Absolutely independent axiomatizations for countable sets in classical logic

The notion of absolute independence, considered in this paper has a clear algebraic meaning and is a strengthening of the usual notion of logical independence. We prove that any consistent and countable set in classical prepositional logic has an absolutely independent axiornatization.     

Two notions of compactness in Gödel logics

Compactness is an important property of classical propositional logic. It can be defined in two equivalent ways. The first one states that simultaneous satisfiability of an infinite set of formulae is equivalent to the satisfiability of all its finite subsets. The second one states that if a set of formulae entails a formula, then there is a finite subset entailing this formula as well.In propositional many-valued logic, we have different degrees of satisfiability and different possible definitions of entailment, hence the questions of compactness is more complex. In this paper we will deal with compactness of Gödel, Gödel_Δ, and Gödel_～ logics.There are several results (all for the countable set of propositional variables) concerning the compactness (based on satisfiability) of these logic by Cintula and Navara, and the question of compactness (based on entailment) for Gödel logic was fully answered by Baaz and Zach (see papers [3] and [2]).In this paper we give a nearly complete answer to the problem of compactness based on both concepts for all three logics and for an arbitrary cardinality of the set of propositional variables. Finally, we show a tight correspondence between these two concepts     

A Proof of Standard Completeness for Esteva and Godo's Logic MTL

In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godo's logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in.     

To Be is to Be the Object of a Possible Act of Choice

Aim of the paper is to revise Boolos’ reinterpretation of second-order monadic logic in terms of plural quantification ([4], [5]) and expand it to full second order logic. Introducing the idealization of plural acts of choice, performed by a suitable team of agents, we will develop a notion of plural reference. Plural quantification will be then explained in terms of plural reference. As an application, we will sketch a structuralist reconstruction of second-order arithmetic based on the axiom of infinite à la Dedekind, as the unique non-logical axiom. We will also sketch a virtual interpretation of the classical continuum involving no other infinite than a countable plurality of individuals.     

Agreement Theorems in Dynamic-Epistemic Logic

This paper introduces Agreement Theorems to dynamic-epistemic logic. We show first that common belief of posteriors is sufficient for agreement in epistemic-plausibility models, under common and well-founded priors. We do not restrict ourselves to the finite case, showing that in countable structures the results hold if and only if the underlying plausibility ordering is well-founded. We then show that neither well-foundedness nor common priors are expressible in the language commonly used to describe and reason about epistemic-plausibility models. The static agreement result is, however, finitely derivable in an extended modal logic. We provide the full derivation. We finally consider dynamic agreement results. We show they have a counterpart in epistemic-plausibility models, and provide a new form of agreements via public announcements.     

On the extension of additive utilities to infinite sets

Independence condition C is known as necessary and sufficient for the existence of an additive utility on a finite subset X of a Cartesian product. A stronger necessary condition, H, interpreted as both an independence and Archimedean condition, is derived. It is shown to be sufficient when X is countable by constructing an additive utility as the limit of a sequence of additive utilities on finite subsets of X. When X is not countable, but is a Cartesian product, another necessary condition, the existence of A, a countable perfectly (order-) dense subset of X, is added to H; an additive utility is constructed by extension to X of an additive utility on a countable set linked to A. An application to a no-solvability case is given.     

On the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics.     

On structural completeness of implicational logics

We consider the notion of structural completeness with respect to arbitrary (finitary and/or infinitary) inferential rules. Our main task is to characterize structurally complete intermediate logics. We prove that the structurally complete extension of any pure implicational in termediate logic C can be given as an extension of C with a certain family of schematically denned infinitary rules; the same rules are used for each C. The cardinality of the family is continuum and, in the case of (the pure implicational fragment of) intuitionistic logic, the family cannot be reduced to a countable one. It means that the structurally complete extension of the intuitionistic logic is not countably axiomatizable by schematic rules.This work was supported by the Polish Academy of Sciences, CPBP 08.15, Struktura logiczna rozumowa niesformalizowanych.     

Galois structures

This paper is a continuation of investigations on Galois connections from [1], [3], [10]. It is a continuation of [2]. We have shown many results that link properties of a given closure space with that of the dual space. For example: for every -disjunctive closure space X the dual closure space is topological iff the base of X generated by this dual space consists of the -prime sets in X (Theorem 2). Moreover the characterizations of the satisfiability relation for classical logic are shown. Roughly speaking our main result here is the following: a satisfiability relation in a logic L with, a countable language is a fragment of the classical one iff the compactness theorem for L holds (Theorems 3–8).     

All roads lead to violations of countable additivity

This paper defends the claim that there is a deep tension between the principle of countable additivity and the one-third solution to the Sleeping Beauty problem. The claim that such a tension exists has recently been challenged by Brian Weatherson, who has attempted to provide a countable additivity-friendly argument for the one-third solution. This attempt is shown to be unsuccessful. And it is argued that the failure of this attempt sheds light on the status of the principle of indifference that underlies the tension between countable additivity and the one-third solution.     

Non-verbal numerical cognition: from reals to integers

Data on numerical processing by verbal (human) and non-verbal (animal and human) subjects are integrated by the hypothesis that a non-verbal counting process represents discrete (countable) quantities by means of magnitudes with scalar variability. These appear to be identical to the magnitudes that represent continuous (uncountable) quantities such as duration. The magnitudes representing countable quantity are generated by a discrete incrementing process, which defines next magnitudes and yields a discrete ordering. In the case of continuous quantities, the continuous accumulation process does not define next magnitudes, so the ordering is also continuous ('dense'). The magnitudes representing both countable and uncountable quantity are arithmetically combined in, for example, the computation of the income to be expected from a foraging patch. Thus, on the hypothesis presented here, the primitive machinery for arithmetic processing works with real numbers (magnitudes).     

