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A Binary Quantifier for Definite Descriptions in Intuitionist Negative Free Logic: Natural Deduction and Normalisation

100%

Kürbis N.

nr 2

81-97

This paper presents a way of formalising definite descriptions with a binary quantifier ℩, where ℩x[F, G] is read as `The F is G'. Introduction and elimination rules for ℩ in a system of intuitionist negative free logic are formulated. Procedures for removing maximal formulas of the form ℩x[F, G] are given, and it is shown that deductions in the system can be brought into normal form.

Two Treatments of Definite Descriptions in Intuitionist Negative Free Logic

75%

Kürbis N.

nr 4

Sentences containing definite descriptions, expressions of the form `The F', can be formalised using a binary quantier that forms a formula out of two predicates, where ℩x[F;G] is read as `The F is G'. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INF℩ of intuitionist negative free logic extended by such a quantier, which was presented in [4], INF℩ is first compared to a system of Tennant's and an axiomatic treatment of a term forming ℩ operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of INF℩ in which the G of ℩x[F;G] is restricted to identity. INF℩ is then compared to an intuitionist version of a system of Lambert's which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion.

IDENTYCZNOŚĆ, PEWNE ZAIMKI FUNKTOROWE I DESKRYPCJE

51%

Wojciechowski E.

Roczniki Filozoficzne

2013

tom 61

nr 3

125-141

From the logical point of view, the most interesting among the pronouns are demonstrative pronouns (especially: this/that), indefinite pronouns (a/an), definite pronoun (the) and quantifying pronouns (every, all, some). Unlike personal pronouns (e.g. I/you/he) they are in fact functors (of the n/n category). The differentiation between personal pronouns (n) and functor pronouns (n/n) is vital here. This differentiation does not exist in traditional grammar. The study is limited to determining functor pronouns with the use of logical properties of quantifying expressions, which are functor pronouns themselves – all (n) and some (cr) – formally expressed in the quantifier-less calculus of names (BRN). The calculus is properly enriched with demonstrative pronouns (demonstrativa), in connection to certain studies by Toshiharu Waragai (LID). An attempt to employ this system (BRND) in the analysis of some fragments of Ockham’s Summa Logicae is shown here. The work is concluded with the analysis of a functor pronoun the only (t), being a special case of a definite pronoun, which is characterised here by means of rules. The work reveals the connection between this pronoun and the operator of definite descriptions (marked in the same way) in relation to a certain Ludwik Borkowski’s conception.