The Bricot–Mair Dispute: Scholastic Prolegomena to Non-Compositional Semantics
2013; Taylor & Francis; Volume: 35; Issue: 2 Linguagem: Inglês
10.1080/01445340.2013.856101
ISSN1464-5149
Autores Tópico(s)Classical Philosophy and Thought
ResumoAbstractFrom a general semantic point of view, Thomas Bricot (d. 1516) and John Mair (1467–1550) are proponents of the solution to semantic paradoxes based on appreciation of the contextuality of truth, who differ in their approach to the relations of logical consequence and contradiction. The core of the study is the analysis of Mair's criticism of Bricot presented in the sixth quaestio of his Tractatus insolubilium where the consequences of non-compositional semantics for the concepts of synonymy and logical form are addressed. The polemic between John Mair and Thomas Bricot is construed as having immediate consequences for research in the area of non-compositional semantics. AcknowledgementsThis work was supported by the grant of Czech Science Foundation (GAČR) ‘Semantic paradoxes between first and second scholasticism’, registration no. 13-08389P.Notes1For biographical information, cf. Weisheipl 1964.2The end of the Swyneshedian tradition was due to the abrupt loss of interest in the traditional scholastic topics which took place in 1530s (with the exception of Spain) rather than theoretically relevant reasons (Ashworth 2008).3De Rijk 1975; Ashworth 1972, 1974, 1977, 1978, 1979; Ashworth and Spade 1992.4Bricot's academic career at the University of Paris lasted approximately 30 years between 1470s and 1500s. His version of Swyneshedian semantics incorporates several additional clauses to retain the ‘classical’ principles, i.e., the classical square of oppositions and truth-preservation of valid inferences. Ashworth 1977, 267–8; Bricot 1986, xiii–iv; Deutscher and Bietenholz 1985, 199–200.5Mair's academic career started in 1491 at Cambridge and continued in Paris between 1494 and 1518. In 1518, he became the principal of the University of Glasgow and 1523 transferred to the University of St. Andrews. After spending another period in Paris, he returned to Scotland in 1531 and died there in 1550. For the further development of Swyneshedian semantics, his circle of pupils became one of the important factors. For Mair's biography, cf. Broadie 1985, 1–7; 1990, 20–23; 2008, 5–8; 2009, 47–61; for his bibliography, cf. Durkan 1950, 140–157.6‘Propositio vera est propositio non falsificans se principaliter sicut est significans naturaliter aut ex impositione vel impositionibus qua vel quibus ultimo fuit imposita ad significandum. ( … ) Propositio falsa est oratio falsificans se vel oratio non falsificans se principaliter aliter quam est significans naturaliter, ex impositione, vel impositionibus qua vel quibus ultimo fuit imposita ad significandum.’ (Swyneshed 1979, 185–186)\newline The phrase ‘signifying as the things are’ is, roughly, equivalent to saying that the sentence in question represents the respective actual state of affairs. For detailed analysis of Swyneshed's semantic terminology, cf. Spade 1983.7Cf. Swyneshed 1979, 182–184. Swyneshed's theory of truth and self-reference is more complicated; in particular, there are sentences which deny their correspondence with reality and are therefore neither true nor false. This aspect of Swyneshedian semantics does not need to be taken into consideration, since the present paper will only deal with Swyneshedian analysis of alethic paradoxes.8Cf. Swyneshed 1979, 188. The term ‘principal signification’ (or, by other authors, ‘total’ or ‘adequate’, which is a term used in some manuscripts of Swyneshed's Insolubilia) as that which a sentence as a totality says is to be distinguished from ‘partial signification’ as follows: ‘if “p ” is a string of phonemes or letters making up a grammatically well-formed sentence, and if “q ” is another string of phonemes or letters making up a grammatically well-formed sentence and containing “p ” as a proper substring, then “q ” principally (totally, adequately) signifies that q, but partially signifies that p ’ (Spade 1983, 106). Some fourteenth-century authors distinguish the ‘principal’ signification from the ‘assertive’ or ‘secondary’ signification, i.e., from signifying the