A Quantified Temporal Logic for Ampliation and Restriction
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Abstract
Temporal logic as a modern discipline is separate from classical logic; it is seen as an addition or expansion of the more basic propositional and predicate logics. This approach is in contrast with logic in the Middle Ages, which was primarily intended as a tool for the analysis of natural language. Because all natural language sentences have tensed verbs, medieval logic is inherently a temporal logic. This fact is most clearly exemplified in medieval theories of supposition. As a case study, we look at the supposition theory of Lambert of Lagny (Auxerre), extracting from it a temporal logic and providing a formalization of that logic.
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Chagrov A. & Zakharyaschev M. Modal Logic 1997 (Oxford 1997)
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Fitting M. & Mendelsohn R.L. First-order Modal Logic 1998 (Synthese Historical Library, 277; Dordrecht 1998)
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Knuuttila S. Kretzmann N., Kenny A., Pinborg J. & Stump E. ‘Modal Logic’ The Cambridge History of Later Medieval Philosophy From the Rediscovery of Aristotle to the Desintegration of Scholasticism 1100-1600 1982 (Cambridge-London-New York-New Rochelle-Melbourne-Sydney 1982, 342-357)
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Knuuttila S. Modalities in Medieval Philosophy 1993 (London-New York 1993)
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de Libera A. see Lambertus Autissiodorensis; Roger Bacon
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Prior A.N. Time and Modality 1957 (Oxford 1957)
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de Rijk L.M. ‘Some Thirteenth Century Tracts on the Game of Obligation, III’ Vivarium 1976 14 26 49 (1976),
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3)
Lambert of Auxerre, ‘Properties of Terms’, 104; idem, Logica (Summa Lamberti) VIII, 205. ‘Significatio termini est intellectus rei ad quem intellectum representandum [representandum om. Alessio] rei vox imponitur ad voluntatem instituentis.’
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4)
Lambert of Auxerre, ‘Properties of Terms’, 106; Lambert of Auxerre, Logica (Summa Lamberti), VIII, 206: ‘Quarto modo dicitur suppositio acceptio termini pro [pro: S.L. Uckelman, per: Alessio] se sive pro re sua, vel pro aliquo supposito contempto sub re sua vel pro aliquibus suppositis contemptis sub re sua.’
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5)
Lambert of Auxerre, ‘Properties of Terms’, 109; Lambert of Auxerre, Logica (Summa Lamberti), VIII, 208: ‘Naturalis suppositio est quam habet terminus a se et a natura se.’
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9)
Lambert of Auxerre, ‘Properties of Terms’, 110; idem, Logica (Summa Lamberti), VIII, 209: ‘Simplex suppositio est illa secundum quam tenetur terminus pro se vel pro re sua, non habito respectu ad supposita sub se contempta.’
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12)
Lambert of Auxerre, ‘Properties of Terms’, 111; idem, Logica (Summa Lamberti), VIII, 209: ‘Discreta est illa quam habet terminus discretus in se, [. . .] ut quando sumitur terminus communis cum pronomine determinato.’
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13)
Lambert of Auxerre, ‘Properties of Terms’, 111; idem, Logica (Summa Lamberti), VIII, 210: ‘Communis est illa que termino communi convenit.’
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19)
Lambert of Auxerre, ‘Properties of Terms’, 112; idem, Logica (Summa Lamberti), VIII, 211: ‘Exilis immobilis est illa quam habet terminus communis quando de necessitate tenetur pro pluribus suppositis sub se contemptis, non tamen pro omnibus, nec sub ipso potest fieri descensus.’
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20)
Lambert of Auxerre, ‘Properties of Terms’, 116; idem, Logica (Summa Lamberti), VIII, 213: ‘Ad explanationem istius regule secundum quod terminus communis ibi ponitur ad differentiam termini discreti, qui non potest restringi nec ampliari.’
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21)
Lambert of Auxerre, ‘Properties of Terms’, 114; idem, Logica (Summa Lamberti), VIII, 212: ‘Quarto modo dicitur appellatio acceptio termini pro supposito vel pro suppositis actu existentibus.’
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22)
Lambert of Auxerre, ‘Properties of Terms’, 115; idem, Logica (Summa Lamberti), VIII, 213: ‘Sciendum autem quod proprie loquendo non dicuntur appellata nisi sint actualiter existentia; appellatur enim proprie quod est, et non quod non est, et ideo bene dicitur quod appellatio est pro existentibus suppositis vel pro supposito.’
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Lambert of Auxerre, ‘Properties of Terms’, 116; idem, Logica (Summa Lamberti), VIII, 213: ‘Terminus communis substantialis vel accidentalis non restrictus aliunde supponens vel apponens verbo presentis temporis non habenti vim ampliandi a se nec ab alio, restringitur ad supponendum pro presentibus, si appellata habeat; si vero non, recurrit ad non existentes [existentiam]’.
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Lambert of Auxerre, ‘Properties of Terms’, 134; idem, Logica (Summa Lamberti), VIII, 226: ‘Restrictio est minoratio ambitus termini communis, secundum quam pro paucioribus suppositis teneter terminus communis quam exigat sua actualis suppositio.’
