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1.
Steffen Lewitzka 《Logica Universalis》2007,1(2):243-276
What is a logic? Which properties are preserved by maps between logics? What is the right notion for equivalence of logics?
In order to give satisfactory answers we generalize and further develop the topological approach of [4] and present the foundations
of a general theory of abstract logics which is based on the abstract concept of a theory. Each abstract logic determines
a topology on the set of theories. We develop a theory of logic maps and show in what way they induce (continuous, open) functions
on the corresponding topological spaces. We also establish connections to well-known notions such as translations of logics
and the satisfaction axiom of institutions [5]. Logic homomorphisms are maps that behave in some sense like continuous functions
and preserve more topological structure than logic maps in general. We introduce the notion of a logic isomorphism as a (not
necessarily bijective) function on the sets of formulas that induces a homeomorphism between the respective topological spaces
and gives rise to an equivalence relation on abstract logics. Therefore, we propose logic isomorphisms as an adequate and
precise notion for equivalence of logics. Finally, we compare this concept with another recent proposal presented in [2].
This research was supported by the grant CNPq/FAPESB 350092/2006-0. 相似文献
2.
Valentin A. Bazhanov 《Logica Universalis》2008,2(1):71-76
In the XIXth century there was a persistent opposition to Aristotelian logic. Nicolai A. Vasiliev (1880–1940) noted this opposition
and stressed that the way for the novel – non-Aristotelian – logic was already paved. He made an attempt to construct non-Aristotelian
logic (1910) within, so to speak, the form (but not in the spirit) of the Aristotelian paradigm (mode of reasoning). What
reasons forced him to reassess the status of particular propositions and to replace the square of opposition by the triangle
of opposition? What arguments did Vasiliev use for the introduction of new classes of propositions and statement of existence
of various levels in logic? What was the meaning and role of the “method of Lobachevsky” which was implemented in construction
of imaginary logic? Why did psychologism in the case of Vasiliev happen to be an important factor in the composition of the
new ‘imaginary’ logic, as he called it?
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3.
Alessio Moretti 《Logica Universalis》2009,3(1):19-57
Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic
(both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”,
“permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s
“logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic
oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical
representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie
73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter,
Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic
internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic
has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper,
by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s
unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”,
“deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities
is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra),
whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional
very regular solid).
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4.
Peter Bernhard 《Logica Universalis》2008,2(1):31-41
In logic, diagrams have been used for a very long time. Nevertheless philosophers and logicians are not quite clear about
the logical status of diagrammatical representations. Fact is that there is a close relationship between particular visual
(resp. graphical) properties of diagrams and logical properties. This is why the representation of the four categorical propositions
by different diagram systems allows a deeper insight into the relations of the logical square. In this paper I want to give
some examples.
I would like to thank the two anonymous reviewers for helpful comments and criticisms. 相似文献
5.
In this paper we reconsider the basic topological and metric structures on spaces of probability measures and random variables, such as e.g. the weak topology and the total variation metric, replacing them with more intrinsic and richer approach structures. We comprehensibly investigate the relationships among, and basic facts about these structures, and prove that fundamental results, such as e.g. the portmanteau theorem and Prokhorov?s theorem, can be recaptured in a considerably stronger form in the new setting. 相似文献
6.
In 1971, Peter Buneman proposed a way to construct a tree from a collection of pairwise compatible splits. This construction
immediately generalizes to arbitrary collections of splits, and yields a connected median graph, called the Buneman graph.
In this paper, we prove that the vertices and the edges of this graph can be described in a very simple way: given a collection
of splitsS, the vertices of the Buneman graph correspond precisely to the subsetsS′ ofS such that the splits inS′ are pairwise incompatible and the edges correspond to pairs (S′, S) withS′ as above andS∈S′. Using this characterization, it is much more straightforward to construct the vertices of the Buneman graph than using prior
constructions. We also recover as an immediate consequence of this enumeration that the Buneman graph is a tree, that is,
that the number of vertices exceeds the number of edges (by one), if and only if any two distinct splits inS are compatible. 相似文献
7.
It is possible to consider two variants of cluster theory: Inaffine cluster theory, one considers collections ofsubsets of a given setX of objects or states, whereas inprojective cluster theory, one considers collections ofsplits (orbipartitions) of that set. In both contexts, it can be desirable to produce acontinuous model, that is, a spaceT encompassing the given setX which represents in a well-specified and more or less parsimonious way all possibleintermediate objects ortransition states compatible with certain restrictions derived from the given collection of subsets or splits. We investigate an interesting and intriguing relationship between two such constructions that appear in the context of projective cluster theory: TheBuneman construction and thetight-span (or justT)construction. 相似文献
8.
Is it possible to give an explicit definition of belief (simpliciter) in terms of subjective probability, such that believed propositions are guaranteed to have a sufficiently high probability, and yet it is neither the case that belief is stripped of any of its usual logical properties, nor is it the case that believed propositions are bound to have probability 1? We prove the answer is ‘yes’, and that given some plausible logical postulates on belief that involve a contextual “cautiousness” threshold, there is but one way of determining the extension of the concept of belief that does the job. The qualitative concept of belief is not to be eliminated from scientific or philosophical discourse, rather, by reducing qualitative belief to assignments of resiliently high degrees of belief and a “cautiousness” threshold, qualitative and quantitative belief turn out to be governed by one unified theory that offers the prospects of a huge range of applications. Within that theory, logic and probability theory are not opposed to each other but go hand in hand. 相似文献
9.