logical consequences of a sentence by the sentence in question (Nuchelmans 1980, 45–46). However, this concept of implied meaning plays no role in Swyneshed's semantics, which is emphasised by some manuscripts of Swyneshed's Insolubilia where praecise is used alongside with principaliter in these contexts.9‘Si antecedens significat sicut est et nec antecedens nec consequens est pertinens ad inferendum se ipsum non significare sicut est, igitur consequens significat sicut est’ (Swyneshed 1979, 193).10For a general overview of the topic of scholastic approaches to validity in medieval and post-medieval periods, cf. Ashworth 1974; Green-Pedersen 1984; Schupp 1988; Boh Citation2001; King Citation2001; Weber Citation2003; Dutilh Novaes Citation2008, 2012.11Heytesbury 1494, fol. 5rb; Pozzi 1987, 222; Spade 1979, 29–30; for general information about the Heytesburian tradition, cf. Pironet 2008.12Spade 1978, 65–66.13Spade 1976, 57.14Spade 1978, 77.15Seaton 1973, 2.16The Parisian origin of his treatise is witnessed by the explicit of his treatise Ergo expletae sunt quaestiones insolubilium a Johanne de Vesalia Parixius disputatae. Based on this information, Spade offered a conjectural indentification of the author of this treatise with the John of Wesel who was at Paris in 1344–1353. Regardless of Wesel's identity, his treatise displays strong influences of both Swyneshed and Heytesbury and provides evidence for the reception of Swyneshed in Paris. Cf. Spade 1996.17Spade 1996.18Cf. Paul of Venice 1499, fols. 192vb–95va. Paul knew Heytesbury's position (which he presents as ‘the twelfth opinion’) and his analysis of σ could have been derived from his Regulae solvendi sophismata, but given his own definition of validity this similarity can be merely coincidental.19Cf. Paul of Venice 1990, 92.20At the turn of the fifteenth and sixteenth centuries, two anonymous British authors produced short treatises which summarise and further develop Swyneshed's position (Anonymus1 1510; Anonymus2 1517; Ashworth 1979) but since they are contemporary or even posterior to the Bricot-Mair dispute, they are not discussed here, nor is the Italian reception of Paul of Venice's logic.21Cf. George of Brussels [ca. 1494] (no pagination); George of Brussels 1496, fols. 271vb–72vb; George of Brussels 1504, fols. 274.22Cf. Ashworth 1977. The analysis of affirmative sentences is shared by Bricot and Mair, Bricot describes the truth-conditions of affirmative sentences as follows: ‘its truth requires that it is the case exactly as it signifies to be, and that it does not signify that it is false’ ( … ad ipsam esse veram requiritur quod omnino taliter sit qualiter ipsa significat, et quod ipsa non significet seipsam esse falsam.) (Bricot 1986, 21), Mair as follows: ‘[T]he truth of an affirmative sentence requires two things: First, that it signifies to be the case as it is or was of will be the case (depending on its copula). Second, that it does not falsify itself.’ ([A]d veritatem affirmativae duo requiruntur: Unum est quod significet taliter esse qualiter est vel fuisse qualiter fuit vel fore qualiter erit secundum exigentiam copularum. Secundum est quod non se falsificet.) Mair 1508, fol. 64. The analysis of negative sentences will be discussed later.23George of Brussels 1496, fols. 272rb–74rb; 1504, fols. 274vb–75vb; for a transcription of the text, cf. Bricot 1986, 129–37.24Bricot 1986, 32–38.25Bricot Citation1986, 135.26‘The proper reason why an inference is valid is that it cannot be the case exactly as its antecedent signifies or can signify to be (the cases of meaning-reimposition apart) unless it is the case exactly as its consequent signifies or can signify to be, and that the antecedent signifies about itself that it is false if the consequent signifies about itself that it is false’ (Causa adaequata bonitatis consequentiae est scilicet quod requiritur quod non possit ita omnino esse sicut per antecedens significatur vel significari potest sine nova impositione terminorum quin ita sit sicut per consequens significatur vel significari potest, et quod antecedens significet se esse falsam si consequens significet se esse falsam.) Bricot 1986, 32.27Bricot 1986, 35–3828Mair 1508, fol. 59ra.29This thesis is based mostly on