Convergence to the Truth Without Countable Additivity

Must probabilities be countably additive? On the one hand, arguably, requiring countable additivity is too restrictive. As de Finetti pointed out, there are situations in which it is reasonable to use merely finitely additive probabilities. On the other hand, countable additivity is fruitful. It can be used to prove deep mathematical theorems that do not follow from finite additivity alone. One of the most philosophically important examples of such a result is the Bayesian convergence to the truth theorem, which says that conditional probabilities converge to 1 for true hypotheses and to 0 for false hypotheses. In view of the long-standing debate about countable additivity, it is natural to ask in what circumstances finitely additive theories deliver the same results as the countably additive theory. This paper addresses that question and initiates a systematic study of convergence to the truth in a finitely additive setting. There is also some discussion of how the formal results can be applied to ongoing debates in epistemology and the philosophy of science.

     

Proof Theory of Paraconsistent Quantum Logic

Paraconsistent quantum logic, a hybrid of minimal quantum logic and paraconsistent four-valued logic, is introduced as Gentzen-type sequent calculi, and the cut-elimination theorems for these calculi are proved. This logic is shown to be decidable through the use of these calculi. A first-order extension of this logic is also shown to be decidable. The relationship between minimal quantum logic and paraconsistent four-valued logic is clarified, and a survey of existing Gentzen-type sequent calculi for these logics and their close relatives is addressed.     

Consistency Defaults

A consistency default is a propositional inference rule that asserts the consistency of a formula in its consequence. Consistency defaults allow for a straightforward encoding of domains in which it is explicitely known when something is possible. The logic of consistency defaults can be seen as a variant of cumulative default logic or as a generalization of justified default logic; it is also able to simulate Reiter default logic in the seminormal case. A semantical characterization of consistency defaults in terms of processes and in terms of a fixpoint equation is given, as well as a normal form. Presented by Melvin Fitting     

On jnd representations of semiorders

This paper determines automorphisms and endomorphisms associated with constant jnd (just noticeable difference) semiorder representations. Building on the constant jnd case, a novel representation theorem is given for countable semiorders which provides insight into the role of more complex jnd functions. The significance of the representation problem for countable structures is clarified by discussion of relations among observations, representations, and theories. The stress is on motivation, and on general methods applicable to many measurement models.     

Cut-Elimination for Quantified Conditional Logic

A semantic embedding of quantified conditional logic in classical higher-order logic is utilized for reducing cut-elimination in the former logic to existing results for the latter logic. The presented embedding approach is adaptable to a wide range of other logics, for many of which cut-elimination is still open. However, special attention has to be payed to cut-simulation, which may render cut-elimination as a pointless criterion.     

可数图灵理想的恰对和一致上界

一个（图灵）理想,是满足两个封闭条件的图灵度集合：向下封闭;任意,中一对图灵度的上确界也在,中。可数理想不仅在图灵度整体性质的研究中有着重要意义,而且在对哥德尔可构成集合L精细结构的早期研究中也发挥过重要作用。研究可数理想的两个重要概念是：恰对和一致上界。借助这两个概念,我们可以将可数理想简化为一个（一致上界）或者一对（恰对）图灵度。通过前人的研究,我们可以发现这两个概念是紧密相连的,同时我们也可以对它们的关系提出进一步的问题。在本文中,我们证明以下定理：任给一个可数理想I,都存在两个I的一致上界a0和a1,同时a0和a1构成,的一个恰对。此定理从正面回答了Lerman提出的关于算术图灵度构成的理想的一个问题。此定理的证明实际上是经过小心修改的、典型的恰对构造。我们在典型恰对构造的过程中,加入一些微妙的限制,使得形成恰对的两个图灵度a0和a1可以各自独立地在一定程度上用逼近的办法还原整个构造,从而分别给出可数理想I的一致枚举。在a0和a1分别的逼近中,我们引入了有穷损坏方法。本文的最后指出a0和a1的图灵跃迁的一些性质。     

Finite additivity,another lottery paradox and conditionalisation

In this paper I argue that de Finetti provided compelling reasons for rejecting countable additivity. It is ironical therefore that the main argument advanced by Bayesians against following his recommendation is based on the consistency criterion, coherence, he himself developed. I will show that this argument is mistaken. Nevertheless, there remain some counter-intuitive consequences of rejecting countable additivity, and one in particular has all the appearances of a full-blown paradox. I will end by arguing that in fact it is no paradox, and that what it shows is that conditionalisation, often claimed to be integral to the Bayesian canon, has to be rejected as a general rule in a finitely additive environment.     

Reconstructing the Topology of the Elementary Self-embedding Monoids of Countable Saturated Structures

Every transformation monoid comes equipped with a canonical topology, the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity. In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too. As a second result we strengthen a result by Lascar by showing that whenever \({\mathbf {A}}\) is a countable \(\aleph _0\)-categorical G-finite structure whose automorphism group has a trivial center and if \({\mathbf {B}}\) is any other countable structure, then every isomorphism between the monoids of elementary self-embeddings is a homeomorphism.     

Conditional Probability and Defeasible Inference

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey–Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of non-monotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).Expectation is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. The research of both authors was supported in part by a grant from NSF, and, for Parikh, also by support from the research foundation of CUNY.