Contextual Deduction Theorems

Raftery, James

doi:10.1007/s11225-011-9353-z

Cited by 7 publications

(5 citation statements)

References 42 publications

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“…Finding this function involves determining the atomic structure of the lattice of theories of

I

, which is the object of . Corollary The logic

I

is not filter‐distributive, does not satisfy any Contextual Deduction‐Detachment Theorem ( CDDT ) , and does not satisfy any Deduction‐Detachment Theorem ( DDT ) . Proof For finitary protoalgebraic logics, being filter‐distributive is equivalent to having a parameterized disjunction [, Theorem 2.5.17]; by [, Theorem 6.8], having a CDDT implies being filter‐distributive. Finally, the DDT is a stronger form of the CDDT.…”

Section: Some Metalogical Resultsmentioning

confidence: 99%

The simplest protoalgebraic logic

Font

2013

Math. Log. Quart.

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Key words abstract algebraic logic, implication, paremeterized deduction theorem, Leibniz hierarchy, Frege hierarchy, protoalgebraic logic, implicative logic. MSC (2010) 03G27, 03B22, 03B47The logic I is the sentential logic defined in the language with just implication → by the axiom of reflexivity or identity "ϕ → ϕ" and the rule of Modus Ponens "from ϕ and ϕ → ψ to infer ψ". The theorems of this logic are exactly all formulas of the form ϕ → ϕ. We argue that this is the simplest protoalgebraic logic, and that in it every set of assumptions encodes in itself not only all its consequences but also their proofs. In this paper we study this logic from the point of view of abstract algebraic logic, and in particular we use it as a relatively natural counterexample to settle some open problems in this theory. It appears that this logic has almost no properties: it is neither equivalential nor weakly algebraizable; it does not have an algebraic semantics; it does not satisfy any form of the Deduction Theorem, other than the most general parameterized and local one that all protoalgebraic logics satisfy; it is not filter-distributive; and so on. It satisfies some forms of the interpolation property but in a rather trivial way. Very few things are known about its algebraic counterpart, save that its intrinsic variety is the class of all algebras of the similarity type.

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“…Finding this function involves determining the atomic structure of the lattice of theories of

I

, which is the object of . Corollary The logic

I

Section: Some Metalogical Resultsmentioning

confidence: 99%

The simplest protoalgebraic logic

Font

2013

Math. Log. Quart.

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“…The main motivation for RA is that it algebraizes the logic R. The algebraization process for R t and DMM carries over verbatim to R and RA, provided we use (12) as a formal device for eliminating all mention of e. Further work on relevant algebras can be found in [18,21,37,38,52,54,65,66].…”

Section: Relevant Algebrasmentioning

confidence: 99%

Varieties of De Morgan monoids: Minimality and irreducible algebras

Moraschini

Raftery

Wannenburg

2019

Journal of Pure and Applied Algebra

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It is proved that every finitely subdirectly irreducible De Morgan monoid A (with neutral element e) is either (i) a Sugihara chain in which e covers ¬e or (ii) the union of an interval subalgebra [¬a, a] and two chains of idempotents, (¬a] and [a), where a = (¬e) 2 . In the latter case, the variety generated by [¬a, a] has no nontrivial idempotent member, and A/[¬a) is a Sugihara chain in which ¬e = e. It is also proved that there are just four minimal varieties of De Morgan monoids. This theorem is then used to simplify the proof of a description (due to K.Świrydowicz) of the lower part of the subvariety lattice of relevant algebras. The results throw light on the models and the axiomatic extensions of fundamental relevance logics.

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“…The main result of the present paper generalizes this characterization of

\mathsf {KC}

to a signature‐independent framework. It is in the spirit of the ‘bridge theorems’ of abstract algebraic logic [13, 17] that correlate, for instance, syntactic interpolation or definability properties with model‐theoretic amalgamation or epimorphism‐surjectivity demands [2, 14, 34], and deduction‐like theorems with congruence extensibility properties [4, 6, 13, 37].…”

Section: Introductionmentioning

confidence: 99%

The algebraic significance of weak excluded middle laws

Lávička

Moraschini

Raftery

2022

Mathematical Logic Qtrly

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For (finitary) deductive systems, we formulate a signature‐independent abstraction of the weak excluded middle law (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety K$\mathsf {K}$ algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of K$\mathsf {K}$ has a greatest proper K$\mathsf {K}$‐congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, in a suitable form, to all protoalgebraic logics. A super‐intuitionistic logic possesses a WEML iff it extends KC$\mathsf {KC}$. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of sans-serifSsans-serif4$\mathsf {S4}$ has a global consequence relation with a WEML iff it extends sans-serifSsans-serif4.sans-serif2$\mathsf {S4.2}$, while every axiomatic extension of sans-serifRsans-serift$\mathsf {R^t}$ with an IL has a WEML.

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Contextual Deduction Theorems

Cited by 7 publications

References 42 publications

The simplest protoalgebraic logic

The simplest protoalgebraic logic

Varieties of De Morgan monoids: Minimality and irreducible algebras

The algebraic significance of weak excluded middle laws

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