Logical Strength Research Articles

Connectedness as a constraint on exhaustification

“Scalar implicatures” is a phrase used to refer to some inferences arising from the competition between alternatives: typically, “Mary read some of the books” ends up conveying that Mary did not read all books, because one could have said “Mary read all books”. The so-called grammatical theory argues that these inferences obtain from the application of a covert operator $$ exh $$ , which not only has the capability to negate alternative sentences, but also the capability to be embedded within sentences under other linguistic operators, i.e. $$ exh $$ has the potential to add to the meaning of expressions (not necessarily full sentences), the negation of their alternatives. This view typically seeks support from the existence of readings that could not be explained without the extra-capability of $$ exh $$ to occur in embedded positions. However, if some embedded positions seem to be accessible to $$ exh $$ , not all conceivable positions that $$ exh $$ could occupy yield sensible results. In short: the $$ exh $$ approach is powerful, maybe too powerful. Various approaches based on logical strength and monotonicity have been proposed to justify on principled grounds the limited distribution of $$ exh $$ ; these approaches are mostly based on a comparison between possible parses, and considerations of monotonicity (e.g., the Strongest Meaning Hypothesis). We propose a new constraint based instead on “connectedness”, ruling out parses because of inherent problems their outcome may raise. Connectedness is a sister notion of monotonicity, which has been recruited to explain certain lexical restrictions on nouns, adjectives and more recently quantifiers; we propose here that connectedness could play a similar role at the level of propositional meanings.

Dealing with Incompatibilities among Procedural Goals under Uncertainty

By considering rational agents, we focus on the problem of selecting goals out of a set of incompatible ones. We consider three forms of incompatibility introduced by Castelfranchi and Paglieri, namely the terminal, the instrumental (or based on resources), and the superfluity. We represent the agent's plans by means of structured arguments whose premises are pervaded with uncertainty. We measure the strength of these arguments in order to determine the set of compatible goals. We propose two novel ways for calculating the strength of these arguments, depending on the kind of incompatibility thatexists between them. The first one is the logical strength value, it is denoted by a three-dimensional vector, which is calculated from a probabilistic interval associated with each argument. The vector represents the precision of the interval, the location of it, and the combination of precision and location. This type of representation and treatment of the strength of a structured argument has not been defined before by the state of the art. The second way for calculating the strength of the argument is based on the cost of the plans (regarding the necessary resources) and the preference of the goals associated with the plans. Considering our novel approach for measuring the strength of structured arguments, we propose a semantics for the selection of plans and goals that is based on Dung's abstract argumentation theory. Finally, we make a theoretical evaluation of our proposal.

Sextus Empiricus' Fourth Conditional and Containment Logic

In his Outlines of Pyrrhonism 2.110–113, Sextus Empiricus presents four different accounts of the conditional, presumably all from the Hellenistic period, in increasing logical strength. While the interpretation and provenance of the first three accounts is relatively secure, the fourth account has perplexed and frustrated interpreters for decades or longer. Most interpreters have ultimately taken a dismissive attitude towards the fourth account and discounted it as being of both little historical and logical interest. We argue that this attitude is unwarranted and demonstrate that the conditional expressed in the fourth account can profitably be understood as a precursor of analytic entailment, familiar from containment logics such as those developed by Parry and Angell. Exploiting recent work by Fine, we present a formal truthmaker semantics for this conditional and show how it sheds light on a number of longstanding issues in the interpretation of this passage.

THE LARGE STRUCTURES OF GROTHENDIECK FOUNDED ON FINITE-ORDER ARITHMETIC

AbstractThe large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck’s SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong.

