Remarks on the Scott–Lindenbaum Theorem

Payette, Gillman; Schotch, Peter Κ.

doi:10.1007/s11225-013-9519-y

Cited by 10 publications

(3 citation statements)

References 7 publications

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“…The relevance of the notion of entailment relation to point-free topology and constructive algebra has been pointed out in [14], and has been used very widely, e.g., in [20–22, 24, 25, 29, 32, 80, 89, 93, 100, 114, 115]. Consequence and entailment have further caught interest from various other angles [6, 35, 41, 52–54, 83, 99, 104, 117].…”

Section: Entailment Relationsmentioning

confidence: 99%

The Jacobson Radical of a Propositional Theory

Fellin¹,

Schuster²,

Wessel³

2021

Bull. symb. log

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Alongside the analogy between maximal ideals and complete theories, the Jacobson radical carries over from ideals of commutative rings to theories of propositional calculi. This prompts a variant of Lindenbaum's Lemma that relates classical validity and intuitionistic provability, and the syntactical counterpart of which is Glivenko's Theorem. The Jacobson radical in fact turns out to coincide with the classical deductive closure. As a by-product we obtain a possible interpretation in logic of the axioms-as-rules conservation criterion for a multi-conclusion Scott-style entailment relation over a single-conclusion one. MSC 2010: 03F03, 03F65.

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Section: Entailment Relationsmentioning

confidence: 99%

The Jacobson Radical of a Propositional Theory

Fellin¹,

Schuster²,

Wessel³

2021

Bull. symb. log

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“…4 The relevance of the notion of entailment relation to point-free topology and constructive algebra has been pointed out in [11]; this has been used e.g., in [16,17,20,24,25,58,62]. Consequence and entailment have caught interest from various angles [6,29,30,37,38,59,72,77].…”

Section: Entailmentmentioning

confidence: 99%

Eliminating Disjunctions by Disjunction Elimination

2017

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Completeness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation given by Scott in 1974.To this end we work with multi-conclusion entailment relations as extending single-conclusion entailment relations. In a nutshell, the additional axioms with disjunctions in positive position can be eliminated by reducing them to the corresponding disjunction elimination rules, which in turn prove admissible in all known mathematical instances. In deduction terms this means to fold up branchings of proof trees by way of properties of the relevant mathematical structures.Applications include the syntactical counterparts of the theorems or lemmas known under the names of Artin–Schreier, Krull–Lindenbaum, and Szpilrajn. Related work has been done before on individual instances, e.g., in locale theory, dynamical algebra, formal topology and proof analysis.

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“…14 For a critical discussion and analysis of this received view, which is in fact one of the tenets of logical empiricism, see for instance(Sloman 1965).15 This theorem is a result proved by(Scott 1974), who saw it as generalization of Lindenbaum's Theorem (see(Scott 1974: 416). This is why some authors mention it as Lindenbaum-Scott Theorem (see(Koslow 1992) and(Rumfitt 2015)) and some others as Scott-Lindenbaum Theorem (seePayette and Schotch 2014).16 [(Russell 1919): 153-154], for instance, argued that logical consequence has no modal ingredient and that its essential feature is only truth-preservingness. 17 See(Etchemendy 1990 andPrawitz 2005) for a criticism of Tarski's account on this aspect and (García-Carpintero 2003) for an analysis of the debates generated by Etchemendy's book.Philosophical Accounts of First-Order Logical Truths…”

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confidence: 99%

Philosophical Accounts of First-Order Logical Truths

Brîncuş

2019

Acta Anal

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Starting from certain metalogical results (the completeness theorem, the soundness theorem, and Lindenbaum-Scott theorem), I argue that first-order logical truths of classical logic are a priori and necessary. Afterwards, I formulate two arguments for the idea that first-order logical truths are also analytic, namely, I first argue that there is a conceptual connection between aprioricity, necessity, and analyticity, such that aprioricity together with necessity entails analyticity; then, I argue that the structure of natural deduction systems for FOL displays the analyticity of its truths. Consequently, each philosophical approach to these truths should account for this evidence, i.e., that first-order logical truths are a priori, necessary, and analytic, and it is my contention that the semantic account is a better candidate.

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Remarks on the Scott–Lindenbaum Theorem

Cited by 10 publications

References 7 publications

The Jacobson Radical of a Propositional Theory

The Jacobson Radical of a Propositional Theory

Eliminating Disjunctions by Disjunction Elimination

Philosophical Accounts of First-Order Logical Truths

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