Equational logic of partial functions under Kleene equality: a complete and an incomplete set of rules

1989 ◽  
Vol 54 (2) ◽  
pp. 354-362 ◽  
Author(s):  
Anthony Robinson
Keyword(s):  
Inference Rule ◽  
Logical Consequence ◽  
Equational Logic ◽  
Inference Rules ◽  
Partial Functions ◽  
Substitution Rule ◽  
Valid Inference ◽  
Recursively Enumerable ◽  
Complete Set ◽  
Recursive Set

When equational logic for partial functions is interpreted using Kleene equality as the predicate, the relation of logical consequence may be said to express what identities of partial functions follow from a given set of identities. In the analogous situation for total functions, there is a complete set of inference rules consisting of reflexivity, symmetry, transitivity, replacement, and substitution; in the case of partial functions, unrestricted substitution fails to be a valid inference rule, and there remains the question of how to obtain a complete set of rules. The first part of the present paper shows that completeness cannot be obtained by a mere restriction of the substitution rule, for a counterexample shows that even a rule allowing substitution in all consequentially valid instances fails, in conjunction with the other four rules, to yield a complete set of rules.The second part of the paper defines a combined rule of transitivity-substitution which, in conjunction with reflexivity, symmetry, replacement, and substitution only of variables, yields a complete set of rules. The new rule is first stated in a form that allows an unbounded number of premises, and then is altered to a three-premise form. In both forms, the rule suffers from the shortcoming that in its formulation an auxiliary notion of conditional existence is involved, which is given by a recursive syntactic definition. As a result, the set of instantiations of the rule is recursively enumerable, but not (apparently) recursive (assuming a recursive set of premises).

A system of logic for partial functions under existence-dependent kleene equality

1988 ◽  
Vol 53 (3) ◽  
pp. 834-839 ◽  
Author(s):  
H. Andréka ◽  
W. Craig ◽  
I. Németi
Keyword(s):  
Practical Interest ◽  
Order Logic ◽  
First Order Logic ◽  
Basic Relation ◽  
Equational Logic ◽  
Partial Functions ◽  
First Order ◽  
Complete Set ◽  
The Subject ◽  
Relationship Of

Ordinary equational logic is a connective-free fragment of first-order logic which is concerned with total functions under the relation of ordinary equality. In [AN] (see also [AN1]) and in [Cr] it has been extended in two equivalent ways into a near-equational system of logic for partial functions. The extension given in [Cr] deals with partial functions under two relationships: a relationship of existence-dependent existence and one of existence-dependent Kleene equality. For the language that involves both relationships a set of rules was given that is complete. Those rules in the set that involve only existence-dependent existence turned out to be complete for the sublanguage that involves this relationship only. In the present paper we give a set of rules that is complete for the other sublanguage, namely the language of partial functions under existence-dependent Kleene equality.This language lacks a certain, often needed, power of expressing existence and fails, in particular, to be an extension of the language that underlies ordinary equational logic. That it possesses a fairly simple complete set of rules is therefore perhaps more of theoretical than of practical interest. The present paper is thus intended to serve as a supplement to [Cr] and, less directly, to [AN]. The subject is further rounded out, and some contrast is provided, by [Rob]. The systems of logic treated there are based on the weaker language in which partial functions are considered under the more basic relation of Kleene equality.

Logical consecutions in discrete linear temporal logic

2005 ◽  
Vol 70 (4) ◽  
pp. 1137-1149 ◽  
Author(s):  
V. V. Rybakov
Keyword(s):  
Temporal Logic ◽  
Propositional Logic ◽  
Inference Rule ◽  
Logical Consequence ◽  
Linear Temporal Logic ◽  
Inference Rules ◽  
Temporal Logics ◽  
Temporal Inference ◽  
Semantic Techniques ◽  
The Given

AbstractWe investigate logical consequence in temporal logics in terms of logical consecutions, i.e., inference rules. First, we discuss the question: what does it mean for a logical consecution to be ‘correct’ in a propositional logic. We consider both valid and admissible consecutions in linear temporal logics and discuss the distinction between these two notions. The linear temporal logic LDTL, consisting of all formulas valid in the frame 〈L ≤, ≥〉 of all integer numbers, is the prime object of our investigation. We describe consecutions admissible in LDTL in a semantic way—via consecutions valid in special temporal Kripke/Hintikka models. Then we state that any temporal inference rule has a reduced normal form which is given in terms of uniform formulas of temporal degree 1. Using these facts and enhanced semantic techniques we construct an algorithm, which recognizes consecutions admissible in LDTL. Also, we note that using the same technique it follows that the linear temporal logic L(N) of all natural numbers is also decidable w.r.t. inference rules. So, we prove that both logics LDTL and L(N) are decidable w.r.t. admissible consecutions. In particular, as a consequence, they both are decidable (known fact), and the given deciding algorithms are explicit.

