Implementation and Evaluation of Contextual Natural Deduction for Minimal Logic

2016 ◽  
pp. 314-324 ◽  
Author(s):  
Bruno Woltzenlogel Paleo
Keyword(s):  
Natural Deduction ◽  
Minimal Logic

Natural deduction and sequent calculus for intuitionistic relevant logic

1987 ◽  
Vol 52 (3) ◽  
pp. 665-680 ◽  
Author(s):  
Neil Tennant
Keyword(s):  
Natural Deduction ◽  
Sequent Calculus ◽  
Deductive System ◽  
Relevant Logic ◽  
Relevance Logic ◽  
Minimal Logic ◽  
Disjunctive Syllogism

Relevance logic began in an attempt to avoid the so-called fallacies of relevance. These fallacies can be in implicational form or in deductive form. For example, Lewis's first paradox can beset a system in implicational form, in that the system contains as a theorem the formula (A & ∼A) → B; or it can beset it in deductive form, in that the system allows one to deduce B from the premisses A, ∼A.Relevance logic in the tradition of Anderson and Belnap has been almost exclusively concerned with characterizing a relevant conditional. Thus it has attacked the problem of relevance in its implicational form. Accordingly for a relevant conditional → one would not have as a theorem the formula (A & ∼A) → B. Other theorems even of minimal logic would also be lacking. Perhaps most important among these is the formula (A → (B → A)). It is also a well-known feature of their system R that it lacks the intuitionistically valid formula ((A ∨ B) & ∼A) → B (disjunctive syllogism).But it is not the case that any relevance logic worth the title even has to concern itself with the conditional, and hence with the problem in its implicational form. The problem arises even for a system without the conditional primitive. It would still be an exercise in relevance logic, broadly construed, to formulate a deductive system free of the fallacies of relevance in deductive form even if this were done in a language whose only connectives were, say, &, ∨ and ∼. Solving the problem of relevance in this more basic deductive form is arguably a precondition for solving it for the conditional, if we suppose (as is reasonable) that the relevant conditional is to be governed by anything like the rule of conditional proof.

scholarly journals Proof Compression and NP Versus PSPACE II: Addendum

2022 ◽  
Author(s):  
Lew Gordeev ◽  
Edward Hermann Haeusler
Keyword(s):  
Natural Deduction ◽  
Sequent Calculus ◽  
Hamiltonian Cycles ◽  
Hamiltonian Graph ◽  
Minimal Logic ◽  
Hamiltonian Graphs ◽  
Graph Problem ◽  
Np Complete

In [3] we proved the conjecture NP = PSPACE by advanced proof theoretic methods that combined Hudelmaier's cut-free sequent calculus for minimal logic (HSC) [5] with the horizontal compressing in the corresponding minimal Prawitz-style natural deduction (ND) [6]. In this Addendum we show how to prove a weaker result NP = coNP without referring to HSC. The underlying idea (due to the second author) is to omit full minimal logic and compress only \naive" normal tree-like ND refutations of the existence of Hamiltonian cycles in given non-Hamiltonian graphs, since the Hamiltonian graph problem in NP-complete. Thus, loosely speaking, the proof of NP = coNP can be obtained by HSC-elimination from our proof of NP = PSPACE [3].

Suppose and Tell

2020 ◽  
Author(s):  
Timothy Williamson
Keyword(s):  
Conditional Probability ◽  
Natural Deduction ◽  
Special Kind ◽  
Context Sensitivity ◽  
Modal Operator ◽  
Counterfactual Conditionals ◽  
Deduction Rules ◽  
Psychological Sense

The book argues that our use of conditionals is governed by imperfectly reliable heuristics, in the psychological sense of fast and frugal (or quick and dirty) ways of assessing them. The primary heuristic is this: to assess ‘If A, C’, suppose A and on that basis assess C; whatever attitude you take to C conditionally on A (such as acceptance, rejection, or something in between) take unconditionally to ‘If A, C’. This heuristic yields both the equation of the probability of ‘If A, C’ with the conditional probability of C on A and standard natural deduction rules for the conditional. However, these results can be shown to make the heuristic implicitly inconsistent, and so less than fully reliable. There is also a secondary heuristic: pass conditionals freely from one context to another under normal conditions for acceptance of sentences on the basis of memory and testimony. The effect of the secondary heuristic is to undermine interpretations on which ‘if’ introduces a special kind of context-sensitivity. On the interpretation which makes best sense of the two heuristics, ‘if’ is simply the truth-functional conditional. Apparent counterexamples to truth-functionality are artefacts of reliance on the primary heuristic in cases where it is unreliable. The second half of the book concerns counterfactual conditionals, as expressed with ‘if’ and ‘would’. It argues that ‘would’ is an independently meaningful modal operator for contextually restricted necessity: the meaning of counterfactuals is simply that derived compositionally from the meanings of their constituents, including ‘if’ and ‘would’, making them contextually restricted strict conditionals.

