A Quantified Temporal Logic for Ampliation and Restriction

In: Vivarium

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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  • KnuuttilaS. Modalities in Medieval Philosophy 1993 (London-New York 1993)

  • de LiberaA. see Lambertus Autissiodorensis; Roger Bacon

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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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  • 24)

    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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  • 25)

    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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  • 27)

    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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  • 29)

    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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  • 31)

    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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  • 32)

    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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  • 35)

    Cf. Knuuttila (1993), passim.

  • 36)

    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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  • 37)

    Such as Chagrov and Zakharyaschev (1997), §3.5.

  • 38)

    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.

  • 39)

    Fitting and Mendelsohn (1998), 94.

  • 43)

    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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  • 44)

    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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  • 47)

    See, e.g., Knuuttila (1982), 347-348, 354-357.

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