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Lambert of Auxerre, ‘Properties of Terms’, 137; idem, Logica (Summa Lamberti), VIII, 228: ‘Ampliatio est extensio ambitus termini communis secundum quod teneri potest terminus communis pro pluribus suppositis quam exigit sua actualis suppositio.’
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Lambert of Auxerre, ‘Properties of Terms’, 138; idem, Logica (Summa Lamberti), VIII, 228: ‘Quedam enim sunt nomina que habent virtutem ampliandi ut possibile, necessarium; et similiter quedam verba ut potest [. . .]; similiter et quedam adverbia ut potentialiter necessario [. . .]’
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Lambert of Auxerre, ‘Properties of Terms’, 138; idem, Logica (Summa Lamberti), VIII, 229: ‘Verba quorum actus comparatus ad subiectum de subiecto dicitur, in subiecto tamen non est, ut sunt ista: potest, opinatur, laudatur.’
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Lambert of Auxerre, ‘Properties of Terms’, 138; idem, Logica (Summa Lamberti), VIII, 229: ‘Illa dicuntur ampliare ad tempora que faciunt terminum extendi ad omnes differentias temporis.’
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33)
Lambert of Auxerre, ‘Properties of Terms’, 129; idem, Logica (Summa Lamberti), VIII, 223: ‘Terminus communis accidentalis non restrictus aliunde supponens verbo preteriti temporis, supponere potest pro presentibus et preteritis; apponens vero solum [Kretzmann and Stump, terminum Alessio] supponit pro preteritis; si vero fuerit terminus substantialis supponens vel apponens verbo preteriti temporis, semper pro preteritis supponit.’
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34)
Lambert of Auxerre, ‘Properties of Terms’, 129; idem, Logica (Summa Lamberti), VIII, 223: ‘Terminus communis accidentalis non restrictus aliunde supponens verbo futuri temporis, supponere potest pro presentibus et futuris; apponens vero solum tenetur pro futuris; si vero fuerit terminus substantialis supponens vel apponens verbo futuri temporis, semper tenetur pro futuris.’
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Cf. Knuuttila (1993), passim.
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Lambert of Auxerre, ‘Properties of Terms’, 138; idem, Logica (Summa Lamberti), VIII, 229: ‘Unde cum dicitur: ‘home est animal necessario’ aliud est ac si diceretur: ‘id est, in omni tempore convenit homini esse animal’, scilicet in presenti preterito et futuro.’
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Such as Chagrov and Zakharyaschev (1997), §3.5.
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See Fitting and Mendelsohn (1998), 89. One exception to this view is Arthur Prior (1957), who believes that a coherent philosophical treatment of quantified modal logic can be given.
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39)
Fitting and Mendelsohn (1998), 94.
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Lambert of Auxerre, ‘Properties of Terms’, 123; idem, Logica (Summa Lamberti), VIII, 219: “Nam nullo homine existente hec est falsa: ‘omnis homo est’, ergo sua contradictoria erit vera, hec sciliect: ‘aliquis homo non est’.”
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Lambert of Auxerre, ‘Properties of Terms’, 138; idem, Logica (Summa Lamberti), VIII, 229: ‘Terminum [tenet] pro suppositis actu et non existentibus.’
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45)
Lambert of Auxerre, ‘Properties of Terms’, 117; idem, Logica (Summa Lamberti), VIII, 214: ‘Ad cognoscendum autem que verba ampliant et que non, sciendum quod ad substantiam actus potest comparari duplicitier: uno modo quantum ad illud in quo est et de quo enunciatur[. . .] alio modo tamquam ad id de quo enunciatur non tamen in ipso est.’
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46)
Lambert of Auxerre, ‘Properties of Terms’, 133; idem, Logica (Summa Lamberti), VIII, 225: ‘Dicendum quod: ‘album erit Socrates’ habet duas acceptiones: potest enim accipi sub hoc sensu: id quod erit album erit Sortes; vel sub isto: quod est album erit Sortes.’
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See, e.g., Knuuttila (1982), 347-348, 354-357.
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A Quantified Temporal Logic for Ampliation and Restriction
In: VivariumSearch for other papers by Sara L. Uckelman in
Current site
Google Scholar
PubMed
Abstract
Temporal logic as a modern discipline is separate from classical logic; it is seen as an addition or expansion of the more basic propositional and predicate logics. This approach is in contrast with logic in the Middle Ages, which was primarily intended as a tool for the analysis of natural language. Because all natural language sentences have tensed verbs, medieval logic is inherently a temporal logic. This fact is most clearly exemplified in medieval theories of supposition. As a case study, we look at the supposition theory of Lambert of Lagny (Auxerre), extracting from it a temporal logic and providing a formalization of that logic.
| All Time | Past Year | Past 30 Days | |
|---|---|---|---|
| Abstract Views | 429 | 35 | 5 |
| Full Text Views | 124 | 1 | 0 |
| PDF Views & Downloads | 39 | 3 | 0 |