51 research outputs found
An Observation Concerning Porte’s Rule in Modal Logic
It is well known that no consistent normal modal logic contains (as theorems) both ◊A and ◊¬A (for any formula A). Here we observe that this claim can be strengthened to the following: for any formula A, either no consistent normal modal logic contains ◊A, or else no consistent normal modal logic contains ◊¬A
Jerzy Łoś Positional Calculus and the Origin of Temporal Logic
Most accounts, including leading textbooks, credit Arthur Norman Prior with the invention of temporal (tense logic). However, (i) Jerzy Łoś delivered his version of temporal logic in 1947, several years before Prior; (ii) Henrk Hiż’s review of Łoś’s system in Journal of Symbolic Logic was published as early as 1951; (iii) there is evidence to the effect that, when constructing his tense calculi, Prior was aware of Łoś’s system. Therefore, although Prior is certainly a key figure in the history tense logic, as well as modal logic in general, it should be accepted both in the literature that temporal logic was invented by Jerzy Łoś
On the notion of negation in certain non-classical propositional logics
The purpose of this study is to investigate some aspects of how negation functions in certain non-classical propositional logics. These include the intuitionistic system developed by Heyting, the minimal calculus proposed by Johansson, and various intermediate logics between the minimal and the classical systems. Part I contains the new results which can be grouped into two classes: extension-criteria results and infinite chain results. In the first group criteria are given for answering the question: when do formulae added to the axioms of the minimal calculus as extra axioms extend the minimal calculus to various known intermediate logics? One of the results in this group (THEOREM 1 in Chapter II, Section 1) is a generalization of a result of Jankov. In the second group certain intermediate logics are defined which form infinite chains between well-known logical systems. One of the results here (THEOREM 1 in Chapter II, Section 2) is a generalization of a result of McKay. In Part II the new results are discussed from the viewpoint of negation. It is rather difficult, however, to draw definite conclusions which are acceptable to all. For these depend on, and are closely bound up with, certain basic philosophical presuppositions which are neither provable, nor disprovable in a strict sense. Taking an essentially classical position, it is argued that the logics appearing in the defined infinite chains are such that they diverge only in the vicinity of negation, and the notions of negation in them are simply ordered in a sense which is specified during the discussion. In Appendix I a number of conjectures are formulated in connection with the new results.<p
Spoiled for choice?
The transition from a theory that turned out trivial to a consistent replacement need not proceed in terms of inconsistencies, which are negation gluts. Logics that tolerate gluts or gaps (or both) with respect to any logical symbol may serve as the lower limit for adaptive logics that assign a minimally abnormal consequence set to a given premise set. The same obtains for logics that tolerate a combination of kinds of gluts and gaps. This result runs counter to the obsession with inconsistency that classical logicians and paraconsistent logicians share.\\ All such basic logics will be systematically reviewed, some variants will be outlined, and the claim will be argued for. While those logics tolerate gluts and gaps with respect to logical symbols, ambiguity logic tolerates ambiguities in non-logical symbols. Moreover, forms of tolerance may be combined, with zero logic as an extreme.\\ In the baffling plethora of corrective adaptive logics (roads from trivial theories to consistent replacements), adaptive zero logic turns out theoretically interesting as well as practically useful. On the one hand all meaning becomes contingent, depending on the premise set. On the other hand, precisely adaptive zero logic provides one with an excellent analyzing instrument. For example, it enables one to figure out which corrective adaptive logics lead, for a specific trivial theory, to a suitable and interesting minimally abnormal consequence set
Computers and relevant logic : a project in computing matrix model structures for propositional logics
I present and discuss four classes of algorithm
designed as solutions to the problem of generating matrix
representations of model structures for some non-classical
propositional logics. I then go on to survey the output
from implementations of these algorithms and finally exhibit
some logical investigations suggested by that output.
All four algorithms traverse a search tree depthfirst.
In the case of the first and fourth methods the
tree is fixed by imposing a lexicographic order on possible
matrices, while the second and third create their search tree
dynamically as the job progresses. The first algorithm is a
simple "backtrack" with some pruning of the tree in response
to refutations of possible matrices. The fourth, the most
efficient we have for time, maximises the amount of pruning
while keeping the same basic form. The second, which uses
a large number of special properties of the logics in question,
and so requires some logical and algebraic knowledge on the
part of the programmer, finds the matrices at the tips of
branches only, while the third, due to P.A. Pritchard, is far
easier to program and tests a matrix at every node of the search
tree.
The logics with which I am concerned are in the "relevant"
group first seriously investigated by A.R. Anderson and N.D.
Belnap (see their Entailment: the logic of relevance and
necessity, 1975). The most surprising observation in my
preliminary survey of the numbers of matrices validating such
systems is that the typical models are not much like the models
normally taken as canonical for the logics. In particular the proportion of inconsistent models (validating some cases of the
scheme 'A & ~A') is much higher than might have been expected.
Among the logical investigations already suggested by the
quasi-empirical data now available in the form of matrices are
some work on the system R-W, including my theorem, proved in
chapter 2.3, that with the law of excluded middle it suffices
to trivialise naive set theory, and the little-noticed subject
of Ackermann constants (sentential constants) in these logics.
The formula which collapses naive set theory in R-W plus
A v ~A
is the most damaging set-theoretic antinomy known. The theorem
that there are at least 3088 Ackermann constants in the logic R
(chapter 2.4) could not reasonably have been proved without the
aid of a computer.
My major conclusion is that this work on applications of
computers in logical research has reached a point where we are
able not only to relieve logicians of some drudgery, but to
suggest theorems and insights of new and possibly important
kinds
Deductive Systems in Traditional and Modern Logic
The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic
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