TRANSLATIONS BETWEEN LINEAR AND TREE NATURAL DEDUCTION SYSTEMS FOR RELEVANT LOGICS

2021 ◽  
pp. 1-22
Author(s):  
SHAWN STANDEFER
Keyword(s):  
Natural Deduction ◽  
The Other ◽  
Relevant Logics

Abstract Anderson and Belnap presented indexed Fitch-style natural deduction systems for the relevant logics R, E, and T. This work was extended by Brady to cover a range of relevant logics. In this paper I present indexed tree natural deduction systems for the Anderson–Belnap–Brady systems and show how to translate proofs in one format into proofs in the other, which establishes the adequacy of the tree systems.

scholarly journals On Polymorphic Sessions and Functions

2021 ◽  
Vol 43 (2) ◽  
pp. 1-55
Author(s):  
Bernardo Toninho ◽  
Nobuko Yoshida
Keyword(s):  
Natural Deduction ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Linear Formulation ◽  
Session Types ◽  
Logical Foundation ◽  
Π Calculus ◽  
Soundness And Completeness ◽  
System F ◽  
Algebraic Representations

This work exploits the logical foundation of session types to determine what kind of type discipline for the Λ-calculus can exactly capture, and is captured by, Λ-calculus behaviours. Leveraging the proof theoretic content of the soundness and completeness of sequent calculus and natural deduction presentations of linear logic, we develop the first mutually inverse and fully abstract processes-as-functions and functions-as-processes encodings between a polymorphic session π-calculus and a linear formulation of System F. We are then able to derive results of the session calculus from the theory of the Λ-calculus: (1) we obtain a characterisation of inductive and coinductive session types via their algebraic representations in System F; and (2) we extend our results to account for value and process passing, entailing strong normalisation.

ON THE NOTION OF CANONICAL DERIVATIONS FROM OPEN ASSUMPTIONS AND ITS ROLE IN PROOF-THEORETIC SEMANTICS

2015 ◽  
Vol 8 (2) ◽  
pp. 296-305 ◽  
Author(s):  
NISSIM FRANCEZ
Keyword(s):  
Natural Deduction ◽  
Proof System ◽  
Logical Constants ◽  
Local Completeness ◽  
Definition Of ◽  
The Impact

AbstractThe paper proposes an extension of the definition of a canonical proof, central to proof-theoretic semantics, to a definition of a canonical derivation from open assumptions. The impact of the extension on the definition of (reified) proof-theoretic meaning of logical constants is discussed. The extended definition also sheds light on a puzzle regarding the definition of local-completeness of a natural-deduction proof-system, underlying its harmony.

AN INTENSIONAL LEIBNIZ SEMANTICS FOR ARISTOTELIAN LOGIC

2010 ◽  
Vol 3 (2) ◽  
pp. 262-272 ◽  
Author(s):  
KLAUS GLASHOFF
Keyword(s):  
Natural Deduction ◽  
Predicate Logic ◽  
Formal System ◽  
Aristotelian Logic ◽  
Categorical Logic ◽  
Characteristic Numbers ◽  
Formal Syntax ◽  
Philosophical Standpoint ◽  
Classical Philosophy

Since Frege’s predicate logical transcription of Aristotelian categorical logic, the standard semantics of Aristotelian logic considers terms as standing for sets of individuals. From a philosophical standpoint, this extensional model poses problems: There exist serious doubts that Aristotle’s terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this question—they looked at terms as standing for concepts (“Begriffe”). In 1972, Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction, while, with respect to semantics, he still made use of an extensional interpretation. In this paper we deal with a simple intensional semantics for Corcoran’s syntax—intensional in the sense that no individuals are needed for the construction of a complete Tarski model of Aristotelian syntax. Instead, we view concepts as containing or excluding other, “higher” concepts—corresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcoran’s work, furnishing his formal syntax with an adequate semantics which is free from presuppositions which have entered into modern interpretations of Aristotle’s theory via predicate logic.

The lattice of extensions of the minimal logic

2007 ◽  
Vol 17 (2) ◽  
pp. 112-143
Author(s):  
S. P. Odintsov
Keyword(s):  
Minimal Logic
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