The strength of compactness in Computability Theory and Nonstandard Analysis

Compactness is one of the core notions of analysis: it connects local properties to global ones and makes limits well-behaved. We study the computational properties of the compactness of Cantor space 2Nfor uncountable covers. The most basic question is: how hard is it to compute a finite sub-cover from such a cover of2N? Another natural question is: how hard is it to compute a sequence that covers2Nminus a measure zero set from such a cover? The special and weak fan functionals respectively compute such finite sub-covers and sequences. In this paper, we establish the connection between these new fan functionals on one hand, and various well-known comprehension axioms on the other hand, including arithmetical comprehension, transfinite recursion, and the Suslin functional. In the spirit of Reverse Mathematics, we also analyse the logical strength of compactness in Nonstandard Analysis. Perhaps surprisingly, the results in the latter mirror (often perfectly) the computational properties of the special and weak fan functionals. In particular, we show that compactness (nonstandard or otherwise) readily brings us to the outer edges of Reverse Mathematics (namely Π21-CA0), and even into Schweber's higher-order framework (namely Σ12-separation).

Is there a Bayesian justification of hypothetico‐deductive inference?

AbstractMany philosophers have claimed that Bayesianism can provide a simple justification for hypothetico‐deductive (H‐D) inference, long regarded as a cornerstone of the scientific method. Following up a remark of van Fraassen (1985), we analyze a problem for the putative Bayesian justification of H‐D inference in the case where what we learn from observation is logically stronger than what our theory implies. Firstly, we demonstrate that in such cases the simple Bayesian justification does not necessarily apply. Secondly, we identify a set of sufficient conditions for the mismatch in logical strength to be justifiably ignored as a “harmless idealization”. Thirdly, we argue, based upon scientific examples, that the pattern of H‐D inference of which there is a ready Bayesian justification is only rarely the pattern that one actually finds at work in science. Whatever the other virtues of Bayesianism, the idea that it yields a simple justification of a pervasive pattern of scientific inference appears to have been oversold.

Forecasting Short-Term Traffic Flow by Fuzzy Wavelet Neural Network with Parameters Optimized by Biogeography-Based Optimization Algorithm.

Forecasting short-term traffic flow is a key task of intelligent transportation systems, which can influence the traveler behaviors and reduce traffic congestion, fuel consumption, and accident risks. This paper proposes a fuzzy wavelet neural network (FWNN) trained by improved biogeography-based optimization (BBO) algorithm for forecasting short-term traffic flow using past traffic data. The original BBO is enhanced by the ring topology and Powell's method to advance the exploration capability and increase the convergence speed. Our presented approach combines the strengths of fuzzy logic, wavelet transform, neural network, and the heuristic algorithm to detect the trends and patterns of transportation data and thus has been successfully applied to transport forecasting. Other different forecasting methods, including ANN-based model, FWNN-based model, and WNN-based model, are also developed to validate the proposed approach. In order to make the comparisons across different methods, the performance evaluation is based on root-mean-squared error (RMSE), mean absolute percentage error (MAPE), and correlation coefficient (R). The performance indexes show that the FWNN model achieves lower RMSE and MAPE, as well as higher R, indicating that the FWNN model is a better predictor.

Deviance and Vice: Strength as a Theoretical Virtue in the Epistemology of Logic

This paper is about the putative theoretical virtue ofstrength, as it might be used in abductive arguments to the correct logic in the epistemology of logic. It argues for three theses. The first is that the well‐defined property oflogical strengthis neither a virtue nor a vice, so that logically weaker theories are not—all other things being equal—worse or better theories than logically stronger ones. The second thesis is that logical strength does not entail the looser characteristic ofscientific strength, and the third is that many modern logics are on a par—or can be made to be on a par—with respect to scientific strength.

The Logical Strength of Compositional Principles

This paper investigates a set of issues connected with the so-called conservativeness argument against deflationism. Although I do not defend that argument, I think the discussion of it has raised some interesting questions about whether what I call “compositional principles,” such as “a conjunction is true iff its conjuncts are true,” have substantial content or are in some sense logically trivial. The paper presents a series of results that purport to show that the compositional principles for a first-order language, taken together, have substantial logical strength, amounting to a kind of abstract consistency statement.