1-genericity in the enumeration degrees

1988 ◽  
Vol 53 (3) ◽  
pp. 878-887 ◽  
Author(s):  
Kate Copestake
Keyword(s):  
Partial Function ◽  
Partial Ordering ◽  
Turing Degree ◽  
Turing Degrees ◽  
Original Formulation ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Generic Set

The structure of the Turing degrees of generic and n-generic sets has been studied fairly extensively, especially for n = 1 and n = 2. The original formulation of 1-generic set in terms of recursively enumerable sets of strings is due to D. Posner [11], and much work has since been done, particularly by C. G. Jockusch and C. T. Chong (see [5] and [6]).In the enumeration degrees (see definition below), attention has previously been restricted to generic sets and functions. J. Case used genericity for many of the results in his thesis [1]. In this paper we develop a notion of 1-generic partial function, and study the structure and characteristics of such functions in the enumeration degrees. We find that the e-degree of a 1-generic function is quasi-minimal. However, there are no e-degrees minimal in the 1-generic e-degrees, since if a 1-generic function is recursively split into finitely or infinitely many parts the resulting functions are e-independent (in the sense defined by K. McEvoy [8]) and 1-generic. This result also shows that any recursively enumerable partial ordering can be embedded below any 1-generic degree.Many results in the Turing degrees have direct parallels in the enumeration degrees. Applying the minimal Turing degree construction to the partial degrees (the e-degrees of partial functions) produces a total partial degree ae which is minimal-like; that is, all functions in degrees below ae have partial recursive extensions.

scholarly journals Gaussoids are two-antecedental approximations of Gaussian conditional independence structures

2021 ◽  
Author(s):  
Tobias Boege
Keyword(s):  
Characteristic Zero ◽  
Conditional Independence ◽  
Inference Rule ◽  
Positive Definite ◽  
Inference Rules ◽  
Identity Matrix ◽  
Ordered Fields ◽  
Finite Set

AbstractThe gaussoid axioms are conditional independence inference rules which characterize regular Gaussian CI structures over a three-element ground set. It is known that no finite set of inference rules completely describes regular Gaussian CI as the ground set grows. In this article we show that the gaussoid axioms logically imply every inference rule of at most two antecedents which is valid for regular Gaussians over any ground set. The proof is accomplished by exhibiting for each inclusion-minimal gaussoid extension of at most two CI statements a regular Gaussian realization. Moreover we prove that all those gaussoids have rational positive-definite realizations inside every ε-ball around the identity matrix. For the proof we introduce the concept of algebraic Gaussians over arbitrary fields and of positive Gaussians over ordered fields and obtain the same two-antecedental completeness of the gaussoid axioms for algebraic and positive Gaussians over all fields of characteristic zero as a byproduct.

Indicative conditionals

2018 ◽  
Author(s):  
Sarah Moss
Keyword(s):  
Conditional Probability ◽  
Inference Rules ◽  
Indicative Conditionals ◽  
Valid Inference ◽  
Logical Operators ◽  
Context Sensitive ◽  
Probabilistic Semantics ◽  
Truth Conditional ◽  
Conditional Credence

This chapter defends a probabilistic semantics for indicative conditionals and other logical operators. This semantics is motivated in part by the observation that indicative conditionals are context sensitive, and that there are contexts in which the probability of a conditional does not match the conditional probability of its consequent given its antecedent. For example, there are contexts in which you believe the content of ‘it is probable that if Jill jumps from this building, she will die’ without having high conditional credence that Jill will die if she jumps. This observation is at odds with many existing non-truth-conditional semantic theories of conditionals, whereas it is explained by the semantics for conditionals defended in this chapter. The chapter concludes by diagnosing several apparent counterexamples to classically valid inference rules embedding epistemic vocabulary.

Splitting and Decomposition by Regressive Sets, II

1967 ◽  
Vol 19 ◽  
pp. 291-311 ◽  
Author(s):  
T. G. McLaughlin
Keyword(s):  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Relationship Of ◽  
The Relationship ◽  
Trivial Fact ◽  
Recursive Set

In (3), Dekker drew attention to an analogy between (a) the relationship of the recursive sets to the recursively enumerable sets, and (b) the relationship of the retraceable sets to the regressive sets. As was to be expected, this analogy limps in some respects. For example, if a number set α is split by a recursive set, then it is decomposed by a pair of recursively enumerable sets; whereas, as we showed in (6, Theorem 2), α may be split by a retraceable set and yet not decomposable (in a liberal sense of the latter term) by a pair of regressive sets. The result for recursive and recursively enumerable sets, of course, follows from the trivial fact that the complement of a recursive set is recursive.

