Per Se Modality and Natural Implication – an Account of Connexive Logic in Robert Kilwardby

Johnston, Spencer

doi:10.12775/llp.2019.033

Cited by 2 publications

(9 citation statements)

References 17 publications

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“…In the case of Doubt 1 and Doubt 2, these conclusions ended up in favour of the connexive principles AT1 and AT2. Doubt 3, in contrast, consists of a main argument (suggesting that ex impossibili quodlibet and necessarium ex quodlibet should be restricted to EIQ ⋆ and NEQ ⋆ ), supplemented by two considerations, [1] and [2], which must be meant to support the former argument since they are introduced by saying that "this is true for the following reason". Next follows a reply to the main argument ("To the contrary") which ends up with the anticonnexive conclusion that "in such inferences one opposite does follow from another".…”

Section: Discussionmentioning

confidence: 99%

“…Some related criteria have been put forward in [9, p. 569]: " (1) No proposition implies its own negation. (2) No proposition implies each of two contradictory propositions. (3) No proposition implies every proposition.…”

Section: At2 ∼(∼P → P)mentioning

confidence: 99%

“…As a matter of fact, this doubt consists of three objections: [1] For it seems that one and the same thing does follow from the same thing's being so and not being so, because if you are sitting then God exists and if you are not sitting then God exists, because the necessary follows from anything. [2] Further, if you are sitting then one of the following is true: that you are sitting, that you are not sitting. And if you are not sitting, one of them is true.…”

Section: Kilwardby's Defence Of At1mentioning

confidence: 99%

“…Without further ado, Kilwardby simply maintains that the full principle necessarium ex quodlibet holds only for accidental implication, while natural implication presumably satisfies NEQ only in a restricted form. In objections [2] and [3], the theologically necessary proposition 'God exists' is replaced by the logically necessary proposition 'p ∨ ∼p'. Thus the pair (p → (p∨∼p)), (∼p → (p∨∼p)) forms another counterexample to AT1.…”

Section: Kilwardby's Defence Of At1mentioning

confidence: 99%

“…The main difference between objections [2] and [3] consists in the underlying justification. In [2], the validity of the two implications seems to be based again on principle necessarium ex quodlibet, while [3] explicitly refers to the principles of disjunction introduction. The latter argument is thus stronger than the former one because it escapes Kilwardby's reply to objection [1].…”

Section: Kilwardby's Defence Of At1mentioning

confidence: 99%

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Kilwardby's 55th Lesson

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2020

LLP

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In "Lectio 55" of his Notule libri Priorum, Robert Kilwardby discussed various objections that had been raised against Aristotle's Theses. The first thesis, AT1, says that no proposition q is implied both by a proposition p and by its negation, ∼p. AT2 says that no proposition p is implied by its own negation. In Prior Analytics, Aristotle had shown that AT2 entails AT1, and he argued that the assumption of a proposition p such that (∼p → p) would be "absurd". The unrestricted validity of AT1, AT2, however, is at odds with other principles which were widely accepted by medieval logicians, namely the law Ex Impossibili Quodlibet, EIQ, and the rules of disjunction introduction. Since, according to EIQ, the impossible proposition (p ∧ ∼p) implies every proposition, it also implies ∼(p∧∼p), in contradiction to AT2. Furthermore, by way of disjunction introduction, the proposition (p ∨ ∼p) is implied both by p and by ∼p, in contradiction to AT1. Kilwardby tried to defend AT1, AT2 against these objections by claiming that EIQ holds only for accidental but not for natural implications. The second argument, however, cannot be refuted in this way because Kilwardby had to admit that every disjunction (p ∨ q) is naturally implied by its disjuncts. He therefore introduced the further requirement that, in order to constitute a genuine counterexample to AT1, (p → q) and (∼p → q) have to hold "by virtue of the same thing". In a recently published paper, Spencer Johnston accepted this futile defence of AT1 and developed a formal semantics that would fit Kilwardby's presumably connexive implication. This procedure, however, is misguided because the remaining considerations of Lesson 55  which were entirely ignored by Johnston  show that Kilwardby eventually recognized that AT2 is bound to fail. After all he concluded: "So it should be granted that from the impossible its opposite follows, and that the necessary follows from its opposite".

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Section: Discussionmentioning

confidence: 99%

Section: At2 ∼(∼P → P)mentioning

confidence: 99%

Section: Kilwardby's Defence Of At1mentioning

confidence: 99%

Section: Kilwardby's Defence Of At1mentioning

confidence: 99%

Section: Kilwardby's Defence Of At1mentioning

confidence: 99%

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Kilwardby's 55th Lesson

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2020

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Rewriting the History of Connexive Logic

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2022

J Philos Logic

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The “official” history of connexive logic was written in 2012 by Storrs McCall who argued that connexive logic was founded by ancient logicians like Aristotle, Chrysippus, and Boethius; that it was further developed by medieval logicians like Abelard, Kilwardby, and Paul of Venice; and that it was rediscovered in the 19th and twentieth century by Lewis Carroll, Hugh MacColl, Frank P. Ramsey, and Everett J. Nelson. From 1960 onwards, connexive logic was finally transformed into non-classical calculi which partly concur with systems of relevance logic and paraconsistent logic. In this paper it will be argued that McCall’s historical analysis is fundamentally mistaken since it doesn’t take into account two versions of connexivism. While “humble” connexivism maintains that connexive properties (like the condition that no proposition implies its own negation) only apply to “normal” (e.g., self-consistent) antecedents, “hardcore” connexivism insists that they also hold for “abnormal” propositions. It is shown that the overwhelming majority of the forerunners of connexive logic were only “humble” connexivists. Their ideas concerning (“humbly”) connexive implication don’t give rise, however, to anything like a non-classical logic.

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Per Se Modality and Natural Implication – an Account of Connexive Logic in Robert Kilwardby

Cited by 2 publications

References 17 publications

Kilwardby's 55th Lesson

Kilwardby's 55th Lesson

Rewriting the History of Connexive Logic

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