Bricot's Tractatus insolubilium and Quaestio addita and is open to discussion. Mair's additions to Bricot's writings could have been based on personal encounter, but the present research does not aim at proving that.30The analysis of σ is part of Mair's argument. Mair introduces two arguments for σ being valid: σ is correspondence-preserving and the negation of c is incompatible with a (Mair 1508, fol. 60va).31‘I will argue against these statements and prove that a true sentence entails a false sentence. The following inference holds: “No conclusion is true, and every conclusion in Celarent is a conclusion; therefore, no conclusion in Celarent is true”, where we argue in Celarent. Also, let there be no other conclusion except for the present one. Then the antecedent is true and the consequent false, therefore etc. The second argument is that the following inference holds in Darii: “every conclusion is false, a conclusion in Darii is a conclusion; therefore, a conclusion in Darii is false”, assuming that there is no other conclusion except for the present one. Then the antecedent is true and the consequent false, therefore a true sentence entails a false sentence.’ (Contra dicta arguitur probando quod ex vero sequitur falsum. Sequitur ‘nulla conclusio est vera et omnis conclusio de Celarent est conclusio, ergo nulla conclusio de Celarent est vera’, arguendo in Celarent et nolo quod sit alia conclusio ab ista. Iam antecedens est verum et consequens falsum. Igitur. Secundo arguitir: sequitur in Darii: ‘omnis conclusio est falsa, conclusio de Darii est conclusio, ergo conclusio de Darii est falsa’, posito quod nulla sit conclusio alia ab ista. Iam antecedens est verum et consequens falsum, ergo ex vero sequitur falsum.) Mair 1508, fol. 59va.32‘And if you say: the consequent falsifies itself but the antecedent does not, hence the inference is not valid, I reply that the inference is valid. That is clear, since the inference in question has the same form as the following one: “Every woman is white, a woman on a bridge is is a woman; therefore, a woman on a bridge is white.” The argument is clear, for the major premises of both inferences have the same form, as do the minor premises and conclusions, and the relation between terms is similar in both cases.’ (Et si dicas: consequens se falsificat, antecedens vero non, propterea non sequitur. Contra hoc arguo quia ista consequentia est bona. Patet quia est eiusdem formae cum ista: ‘omnis mulier est alba, mulier de ponte est mulier, ergo mulier de ponte est alba’. Patet quia maior unius est eiusdem formae cum maiore alterius et minoris cum minore conclusio cum conclusione et similis habitudo terminorum inter se.) Mair 1508, fol. 59v.33‘And we confirm this position, because if the inference in question does not hold, the premises in Darii are an instance of a useless combination.’ (Et confirmatur, quia si non sequitur, sequeretur quod premissae de Darii esset combinatio inutilis.) Mair 1508, fol. 59vb.34‘The third argument: if the position were correct, an inference would be true or false depending on the decision of a single mouse. The consequent of this deduction is absurd, hence so is its antecedent. And I can substantiate the deduction. The following inference holds if there is a mouse in the corner: ‘This consequent is false; therefore, this consequent is false or there is a mouse in the corner’, where [the first part of] the consequent is denoted. The argument infers a disjunction from one of its disjuncts, and the consequent is not self-falsifying but true when there is a mouse in the corner. But as soon as the mouse leaves the corner, the inference turns invalid, since its antecedent is true and its consequent false, therefore etc.’ (Tertio arguitur: sequeretur quod aliqua consequentia esset bona et mala ad nutum unius muris. Consequens est absurdum, ergo et antecedens. Declaro consequentiam, nam bene sequitur ‘hoc consequens est falsum, ergo hoc consequens est falsum vel mus est in angulo’ continue demonstrando primam partem consequentis, posito quod mus sit in angulo. Arguitur a parte disiunctivae ad totam disiunctivam et consequens se non falsificat, immo consequens est verum, quando mus est in angulo. Et tamen mure egrediente angulum non sequitur, quia antecedens est verum et consequens falsum, igitur.) Mair 1508, fol. 59vb. Despite Mair's formulation, it should be the entire consequent and not its first part which is denoted by the demonstrative in his example, hence the insertion in square brackets.35Mair 1508, fol. 59vb.36Bricot 1986, 21–22. Incidentally, Mair's ‘Bricot’ analyses truth as follows: ‘The truth of an affirmative sentence requires two things: First, that it signifies to be the case as is the case. Second, that it does not falsify itself. For the truth of a negative sentence, either of that is sufficient.’ ( … duo requiruntur ad veritatem propositionis affirmativae. Primum est quod significet taliter esse qualiter est. Secundum quod non se falsificet. Et ad veritatem negativae sufficit utrumque istorum.) Mair 1508, fol. 59rb. Assuming that self-falsification can have these indirect forms, these formulations are equivalent.37Mair does not explicitly use this argument which seems to be implied in the structure of the text, since Mair deals with the objection that self-referential sentences cannot be entailed by non-self-referential ones. Mair uses yet another argument of this kind, proving that Bricot's definition of validity is undergenerating: an instance of ex impossibili quodlibet, i.e., an inference with an impossible antecedent and a self-referential consequent should probably be regarded as valid but is not on Bricot's definition (Mair 1508, fol. 60rb).38Mair 1508, fol. 60ra. The same holds with respect to the form of sentences with self-referential and non-self-referential negations, which is motivated by Bricot's concept of self-falsification (propositio cuius contradictoria se falsificat non est eiusdem formae cum propositione cum cuius contradictoria non se falsificat); Mair 1508, fol. 60rb.39Mair does not endorse the relevance concept of formality, hence this formulation aims at a different problem than that of Swyneshed and others who claim that σ is formally valid; Mair's formality is a matter of the structure of sentential meaning and possible inter-substitution in inference-schemata.40Mair 1508, fol. 60r.41Naturally, this term does not suggest that there is a distinction between object language and meta-language in Bricot's or Mair's semantics.42Hodges 1998, 16–19; Hodges Citation2001. Hodges introduces several definitions of what he calls ‘strong’ and ‘weak’ compositionality, but introducing one of them seems sufficient.43Gaifman 2000, 84.44Gaifman Citation1988, 1992, 2000.45The question is whether languages without such a set of instructions are possible; such option seems to be incompatible with conceiving language as a rule-governed system. Regardless, such language would lack organization and a systematic theory of it would be impossible.46Hughes 1982.47Buridanus 1994, 90–93.48Buridanus 2004.49Mair's own interpretation of Buridan seems to be close to the present interpretation. After describing Albert of Saxony's theory, which is identical with Buridan's meaning theory, Buridan's entailment theory is introduced by saying that there is only a small difference between Buridan and Albert (Buridanus parum differens ab Alberti). Also, in the case of Albert, Mair emphasises that Albert construes Liar sentences as hypothetical as a consequence of their implicit meaning, as opposed to Bricot, who is said to construe Liar sentences as categorical sentences. In the case of Buridan, he does not say either, but designates the virtually implied assertion as ‘propositio implicita’. (Mair 1508, fol. 58).50Buridanus 1994, 93.51Ibid, 90. Such solution is close to the position of Albert of Saxony whose analysis is limited to saying that two consimiles sentences (which in this context means sentences with identical explicit meaning) can be assigned different semantic values if one of them is paradoxical (Albert of Saxony 1988, 476–77).52Since Buridanian theory of logical consequence is based on defining validity in terms of truth-preservation and on a substitutional concept of formal validity (Buridanus Citation1976, Dutilh Novaes 2012), mutual (formal) entailment and synonymy are different relations in his logic.53Buridanus 2004, 159.54Mair 1508, fols. 64ff.55This presentation of compositionalist approaches makes no