Linguistic Conventions

2020 ◽  
pp. 21-52
Author(s):  
Jared Warren
Keyword(s):  
Inference Rule ◽  
Linguistic Competence ◽  
Inference Rules ◽  
Rule Following ◽  
Full Account ◽  
Social Conventions ◽  
Linguistic Conventions ◽  
The Way

What are linguistic conventions? This chapter begins by noting and setting aside philosophical accounts of social conventions stemming from Lewis’s influential treatment. It then criticizes accounts that see conventions as explicit stipulations. From there the chapter argues that conventions are syntactic rules of inference, arguing that there are scientific reasons to posit these rules as part of our linguistic competence and that we need to include both bilateralist and open-ended inference rules for a full account. The back half of the chapter aims to naturalize inference rule-following by providing functionalist-dispositionalist approaches to our attitudes, inference, and inference-rule–following, addressing Kripkenstein’s arguments and several other concerns along the way.

Analytic cut

1969 ◽  
Vol 33 (4) ◽  
pp. 560-564 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Propositional Logic ◽  
Inference Rule ◽  
The Other ◽  
Inference Rules ◽  
Quantification Theory ◽  
The Real ◽  
Gentzen Systems ◽  
Real Importance ◽  
Per Se ◽  
Axiom Systems

The real importance of cut-free proofs is not the elimination of cuts per se, but rather that such proofs obey the subformula principle. In this paper we accomplish this latter objective in a different manner.In the usual formulations of Gentzen systems, there is only one axiom scheme; all the other postulates are inference rules. By contrast, we consider here some Gentzen type axiom systems for propositional logic and Quantification Theory in which there is only one inference rule; all the other postulates are axiom schemes. This admits of an unusually elegant axiomatization of logic.

Rules of inference with parameters for intuitionistic logic

1992 ◽  
Vol 57 (3) ◽  
pp. 912-923 ◽  
Author(s):  
Vladimir V. Rybakov
Keyword(s):  
Intuitionistic Logic ◽  
Inference Rule ◽  
Kripke Models ◽  
Inference Rules ◽  
Direct Solution ◽  
Usual Form

AbstractAn algorithm recognizing admissibility of inference rules in generalized form (rules of inference with parameters or metavariables) in the intuitionistic calculus H and, in particular, also in the usual form without parameters, is presented. This algorithm is obtained by means of special intuitionistic Kripke models, which are constructed for a given inference rule. Thus, in particular, the direct solution by intuitionistic techniques of Friedman's problem is found. As a corollary an algorithm for the recognition of the solvability of logical equations in H and for constructing some solutions for solvable equations is obtained. A semantic criterion for admissibility in H is constructed.

Index sets in Ershov's hierarchy

1974 ◽  
Vol 39 (1) ◽  
pp. 97-104 ◽  
Author(s):  
Jacques Grassin
Keyword(s):  
Recursion Theory ◽  
Case Note ◽  
Boolean Combination ◽  
Open Set ◽  
Recursively Enumerable ◽  
Power Set ◽  
Index Sets ◽  
Recursively Enumerable Sets ◽  
Complete Set ◽  
Recursive Isomorphism

This work is an attempt to characterize the index sets of classes of recursively enumerable sets which are expressible in terms of open sets in the Baire topology on the power set of the set N of natural numbers, usual in recursion theory. Let be a class of subsets of N and be the set of indices of recursively enumerable sets Wх belonging to .A well-known theorem of Rice and Myhill (cf. [5, p. 324, Rice-Shapiro Theorem]) states that is recursively enumerable if and only ifis a r.e. open set. In this case, note that if is not empty and does not contain all recursively enumerable sets, is a complete set. This theorem will be partially extended to classes which are boolean combinations of open sets by the following:(i) There is a canonical boolean combination which represents, namely the shortest among boolean combinations which represent.(ii) The recursive isomorphism type of depends on the length n of this canonical boolean combination (and trivial properties of ); for instance, is recursively isomorphic (in the particular case where is a boolean combination of recursive open sets) to an elementary set combination Yn or Un, constructed from {х ∣ х Wх) and depending on the length n. We can say also that is a complete set in the sense of Ershov's hierarchy [1] (in this particular case).

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