Aristotelian Logic Axioms in Propositional Logic: The Pouch Method

Alvarez-Fontecilla, Enrique; Lungenstrass, Tomas

doi:10.1080/01445340.2018.1442172

Cited by 4 publications

(3 citation statements)

References 11 publications

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“…The theory of syllogistic reasoning was investigated by several authors as a generalization of classical Aristotelian syllogisms ( [1][2][3]). Categorical syllogisms ( [4,5]) consist of three main parts: the major premise, the minor premise, and the conclusion.…”

Section: Introductionmentioning

confidence: 99%

A Formal Analysis of Generalized Peterson’s Syllogisms Related to Graded Peterson’s Cube

Fiala

Murinová

2022

Mathematics

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Section: Introductionmentioning

confidence: 99%

A Formal Analysis of Generalized Peterson’s Syllogisms Related to Graded Peterson’s Cube

Fiala

Murinová

2022

Mathematics

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“…This is in agreement with the obversion and contraposition rules and the standard definition of distribution: from I(M,P)=O(M,P')= O(P,M') one realizes that M' and P' are distributed since M,P were not. The same definition of distribution, extended to indefinite terms, was used, e.g., by Alvarez and Correia [3] and was cited in Alvarez-Fontecilla and Lungenstrass [4] as having been used in 1932 by Wilkinson [5]. Note also that the arguments of a statement and of its contradictory one, have opposite distributionsin E(M,P) both M and P are distributed, while in the contradictory statement, I(M,P), both M and P are undistributed, (while M', P' are distributed).…”

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confidence: 99%

An Extension of the Rules of Valid Syllogism to Rules of Conclusive Syllogisms with Indefinite Terms

Rădulescu¹

2020

Preprint

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One extends to indefinite terms the Classical Syllogistics and the Rules of Valid Syllogism (RofVS). The valid syllo-gisms are generalized to conclusive syllogisms, made of conclusive pairs of categorical premises and their logical consequences (LCs), which all may contain indefinite terms: the positive, S,P,M, terms, and their complementary sets in the universe of discourse, U, i.e., the negative terms. A “pattern and type” classification, splits the 32 conclusive syllogisms into four types, Barbara, Darapti or Darii and Disamis – each containing eight syllogisms, which follow only three “patterns of inclusion and intersection”, namely either S⊆M⊆P (Barbara), or, M⊆S, M⊆P (Darapti), or M S≠Ø, M⊆P (Darii). (The Disamis type syllogisms follow the Darii pattern, but switch the roles played by the P and S terms.) By using only four Rules of Conclusive Syllogisms (RofCS), (and abandoning the RofVS “two negative prem-ises are not allowed” and “the middle term has to be dis-tributed in at least one premise”), one can make the RofCS into a “theory” "almost equivalent" with the formulas which describe all the conclusive syllogisms, including existential import syllogisms (to which the RofVS do not make any reference). Very importantly, each precise LC of the Barba-ra, Darapti and Darii patterns pinpoints a unique partitioning subset of U: S=S M P and P'=S' M' P' for Bar-bara’s pattern - which entails two LCs; M=M S P for Darapti’s pattern; S M P≠Ø for Darii’s pattern. The middle term’s elimination from the LC, initiated by Aristo-tle, simplifies, but weakens the LC.

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“…This is in agreement with the obversion and contraposition rules and the standard definition of distribution: from I(M,P)=O(M,P')= O(P,M') one realizes that M' and P' are distributed since M,P were not. The same definition of distributeon, extended to indefinite terms, was used, e.g., by Alvarez and Correia [3] and was cited in Alvarez-Fontecilla and Lungenstrass [4] as having been used in 1932 by Wilkinson [5]. Note also that the arguments of a statement and of its contradictory one, have opposite distributionsin E(M,P) both M and P are distributed, while in the contradictory statement, I(M,P), both M and P are undistributed, (while M', P' are distributed).…”

mentioning

confidence: 99%

A simple extension to indefinite terms of the Classical Syllogistics and of the Rules of Valid Syllogism

Rădulescu¹

2020

Preprint

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One uses a minimal extension to indefinite terms of the Classical Syllogisticsand of the Rules of Valid Syllogism. The valid syllogisms (VS) are replacedby valid categorical arguments (VCAs), made of conclusive pairs of categoricalpremises (PCPs) and their logical consequences (LCs), which can containindefinite (both positive and negative) terms. The Classical Syllogistics isreplaced by explicit formulas reproducing all 32 VCAs and their LCs. One noticesthat all the VCAs (and VS) follow one of the Barbara, Darapti or Darii“patterns of inclusion and intersection”, namely either S⊆M⊆P, or, M⊆S,M⊆P, or M⋂S≠Ø, M⊆P. One shows that by using slight generalizations of therules of valid syllogism (RofVS) one can transform the RofVS into a “theory”equivalent to Classical Syllogistics. By using appropriate rules of valid categoricalargument (RofVCA), (which disregard the RofVS asserting that “two negativepremises are not allowed” and “the middle term has to be distributed in atleast one premise”), one can make the RofVCA into a “theory” "almost equivalent"with the formulas which describe all the VCAs. One also notices that eachprecise LC of the Barbara, Darapti and Darii patterns pinpoints a unique partitioningsubset of the universe of discourse, U: S=S⋂M⋂P and P'=S'⋂ M'⋂P' for Barbara’s pattern - which entails two LCs; M=M⋂S⋂P forDarapti’s pattern; S⋂M⋂P≠Ø for Darii’s pattern. The middle term eliminationfrom the LC, initiated by Aristotle, simplifies but weakens the LC.

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Aristotelian Logic Axioms in Propositional Logic: The Pouch Method

Cited by 4 publications

References 11 publications

A Formal Analysis of Generalized Peterson’s Syllogisms Related to Graded Peterson’s Cube

A Formal Analysis of Generalized Peterson’s Syllogisms Related to Graded Peterson’s Cube

An Extension of the Rules of Valid Syllogism to Rules of Conclusive Syllogisms with Indefinite Terms

A simple extension to indefinite terms of the Classical Syllogistics and of the Rules of Valid Syllogism

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