Equality of proofs for linear equality

Došen, Kosta; Petric, Z.

doi:10.1007/s00153-008-0096-0

Cited by 3 publications

(4 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It should be noted that, despite Lafont's theorem, some authors have been able to find inequivalent proofs in the framework of classical logic. See, in particular, Došen [42] (see details in [43]; see also [44] and references there), Kuznets [117] and references there; and Guglielmi [73], [74] (see also [75] and references there) Remark 2.2. One issue with Γ-transformations is that (Γ L ) does not merely eliminate q, but only does so at the expense of changing a weakening.…”

Section: The Collapse Of Non-constructive Proofsmentioning

confidence: 99%

“…Note that the converse to the ¬¬-Shift Principle is nothing but the intuitionistically derivable principle (46). If ∀ is "specialized" to ∧, the ¬¬-Shift Principle itself becomes intuitionistically derivable (43); and if ∀ is "specialized" to →, the resulting principle…”

Section: ¬¬-Translationmentioning

confidence: 99%

“…The converse implication is equivalent to Jankov's principle ¬α ∨ ¬¬α, see Proposition 5.9 below 43. The converse implication is known as the principle of Double Negation Shift; see §5.6.1.44 The converse implication is known as the Strong Markov Principle; see §5.5.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Mathematical semantics of intuitionistic logic

Melikhov¹

2015

Preprint

View full text Add to dashboard Cite

This work is a mathematician's attempt to understand intuitionistic logic. It can be read in two ways: as a research paper interspersed with lengthy digressions into rethinking of standard material; or as an elementary (but highly unconventional) introduction to first-order intuitionistic logic. For the latter purpose, no training in formal logic is required, but a modest literacy in mathematics, such as topological spaces and posets, is assumed.The main theme of this work is the search for a formal semantics adequate to Kolmogorov's informal interpretation of intuitionistic logic (whose simplest part is more or less the same as the so-called BHK interpretation). This search goes beyond the usual model theory, based on Tarski's notion of semantic consequence, and beyond the usual formalism of first-order logic, based on schemata. Thus we study formal semantics of a simplified version of Paulson's meta-logic, used in the Isabelle prover.By interpreting the meta-logical connectives and quantifiers constructively, we get a generalized model theory, which covers, in particular, realizability-type interpretations of intuitionistic logic. On the other hand, by analyzing Kolmogorov's notion of semantic consequence (which is an alternative to Tarski's standard notion), we get an alternative model theory. By using an extension of the meta-logic, we further get a generalized alternative model theory, which suffices to formalize Kolmogorov's semantics.On the other hand, we also formulate a modification of Kolmogorov's interpretation, which is compatible with the usual, Tarski-style model theory. Namely, it can be formalized by means of sheaf-valued models, which turn out to be a special case of Palmgren's categorical models; intuitionistic logic is complete with respect to this semantics.3.2. Platonism vs. Verificationism 3.3. The issue of understanding 3.4. Solutions of problems 3.5. The Hilbert-Brouwer controversy 3.6. Problems vs. conjectures 3.7. The BHK interpretation 3.8. Understanding the connectives 3.9. Something is missing here 3.10. Classical logic revisited 3.11. Clarified BHK interpretation 3.12. The principle of decidability and its relatives 3.13. Medvedev-Skvortsov problems 3.14. Some intuitionistic validities 4. What is a logic, formally? 4.1. Introduction 4.2. Typed expressions 4.3. Languages and structures 4.4. Meta-logic 4.5. Logics 4.6. Intuitionistic logic (syntax) 4.7. Models 4.8. Classical logic (semantics) 5. Topology, translations and principles 5.1. Deriving a model from BHK 5.2. Tarski models 5.3. Topological completeness 5.4. Jankov's principle 5.5. Markov's principle 5.6. ¬¬-Translation 5.7. Provability translation 5.8. Independence of connectives and quantifiers 6. Sheaf-valued models 6.1. Sheaves 6.2. Operations on sheaves 6.3. Sheaf-valued models 7. Semantics of the meta-logic 7.1. Generalized models 7.2. Alternative semantics 7.3. Generalized alternative semantics References 1 In the introduction to her survey of logic in the USSR up to 1957, S. A. Yanovskaya emphasizes "The difference of t...

show abstract

Section: The Collapse Of Non-constructive Proofsmentioning

confidence: 99%

Section: ¬¬-Translationmentioning

confidence: 99%

See 1 more Smart Citation

Mathematical semantics of intuitionistic logic

Melikhov¹

2015

Preprint

View full text Add to dashboard Cite

show abstract

“…The finite ordinals that are the objects of M in this paper may be replaced by modalities so as to make C isomorphic to a subcategory of M (see the parenthetical remark in the first paragraph of Section 3 of [15]), but when the objects of C are modalities built out only of , or only of ♦, we need no further adjustments of M to have as a consequence of coherence that C is isomorphic to a subcategory of M, and this subcategory may happen to be an important and interesting concrete category. As an example of such an important concrete category, we find in this paper the simplicial category, whose arrows are the order-preserving functions between finite ordinals, which is isomorphic to the category C whose arrows may be taken as the deductions in the modal logic S4 involving the modalities built out only of ♦.…”

mentioning

confidence: 99%