On sequence-conclusion natural deduction systems

Boričić, Branislav R.

doi:10.1007/bf00649481

Cited by 41 publications

(6 citation statements)

References 10 publications

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“…In classical logic, these collections can be sets, in which formulas are tied by an implicit disjunction (cf. [7]). Another possibility is to understand multipleconclusion deductions in the style of Shoesmith and Smiley [39].…”

Section: Thesis [I]mentioning

confidence: 99%

Logical constants as punctuation marks.

Došen¹

1989

Notre Dame J. Formal Logic

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This paper presents a proof-theoretical approach to the question "What is a logical constant?" This approach starts with the assumption that logic is the science of formal deductions, and that basic formal deductions are structural deductions, i.e. deductions independent of any constant of the language to which the premises and conclusions belong. Logical constants, on which the remaining formal deductions are dependent, may be said to serve as "punctuation marks" for some structural features of deductions; this punctuation function, exhibited in equivalences which amount to analyses of logical constants, is taken as a criterion for being a logical constant. The paper presents an account of philosophical analysis which covers the proposed analyses of logical constants. Some related assumptions concerning logic are also considered. In particular, since a logical system is completely determined by its structural deductions, alternative logical systems arise by changing structural deductions while having constants with the same punctuation function. Some other approaches to the question "What is a logical constant?", grammatical, model-theoretical, and proof-theoretical, are briefly considered.

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Section: Thesis [I]mentioning

confidence: 99%

Logical constants as punctuation marks.

Došen¹

1989

Notre Dame J. Formal Logic

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“…Carnap observes that standard axiomatizations of propositional classical logic fail to uniquely specify the class of admissible valuations. 4 In other words, it includes nonstandard interpretations of the logical connectives. In particular, Carnap shows that as far as the proof system is concerned, there is nothing that rules out a valuation in which everything is true.…”

Section: Categoricitymentioning

confidence: 99%

Speech Acts, Categoricity, and the Meanings of Logical Connectives

Hjortland¹

2014

Notre Dame J. Formal Logic

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In bilateral systems for classical logic, assertion and denial occur as primitive signs on formulas. Such systems lend themselves to an inferentialist story about how truth-conditional content of connectives can be determined by inference rules. In particular, for classical logic there is a bilateral proof system which has a property that Carnap in 1943 called categoricity. We show that categorical systems can be given for any finite many-valued logic using n-sided sequent calculus. These systems are understood as a further development of bilateralism-call it multilateralism. The overarching idea is that multilateral proof systems can incorporate the logic of a variety of denial speech acts. So against Frege we say that denial is not the negation of assertion and, with Mark Twain, that denial is more than a river in Egypt.

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“…For example, the usual introduction rule for ∨ fails for his intended meaning. The simultaneous presence of the disjuncts can perhaps be captured by the par connective of linear logic (see the rules for 'par logic' in Hyland & de Paiva, 1993, p. 282), or the structural connective "," in Boricic' (1985) multiple-conclusion natural deduction system . Note that in these cases the elimination rule for disjunction is not the one familiar from classical (or intuitionistic) logic, but the disjunctive syllogism, as in Kant.…”

Section: Proof Ifmentioning

confidence: 99%

A Formalization of Kant’s Transcendental Logic

Achourioti

Lambalgen

2011

The Review of Symbolic Logic

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ABSTRACT. Although Kant envisaged a prominent role for logic in the argumentative structure of his Critique of pure reason [12], logicians and philosophers have generally judged Kant's logic negatively. What Kant called 'general' or 'formal' logic has been dismissed as a fairly arbitrary subsystem of first order logic, and what he called 'transcendental logic' is considered to be not a logic at all: no syntax, no semantics, no definition of validity. Against this, we argue that Kant's 'transcendental logic' is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first order logic. The main technical application of the formalism developed here is a formal proof that Kant's Table of Judgements in §9 of the Critique of pure reason, is indeed, as Kant claimed, complete for the kind of semantics he had in mind. This result implies that Kant's 'general' logic is after all a distinguished subsystem of first order logic, namely what is known as geometric logic.

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On sequence-conclusion natural deduction systems

Cited by 41 publications

References 10 publications

Logical constants as punctuation marks.

Logical constants as punctuation marks.

Speech Acts, Categoricity, and the Meanings of Logical Connectives

A Formalization of Kant’s Transcendental Logic

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