Two kinds of fixed point theorems and reverse mathematics

AbstractIn this paper, we investigate the logical strength of two types of fixed point theorems in the context of reverse mathematics. One is concerned with extensions of the Banach contraction principle. Among theorems in this type, we mainly show that the Caristi fixed point theorem is equivalent to over . The other is dedicated to topological fixed point theorems such as the Brouwer fixed point theorem. We introduce some variants of the Fan‐Browder fixed point theorem and the Kakutani fixed point theorem, which we call and , respectively. Then we show that is equivalent to and is equivalent to , over . In addition, we also study the application of the Fan‐Browder fixed point theorem to game systems.

Euler’s Königsberg: the explanatory power of mathematics

The present paper provides an analysis of Euler’s solutions to the Konigsberg bridges problem. Euler proposes three different solutions to the problem, addressing their strengths and weaknesses along the way. I put the analysis of Euler’s paper to work in the philosophical discussion on mathematical explanations. I propose that the key ingredient to a good explanation is the degree to which it provides relevant information. Providing relevant information is based on knowledge of the structure in question, graphs in the present case. I also propose computational complexity and logical strength as measures of relevant information.

Coloring the rationals in reverse mathematics

Ramsey’s theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey’s theorem for pairs due to Erdős and Rado, which states that every 2-coloring of the pairs of rationals has either an infinite 0-homogeneous set or a 1-homogeneous set of order type η, where η is the order type of the rationals. This theorem is a natural candidate to lie strictly between the arithmetic comprehension axiom and Ramsey’s theorem for pairs. This Erdős–Rado theorem, like the tree theorem for pairs, belongs to a family of Ramsey-type statements whose logical strength remains a challenge.

Vertical Handoff Decision Using Fuzzification and Combinatorial Fusion

This letter proposes a new approach to making network selection simple yet effective. Under consideration is a heterogeneous environment where a mobile user roams within a two-tier network architecture over different radio access technologies. Allowing for multiple determinants, we design a scheme for a user equipment to deduce whether handoff is imminent by fusing several metrics-received signal strength, data rate, and network latency. Our method harnesses strengths of fuzzy logic while maintaining lower computational complexity by means of combinatorial fusion. Our results demonstrate that the new approach outperforms other previous proposed fuzzy logic-based strategies in terms of time efficiency and operational effectiveness with the same level of vertical handoffs.

Fuzzy Reasoning Based Time Series Forecasting Using Weighted Subsethood Model

The one central problem in global forecasting area is to minimize the forecasting error and to have a simple model. This paper focuses on the fuzzy time series (FTS) forecasting which is one branch of fuzzy set (FS). Numerous FTS models have been proposed in literature during the past decades and the number of new models is still growing especially in stock markets and student enrolment problems. Some of the FTS models found in literature do not explain the fuzzy logical relationship strength between Ai and Aj which are the left-hand side (LHS) and right-hand side (RHS) of the fuzzy logical relationship respectively. This paper proposed Weighted Subsethood Fuzzy Time Series (WeSuFTS) to build fuzzy reasoning based to address the drawback in forecasting enrolment of students. In this paper, we create the weighted subsethood fuzzy rules to explain the fuzzy relationship between Ai and Aj . The simple centre mid-point defuzzification technique was introduced in defuzzification step. The results show that WeSuFTS method gives promising results where the average error forecasting is 2.31% which is smaller than the one presented by Song and Chissom in 1993 and Chen in 1996, the pioneer of FTS. The proposed method has an advantage since it can give a new interpretation of FTS model. Furthermore, this method able to present the degree of importance of LHS to RHS in explaining the fuzzy logical relationship in FTS.

A Multitheoretical Approach to Big Text Data: Comparing Expressive and Rhetorical Logics in Yelp Reviews

This article uses a multitheoretical approach to investigate the relationship between language use and opinion expression on Yelp. Using review metadata (e.g., star rating) to observe variation in reviewer feelings and motivations, we test for the strength of different message design logics: expressive logics, where language reflects a reviewer’s underlying opinion, and rhetorical logics, where language reflects a reviewer’s desire to make his or her opinion credible and acceptable to their audience. Results suggest that emotional language is motivated by expression as higher rated businesses are reviewed with more positive and fewer negative emotion terms. Rhetorical logics are associated with the use of abstract and self-focused language, with analysis suggesting this may result from the reviewer’s decision to write either narratively or formally.