Vladimir L. Vasyukov 《Logica Universalis》2007,1(2):277-294
How, why and what for we should combine logics is perfectly well explained in a number of works concerning this issue. But
the interesting question seems to be the nature and the structure of the general universe of possible combinations of logical
systems. Adopting the point of view of universal logic in the paper the categorical constructions are introduced which along
with the coproducts underlying the fibring of logics describe the inner structure of the category of logical systems. It is
shown that categorically the universe of universal logic turns out to be a topos and a paraconsistent complement topos.
This work was supported by Russian Foundation for Humanities via the Project ”The structure of Universal Logics”, grant No
06-03-00195a. 相似文献
10.
The notion of distribution function with respect to a conditional expectation is defined and studied in the framework of Riesz spaces. 相似文献
11.
A nucleotide sequence can be considered as a realization of the non-equal-probability independently and identically distributed (niid) model. In this paper we derive the exact distribution of the occurrence number for each K-tuple with respect to the niid model by means of the Goulden-Jackson cluster method. An application of the probability function to get exact expectation curves [9] is presented, accompanied by comparison between the exact approach and the approximate solution.Received October 31, 2004 相似文献
12.
In 1971, Peter Buneman presented a paper in which he described, amongst other things, a way to construct a tree from a collection
of pairwise compatible splits of a finite set. This construction is immediately generalizable to a collection of arbitrary
splits, in which case it gives rise to theBuneman graph, a certain connected, median graph representing the given collection of splits in a canonical fashion. In this paper, we
look at the complex obtained by filling those faces of the associated hypercube whose vertices consist only of vertices in
the Buneman graph, a complex that we call theBuneman complex. In particular, we give a natural filtration of the Buneman complex, and show that this filtration collapses when the system
of splitis is weakly compatible. In the case where the family of splits is not weakly compatible, we also see that the filtration
naturally gives us a way to associate a graded hierarchy of phylogenetic networks to the collection of splits. 相似文献
13.
We explore the link between combinatorics and probability generated by the question “What does a random parking function look like?” This gives rise to novel probabilistic interpretations of some elegant, known generating functions. It leads to new combinatorics: how many parking functions begin with i? We classify features (e.g., the full descent pattern) of parking functions that have exactly the same distribution among parking functions as among all functions. Finally, we develop the link between parking functions and Brownian excursion theory to give examples where the two ensembles differ. 相似文献
14.
There is a close connection between a variety and its clone. The clone of a variety is a multibased algebra, where the different
universes are the sets of n-ary terms over this variety for every natural number n and where the operations describe the superposition of terms of different arities. All projections are added as nullary operations.
Subvarieties correspond to homomorphic images of clones. Subclones can be described by reducts of varieties, isomorphic clones
by equivalent varieties. Clone identities correspond to hyperidentities and varieties of clones to hypervarieties. Pseudovarieties
are classes of finite algebras which are closed under taking of subalgebras, homomorphic images and finite direct products.
Pseudovarieties are important in the theories of finite state automata, rational languages, finite semigroups and their connections.
In a very natural way, there arises the question for the clone of a pseudovariety. In the present paper, we will describe
this algebraic structure.
Received April 6, 2004; accepted in final form March 28, 2005. 相似文献
15.
Maxim R. Burke 《Journal of Geometry》2004,79(1-2):56-58
We give examples of paradoxical subsets of the plane which do not have Lebesgue
measure zero. 相似文献
16.
We define a global order of norms using strongly optimal strategies in Blackwell games and prove that it is a prewellordering
under the assumption of the Axiom of Blackwell determinacy. 相似文献
17.
Shokoofeh Ghorbani 《Annals of Pure and Applied Logic》2018,169(10):981-996
In this paper, we introduce and study a logic that corresponds to abstract hoop twist-structures and present some results on this logic. We prove the local deductive theorem for this logic and show that this logic is algebraizable with respect to the quasi-variety of abstract hoop twist-structures. 相似文献
18.
Kripke models for classical logic 总被引:1,自引:0,他引:1
We introduce a notion of the Kripke model for classical logic for which we constructively prove the soundness and cut-free completeness. We discuss the novelty of the notion and its potential applications. 相似文献
19.
Wolfgang Lenzen 《Logica Universalis》2008,2(1):43-58
In the 18th century, Gottfried Ploucquet developed a new syllogistic logic where the categorical forms are interpreted as
set-theoretical identities, or diversities, between the full extension, or a non-empty part of the extension, of the subject
and the predicate. With the help of two operators ‘O’ (for “Omne”) and ‘Q’ (for “Quoddam”), the UA and PA are represented
as ‘O(S) – Q(P)’ and ‘Q(S) – Q(P)’, respectively, while UN and PN take the form ‘O(S) > O(P)’ and ‘Q(S) > O(P)’, where ‘>’ denotes set-theoretical disjointness. The use of the symmetric operators ‘–’ and ‘>’ gave rise to a new conception
of conversion which in turn lead Ploucquet to consider also the unorthodox propositions O(S) – O(P), Q(S) – O(P), O(S) > Q(P), and Q(S) > Q(P). Although Ploucquet’s critique of the traditional theory of opposition turns out to be mistaken, his theory of the “Quantification
of the Predicate” is basically sound and involves an interesting “Double Square of Opposition”.
My thanks are due to Hanno von Wulfen for helpful discussions and for transforming the word-document into a Latex-file. 相似文献