Axiomatic inscriptional syntax. Part II. The syntax of protothetic.

Rickey, V. Frederick

doi:10.1305/ndjfl/1093890806

Cited by 11 publications

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“…System MO will contain all the syntactical terms necessary for formulating the directive which incorporates the Axiom of Choice into Ontology. It is not our intent to provide a detailed description of Rickey's system, for complete details can be found in [22] and [23]. What follows presupposes some familiarity with these articles.…”

Section: Axiomatic Inscriptional Syntaxmentioning

confidence: 99%

“…The relationship between Protothetic and Ontology is somewhat similar to that between the propositional calculus and the ω -order functional calculus. Details regarding Protothetic can be found in [5], [15], [23], and [28]. In the present work we will assume that an adequately developed Protothetical base has been provided and no explicit reference to purely Protothetical reasoning is made.…”

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confidence: 99%

“…A formal statement of the Rule of Ontology can be found in [16] which itself refers to [15]. A formal presentation of the Rule of Protothetic which is incorporated into the Rule of Ontology can be found in [21] and [23].…”

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confidence: 99%

“…14. System MO is essentially Rickey's system MP {cf, [21] or [23]) with some minor modifications and adjustments chief among which are the addition of the primitives 'effhp' and 'tho'.…”

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confidence: 99%

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Leśniewski's ontology extended with the axiom of choice.

Kowalski¹

1977

Notre Dame J. Formal Logic

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Section: Axiomatic Inscriptional Syntaxmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

“…14. System MO is essentially Rickey's system MP {cf, [21] or [23]) with some minor modifications and adjustments chief among which are the addition of the primitives 'effhp' and 'tho'.…”

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Leśniewski's ontology extended with the axiom of choice.

Kowalski¹

1977

Notre Dame J. Formal Logic

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“…Condition 1 is necessary, since it is required for the very formulation of AC*, and is, then, trivially required for the proof. 20 Condition 3 allows the complex functors constructed in the course of the proofs to be substituted into contexts where they are required. Theses of extensionality were used extensively throughout the deductions.…”

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An investigation concerning the Hilbert-SierpiÅski logical form of the axiom of choice.

Davis¹

1975

Notre Dame J. Formal Logic

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In [9], Hubert introduced the ε-calculus 1 which ''differs from the ordinary first-order functional calculus by not containing quantifiers among the primitive notation'' ([3], p. 183), as a part of the finitistic examination of the foundations of mathematics. By introducing a new primitive term, ε, by means of the axiom schema:Hubert was able to define the particular quantifier by:and prove all the usual rules of quantification theory ([10], pp. 67-68). The intuitive reading of ( ε x F(x)' is 'an F', or more concretely, ( ε x (x is a man)' is read 'a man'. Thus 'ε' corresponds, more or less, to the indefinite article 'a' ('an'), and A is to be read "If anything has the property F, then an F has the property F".Although Hubert seems to see a connection between A and the axiom of choice, in that he refers to the ε-operator as playing the role of a choice function ([10], p. 68), others find it misleading to refer to A as a "generalized principle of choice". 2 Wang argues that:If the axiom of choice were a special case of the ε-rule, why does the consistency of the axiom of choice not follow from the ε-theorems according to which application of the ε-rules can be dispensed with if the ε-operator occurs in neither the axioms nor the conclusions? Indeed, if the axiom of choice were derivable from the ε-rule, we would, by the ε-theorems, be able to derive the axiom of choice from the other axioms of set theory. 2In [23], Sierpiήski states what he calls Hubert's axiom: B There is a function, ε, associating with every property, P, for which there exists at least one object having that property, an object, ε(P), having the property P. and argues that "From the logical axiom ... it follows that there exists a function associating with every non-empty set a certain element of that set". Sierpiήski carries out the proof by associating with every set, A, the property of being an element of A, P A .

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