claim to completeness. The presented positions were chosen for their affinity to the procedural concept of meaning close to the modern counterpart of Swyneshedian semantics, Gaifman's operational non-compositional semantics. This resemblance supports the strategy of minimal adjustments.56This approach can be refined by adding temporal and possibly other indices (Lewis 1970).57Carnap 1947.58Cresswell Citation1985.59Lewis 1970. The difference from the previous version is that the meanings of simple expressions are one-node trees rather than intensions as such. (Lewis admits that his theory can be thought of as offering a formal counterpart to meaning rather than a theory of it, as opposed to the stronger claims of TIL.)60Tichý 1988; Duží et al. 2010.61Duží et al. 2010, 15–23. The semantic scheme of TIL is used here as a fine-grained model of sentential meaning, but, undoubtedly, there are other conceptual frameworks that could be used just as well and using terminology close to TIL is no more than an option.62The distinction develops two aspects of functions already distinguished by Church, i.e., ‘function in extension’ as mapping (mappings being identical if they have the same range and the same value for each argument in that range) and ‘functions in intension’ or ‘function concept’ as ‘the way in which a function yields or produces its value from its argument’ (Church 1956, 16).63We say that this is impossible ‘in general’ to emphasise that such approach would not explain the semantic behaviour of all sentences that can be formed in a given language; it would, naturally, cover the semantic behaviour of grounded sentences.64Any language with non-compositional semantics is learnable if the number of types of sentence-positions is finite. If possible, the number should be a fairly small number: the smaller the number of positions, the simpler the semantic theory.65To substantiate this position, Bricot endorses a specific theory of indirect self-falsification which was in part presented above (e.g. a sentence saying that its negation is true falsifies itself).66There is, of course, a quite detailed classification of self-falsification in Swyneshed's semantics, but its function is to illustrate rather than explicate the general definition. The same ‘deductive’ approach to network-analysis is presented in Uckelman 2009, 119.67Incidentally, this part of the Bricot–Mair dispute can be regarded as purely terminological: it is a matter of defining the concept of synonymy. Bricot would naturally agree that any two sentences can be synonymous in the sense of having identical ‘primary signification’, but for some reason does not regard this as sufficient for being synonymous. If both authors were willing to accept the other's definition (say, as definitions of synonymy1 and synonymy2), there would be no room for disagreement at this point.68Another way to draw the same distinction would be to say that a Mairian logical form is the structure of a sentence as such, whereas a Bricotian logical form is the structure of the use of a sentence.69Read could actually bite the bullet here, since Bradwardinian semantics agrees that meaning is contingently closed under material implication. Such theory would, however, have undesirable implications for pragmatics and the logic of epistemic contexts, rendering any speaker omniscient about his own world and, even worse, about all necessary truths.70Bricot suggests that inferences where both the antecedent and the consequent are self-referential, e.g. claims its own falsity, can be valid. These inferences, however, are not formally valid, hence the objection is not meant to cover them. They are in fact some kind of paradoxes of validity: they are truth-preserving because their components have the same truth-value but they have the form ‘A↠B’ or ‘A ⊢ B’.71Technically, this makes non-compositional semantics very close to a compositional one, a position in a network being ‘just another index’. The conceptual difference is that the position-indexicality is not concerned with the state of affairs corresponding to sentences and cannot be analysed away by a disambiguation (as required by the analysis of two-line puzzles).
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