The Role of Speaker Knowledge in Children's Pragmatic Inferences.

During communication, conversational partners should offer as much information as is required and relevant. For instance, the statement "Some Xs Y" is infelicitous if one knows that all Xs Y. Do children understand the link between speaker knowledge and utterance strength? In Experiment 1, 5-year-olds (N=32) but not 4-year-olds (N=32) reliably connected statements of different logical strength (e.g., "The girl colored all/some of the star") to observers who were fully or partially informed. Four-year-olds' performance improved when observer knowledge could be assessed more easily (Experiment 2a, N=25) but remained the same in a nonlinguistic version of Experiment 1 that preserved the epistemic requirements of the original study (Experiment 2b, N=26). These findings have implications for the development of early communicative abilities.

CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY

AbstractWe analyze the set-theoretic strength of determinacy for levels of the Borel hierarchy of the form$\Sigma _{1 + \alpha + 3}^0 $, forα<ω1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to requireα+ 1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of weak reflection principles, Π1-RAPα, whose consistency strength corresponds exactly to the logical strength of${\rm{\Sigma }}_{1 + \alpha + 3}^0 $determinacy, for$\alpha < \omega _1^{CK} $. This yields a characterization of the levels ofLby or at which winning strategies in these games must be constructed. Whenα= 0, we have the following concise result: The leastθso that all winning strategies in${\rm{\Sigma }}_4^0 $games belong toLθ+1is the least so that$L_\theta \models {\rm{``}}{\cal P}\left( \omega \right)$exists, and all wellfounded trees are ranked”.

On the logical strengths of partial solutions to mathematical problems

We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood, we say that a Ramsey-type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey-type variants of problems related to K\"onig's lemma, such as restrictions of K\"onig's lemma, Boolean satisfiability problems, and graph coloring problems. We find that sometimes the Ramsey-type variant of a problem is strictly easier than the original problem (as Flood showed with weak K\"onig's lemma) and that sometimes the Ramsey-type variant of a problem is equivalent to the original problem. We show that the Ramsey-type variant of weak K\"onig's lemma is robust in the sense of Montalban: it is equivalent to several perturbations of itself. We also clarify the relationship between Ramsey-type weak K\"onig's lemma and algorithmic randomness by showing that Ramsey-type weak weak K\"onig's lemma is equivalent to the problem of finding diagonally non-recursive functions and that these problems are strictly easier than Ramsey-type weak K\"onig's lemma. This answers a question of Flood.

The strength of infinitary Ramseyan principles can be accessed by their densities

In this article, we conduct a model-theoretic investigation of three infinitary Ramseyan statements: the Infinite Ramsey Theorem for pairs and two colours (RT22), the Canonical Ramsey Theorem for pairs (CRT2) and the Regressive Ramsey Theorem for pairs (RegRT2). We approximate the logical strength of these principles by the strength of their first-order iterated versions, known as density principles. We then investigate their logical strength using strong initial segments of models of Peano Arithmetic, in the spirit of the classical article by Paris and Kirby, hereby re-proving old results model-theoretically. The article is concluded by a discussion of two further outreaches of densities. One is a further investigation of the strength of the Ramsey Theorem for pairs. The other deals with the asymptotics of the standard Ramsey function R22.

Reverse mathematics and marriage problems with finitely many solutions

We analyze the logical strength of theorems on marriage problems with a fixed finite number of solutions via the techniques of reverse mathematics. We show that if a marriage problem has k solutions, then there is a finite set of boys such that the marriage problem restricted to this set has exactly k solutions, each of which extend uniquely to a solution of the original marriage problem. The strength of this assertion depends on whether or not the marriage problem has a bounding function. We also answer three questions from our previous work on marriage problems with unique solutions.