Arnon Avron. Relevance and paraconsistency—a new approach. The journal of symbolic logic, vol. 55 (1990), pp. 707–732. - Arnon Avron. Relevance and paraconsistency—a new approach. Part II: the formal systems. Notre Dame journal of formal logic, vol. 31 (1990), pp. 169–202. - Arnon Avron. Relevance and paraconsistency—a new approach. Part III: cut-free Gentzen-type systems. Notre Dame journal of formal logic, vol. 32 (1991), pp. 147–160.

1992 ◽  
Vol 57 (4) ◽  
pp. 1481-1482
Author(s):  
Alasdair Urquhart
Keyword(s):  
Type Systems ◽  
Formal Logic ◽  
Symbolic Logic ◽  
New Approach ◽  
Formal Systems

Leibniz's interpretation of his logical calculi

1954 ◽  
Vol 19 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Nicholas Rescher
Keyword(s):  
Formal Logic ◽  
Point Of View ◽  
Symbolic Logic ◽  
Logical Calculus ◽  
Formal Systems ◽  
Logical Calculi

The historical researches of Louis Couturat saved the logical work of Leibniz from the oblivion of neglect and forgetfulness. They revealed that Leibniz developed in succession several versions of a “logical calculus” (calculus ratiocinator or calculus universalis). In consequence of Couturat's investigations it has become well known that Leibniz's development of these logical calculi adumbrated the notion of a logistic system; and for these foreshadowings of the logistic treatment of formal logic Leibniz is rightly regarded as the father of symbolic logic.It is clear from what has been said that it is scarcely possible to overestimate the debt which the contemporary student of Leibniz's logic owes to Couturat. This gratitude must, however, be accompanied by the realization that Couturat's own theory of logic is gravely defective. Couturat was persuaded that the extensional point of view in logic is the only one which is correct, an opinion now quite antiquated, and shared by no one. This prejudice of Couturat's marred his exposition of Leibniz's logic. It led him to battle with windmills: he viewed the logic of Leibniz as rife with shortcomings stemming from an intensional approach.The task of this paper is a re-examination of Leibniz's logic. It will consider without prejudgment how Leibniz conceived of the major formal systems he developed as logical calculi – that is, these systems will be studied with a view to the interpretation or interpretations which Leibniz himself intends for them. The aim is to undo some of the damage which Couturat's preconception has done to the just understanding of Leibniz's logic and to the proper evaluation of his contribution.

Disarming Stove’s Paradox: In Defence of Formal Logic

1998 ◽  
pp. 127-134
Author(s):  
R. Rodrigo Soberano
Keyword(s):  
Main Part ◽  
Formal Logic ◽  
Symbolic Logic ◽  
Logical Contradiction ◽  
False Conclusion

The argument (d) ("All arguments with true premises and false conclusions are invalid.") is an argument with true premises and false conclusion. Therefore "(d) is invalid" seems to be formally valid. Thus presumably formal logic has to admit it as valid. But then formal logic finds itself in a bind. For the above argument is problematic and even paradoxical since it involves an internal logical contradiction. The paradox, aptly termed "Stove's paradox," is fully realized by demonstrating with the help of symbolic logic the contradiction within the argument. Then as the main part of this essays shows, the paradox is attacked by exposing the paradox's genesis. It is shown that by appeal to some not so obvious logical considerations regarding sound linguistic construction and usage, the above argument could not have been legitimately construction. For its construction must have involved either equivocation or hiatus of meaningfulness in the use of the symbol (d).

scholarly journals αCheck: A mechanized metatheory model checker

2017 ◽  
Vol 17 (3) ◽  
pp. 311-352 ◽  
Author(s):  
JAMES CHENEY ◽  
ALBERTO MOMIGLIANO
Keyword(s):  
Programming Language ◽  
Dual Problem ◽  
State Of The Art ◽  
Operational Semantics ◽  
Type Systems ◽  
Model Checker ◽  
Formal Systems ◽  
Current State ◽  
Fully Automatic ◽  
Nominal Logic

AbstractThe problem of mechanically formalizing and proving metatheoretic properties of programming language calculi, type systems, operational semantics, and related formal systems has received considerable attention recently. However, the dual problem of searching for errors in such formalizations has attracted comparatively little attention. In this article, we present αCheck, a bounded model checker for metatheoretic properties of formal systems specified using nominal logic. In contrast to the current state of the art for metatheory verification, our approach is fully automatic, does not require expertise in theorem proving on the part of the user, and produces counterexamples in the case that a flaw is detected. We present two implementations of this technique, one based onnegation-as-failureand one based onnegation elimination, along with experimental results showing that these techniques are fast enough to be used interactively to debug systems as they are developed.

The completeness of the first-order functional calculus

1949 ◽  
Vol 14 (3) ◽  
pp. 159-166 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Propositional Calculus ◽  
Higher Order ◽  
New Method ◽  
Completeness Proof ◽  
New Approach ◽  
First Order ◽  
Formal Systems ◽  
The Subject ◽  
Image Position

Although several proofs have been published showing the completeness of the propositional calculus (cf. Quine (1)), for the first-order functional calculus only the original completeness proof of Gödel (2) and a variant due to Hilbert and Bernays have appeared. Aside from novelty and the fact that it requires less formal development of the system from the axioms, the new method of proof which is the subject of this paper possesses two advantages. In the first place an important property of formal systems which is associated with completeness can now be generalized to systems containing a non-denumerable infinity of primitive symbols. While this is not of especial interest when formal systems are considered as logics—i.e., as means for analyzing the structure of languages—it leads to interesting applications in the field of abstract algebra. In the second place the proof suggests a new approach to the problem of completeness for functional calculi of higher order. Both of these matters will be taken up in future papers.The system with which we shall deal here will contain as primitive symbolsand certain sets of symbols as follows:(i) propositional symbols (some of which may be classed as variables, others as constants), and among which the symbol “f” above is to be included as a constant;(ii) for each number n = 1, 2, … a set of functional symbols of degree n (which again may be separated into variables and constants); and(iii) individual symbols among which variables must be distinguished from constants. The set of variables must be infinite.

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