk, as an extension of the formal system characterizing deducibility,and an extension in two senses. (a) The notion of sentence is extended byintroducing A-plonk-B as a sentence, whenever A and B are sentences.(b) We add some axioms or rules governing A-plonk-B as occurring asone of the prernisses or as conclusion of a deducibility-statement. Theseaxioms or rules constitute our definition ofplonk in terms of the role itplays in inference.(3)We may now state the demand for the consistency of the definitionof the new connective, plank, as follows: the extension must be con-servativel; i.e., although the extension may well have new deducibility-statements, these new statements will all involve plonk. The extensionwill not have any new deducibility-statements which do not involveplonk itself. It will not lead to any deducibility-statement A,, ..., A, I- Bnot containing plonk, unless that statement is already provable in theabsence of theplonk-axioms andplonk-rules. The justification for unpack-ing the demand for consistency in terms of conservativeness is preciselyour antecedent assumption that we already had all the universally validdeducibility-statements not involving any special connectives.

    So the trouble with the definition of tonk given by Prior is that it isinconsistent. It gives us an extension of our original characterizationof deducibility which is not conservative, since in the extension (but notin the original) we have, for arbitrary A and ByA t B. Adding a tonkishrole to the deducibility-context would be like adding to cricket a playerwhose role was so specified as to make it impossible to distinguish

  • 8/6/2019 Tonk, Plonk and Plink - Nuel D. Belnap

    5/7

    T O N K , P L O N K A N D PL INK 133Hence, given that our characterization of deducibility is taken ascomplete, we may with propriety say 'There is no such connective astonk'; just as we say that there is no operation, '?', on rational numbers

    a c = a+c On the other hand, it is easily shown thatuch that ( 6 ? a ) wd.the extension got by adding and is conservative, and we may hence say' There is a connective having these properties '.It is good to keep in mind that the question of the existence of aconnective having such and such properties is relative to our character-ization of deducibility. If we had initially allowed A I- B (!), there wouldhave been no objection to tonk, since the extension would then havebeen conservative. Also, there would have been no inconsistency hadwe omitted from our characterization of deducibility the rule of transi-tivity.The mathematical analogy leads us to ask if we ought not also to adduniqtlenessl as a requirement for connectives introduced by definitions interms of deducibility (although clearly this requirement is not as essentialas the first, or at least not in the same way). Suppose, for example, thatI propose to define a comectiveplonk by specifying that B I- A-plonk-B.The extension is easily shown to be conservative, and we may, therefore,say ' There is a connective having these properties '. But is there onlyone? It seems rather odd to say we have defined plonk unless we canshow that A-plonk-B is a function of A and By .e., given A and By hereis only one proposition A-plonk-B. But what do we mean by uniquenesswhen operating from a synthetic, contextualist point of view? Clearlythat at most one inferential role is permitted by the characterization ofplonk; i.e., that there cannot be two connectives which share the charac-terization given to plonk but which otherwise sometimes play differentroles. Formally put, uniqueness means that if exactly the same propertiesare ascribed to some other connective, say plink, then A-plink-B willplay exactly the same role in inference as A-$onk-By both as prernissand as conclusion. To say thatplonk (characterized thus and so) describesa unique way of combining A and B is to say that if plink is given acharacterization formally identical to that ofplonk, then(1)A,, ..., B-plonk-C, ..., A, I- D if and only if A,, . ,B-$link-C, . ,A, I-Dand(2) A,, ..., A, I- B-plonk-C if and only if A,, .. A, I- B-plink-C.Whether or not we can show this will depend, of course, not onlyon the properties ascribed to the comectives, but also on the propertiesascribed to deducibility. Given the characterization of deducibilityabove, it is sufficient and necessary that B-plonk-C I- B-plink-C, andconversely.

  • 8/6/2019 Tonk, Plonk and Plink - Nuel D. Belnap

    6/7

    134 A N A L Y S I SHarking back now to the definition of plonk by: B I- A-plonk-B, it is

    easy to show thatplonk is not unique; that given only: B I- A-plonk-ByandB I- A-plink-B, we cannot show that plonk and plink invariably play thesame role in inference. Hence, the possibility arises that plonk and plinkstand for different connectives : the conditions onplod do not determinea unique connective. On the other hand, if we introduce a connective,et, with the same characterization as and, it will turn out that A-and-Band A-et-B play exactly the same role in inference. The conditions onand therefore do determine a unique connective.

    Though it is difficult to draw a moral from Prior's delightful notewithout being planking, I suppose we might put it like this: one candefine connectives in terms of deducibility, but one bears the onus ofproving at least consistency (existence); and if one wishes further totalk about the connective (instead of a connective) satisfying certainconditions, it is necessary to prove uniqueness as well. But it is notnecessary to have an antecedent idea of the independent meaning of theconnective.Yale UniversiD

    UNQUANTIFIED INDUCTIVE GENERALIZATIONS

    J 0. NELSON argues that the ordinary grammar of inductive. eneralizations is not amenable to quantification and that thisgrammar is preferable to the one that, in his view, logicians have foistedupon them (ANALYSIS,anuary 1962). It seems to me that his diagnosisdeparts from instead of returning to ordinary grammar.

    1. As an example of an unquantified inductive generalization (hence-forth U.I.G.) he gives 'Ants attack spiders '. This, he says, is an ambigu-ous proposition. ''At the point of asserting the proposition, do we meanby it all or some ants? And if ' all ', all ants at present on earth or allants present and future? And if ' some ', merely the ant we haveobserved or others in addition, and what proportion of them? We do

  • 8/6/2019 Tonk, Plonk and Plink - Nuel D. Belnap

    7/7

    You have printed the following article:

    Tonk, Plonk and Plink

    Nuel D. Belnap

    Analysis, Vol. 22, No. 6. (Jun., 1962), pp. 130-134.

    Stable URL:

    http://links.jstor.org/sici?sici=0003-2638%28196206%2922%3A6%3C130%3ATPAP%3E2.0.CO%3B2-G

    This article references the following linked citations. If you are trying to access articles from anoff-campus location, you may be required to first logon via your library web site to access JSTOR. Pleasevisit your library's website or contact a librarian to learn about options for remote access to JSTOR.

    [Footnotes]

    3Roundabout the Runabout Inference-Ticket

    J. T. Stevenson

    Analysis, Vol. 21, No. 6. (Jun., 1961), pp. 124-128.

    Stable URL:http://links.jstor.org/sici?sici=0003-2638%28196106%2921%3A6%3C124%3ARTRI%3E2.0.CO%3B2-D

    http://www.jstor.org

    LINKED CITATIONS- Page 1 of 1 -

    NOTE: The reference numbering from the original has been maintained in this citation list.

    http://links.jstor.org/sici?sici=0003-2638%28196206%2922%3A6%3C130%3ATPAP%3E2.0.CO%3B2-G&origin=JSTOR-pdfhttp://links.jstor.org/sici?sici=0003-2638%28196106%2921%3A6%3C124%3ARTRI%3E2.0.CO%3B2-D&origin=JSTOR-pdfhttp://links.jstor.org/sici?sici=0003-2638%28196106%2921%3A6%3C124%3ARTRI%3E2.0.CO%3B2-D&origin=JSTOR-pdfhttp://links.jstor.org/sici?sici=0003-2638%28196206%2922%3A6%3C130%3ATPAP%3E2.0.CO%3B2-G&origin=JSTOR-pdf