What is an Inference Rule

What is an inference rule? This question does not have a unique answer. One usually finds two distinct standard answers in the literature; validity inference (σ ⊦ v φ for every substitution τ, the validity of τ[σ] entails the validity of τ[φ]), and truth inference (σ⊦ l φ if for every substitution τ, the truth of τ[σ] entails the truth of τ[φ]). In this paper we introduce a general semantic framework that allows us to investigate the notion of inference more carefully. Validity inference and truth inference are in some sense the extremal points in our framework. We investigate the relationship between various types of inference in our general framework, and consider the complexity of deciding if an inference rule is sound, in the context of a number of logics of interest: classical propositional logic, a nonstandard propositional logic, various propositional modal logics, and first-order logic.

[1]  D. Gabbay.  Semantical investigations in Heyting's intuitionistic logic , 1981 .

[2]  Vladimir V. Rybakov,et al.  DECIDABILITY OF ADMISSIBILITY IN THE MODAL SYSTEM Grz AND IN INTUITIONISTIC LOGIC , 1987 .

[3]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[4]  J. Michael Dunn,et al.  Relevance Logic and Entailment , 1986 .

[5]  V. Rybakov,et al.  BASES OF ADMISSIBLE RULES OF THE MODAL SYSTEM $ \mathrm{Grz}$ AND OF INTUITIONISTIC LOGIC , 1987 .

[6]  Willard Van Orman Quine,et al.  Methods of Logic , 1951 .

[7]  Harvey M. Friedman,et al.  One hundred and two problems in mathematical logic , 1975, Journal of Symbolic Logic.

[8]  Vaughan R. Pratt,et al.  Models of program logics , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[9]  V. V. Rybakov,et al.  Logical equations and admissible rules of inference with parameters in modal provability logics , 1990, Stud Logica.

[10]  Joseph Y. Halpern.  USING REASONING ABOUT KNOWLEDGE TO ANALYZE DISTRIBUTED SYSTEMS , 1987 .

[11]  Mihalis Yannakakis,et al.  The complexity of facets (and some facets of complexity) , 1982, STOC '82.

[12]  A. Ehrenfeucht.  An application of games to the completeness problem for formalized theories , 1961 .

[13]  Larry J. Stockmeyer,et al.  Classifying the computational complexity of problems , 1987, The Journal of Symbolic Logic.

[14]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[15]  Richard E. Ladner,et al.  Propositional Dynamic Logic of Regular Programs , 1979, J. Comput. Syst. Sci..

[16]  Gottlob Frege,et al.  Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens , 1879 .

[17]  Albert R. Meyer,et al.  The Deducibility Problem in Propositional Dynamic Logic , 1981, ICALP.

[18]  Richard Routley,et al.  The Semantics of First Degree Entailment , 1972 .

[19]  Alonzo Church,et al.  Introduction to Mathematical Logic , 1991 .

[20]  Nuel D. Belnap,et al.  Entailment : the logic of relevance and necessity , 1975 .

[21]  Richard E. Ladner,et al.  The Computational Complexity of Provability in Systems of Modal Propositional Logic , 1977, SIAM J. Comput..

[22]  Valentin Goranko,et al.  Using the Universal Modality: Gains and Questions , 1992, J. Log. Comput..

[23]  Ronald Fagin,et al.  Finite-Model Theory - A Personal Perspective , 1990, Theor. Comput. Sci..

[24]  Vladimir V. Rybakov,et al.  Problems of Substitution and Admissibility in the Modal System Grz and in Intuitionistic Propositional Calculus , 1990, Ann. Pure Appl. Log..

[25]  J. Neumann.  Zur Hilbertschen Beweistheorie , 1927 .

[26]  Steven K. Thomason,et al.  Reduction of second-order logic to modal logic , 1975, Math. Log. Q..

[27]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[28]  Vladimir V. Rybakov,et al.  Problems of Admissibility and Substitution, Logical Equations and Restricted Theories of free Algebras , 1989 .

[29]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[30]  S. K. Thomason,et al.  The logical consequence relation of propositional tense logic , 1975, Math. Log. Q..

[31]  Ian Hacking,et al.  What is logic , 1979 .

[32]  Max J. Cresswell,et al.  A companion to modal logic , 1984 .

[33]  Arnon Avron,et al.  Simple Consequence Relations , 1988, Inf. Comput..

[34]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[35]  Herbert B. Enderton,et al.  A mathematical introduction to logic , 1972 .

[36]  G. Mints,et al.  Derivability of admissible rules , 1976 .