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There are a number of different approaches one can take to giving the semantics for the quantifiers. The simplest method uses truth value semantics with the substitution interpretation of the quantifiers (Leblanc, 1976). Although the substitution interpretation can be criticized, it provides an excellent starting point for understanding the alternatives, since it avoids a number of annoying technical complications. For students who prefer to learn the adequacy proofs in easy stages, it is best to master the reasoning for the substitution interpretation first. This will provide a core understanding of the basic strategies, which may be embellished (if one wishes) to accommodate more complex treatments of quantification.
Truth Value Semantics with the Substitution Interpretation
The substitution interpretation is based on the idea that a universal sentence ∀xAx is true exactly when each of its instances Aa, Ab, Ac, . . , is true. For classical logic, ∀xAx is T if and only if Ac is T for each constant c of the language. In the case of free logic, the truth condition states that ∀xAx is T if and only if Ac is T for all constants that refer to a real object. Since the sentence Ec indicates that c refers to a real object, the free logic truth condition should say that Ac is T for all those constants c such that Ec is also true.
A pervasive feature of natural languages is that sentences depend for their truth value on the context or situation in which they are evaluated. For example, sentences like ‘It is raining’ and ‘I am glad’ cannot be assigned truth values unless the time, place of utterance, and the identity of the speaker are known. The same sentence may be true in one situation and false in another. In modal language, where we consider how things might have been, sentences may be evaluated in different possible worlds.
In the standard extensional semantics, truth values are assigned directly to sentences, as if the context had no role to play in their determination. This conflicts with what we know about ordinary language. There are two ways to solve the problem. The first is to translate the content of a sentence uttered in a given context into a corresponding sentence whose truth value does not depend on the context. For example, ‘It is raining’ might be converted into ‘It is raining in Houston at 12:00 EST on Dec. 9, 1997..’. The dots here indicate that the attempt to eliminate all context sensitivity may be a never-ending story. For instance, we forgot to say that we are using the Gregorian calendar, or that the sentence is to be evaluated in the real world.
English phrases that begin with ‘the’, such as ‘the man’ and ‘the present king of France’, are called definite descriptions (or descriptions, for short). So far, we have no adequate logical notation for descriptions. It is possible to translate ‘the man is bald’ by choosing a constant c for ‘the man’, a predicate letter P for ‘is bald’, and writing: Pc. However, treating the description as if it were a constant will cause us to classify some valid arguments as invalid.
For example, it should be clear that (1) entails (2).
(1) Aristotle is the philosopher who taught Alexander the Great.
(2) Aristotle taught Alexander the Great.
If we choose the constants: a for Aristotle, and g for Alexander the Great, we might notate (2) as (2′).
(2′) Tag
If we treat ‘the philosopher who taught Alexander the Great’ as a constant g, then (1) is notated by (1′).
(1′) a≈g
However, there is no logical relationship between the atomic sentences (1′) and (2′) that would cause us to recognize that the argument from (1′) to (2′) is valid. Clearly we need a way to notate the internal structure of ‘the philosopher who taught Alexander the Great’ if we are ever to show that (1) entails (2) in logic.
We have already encountered the de re – de dicto distinction at a number of points in this book. In this section, we will investigate the distinction more carefully, explain methods used to notate it, and develop quantified modal logics that are adequate for arguments involving the new notation.
Some of the best illustrations of the de re – de dicto distinction can be found among sentences of tense logic. For example, consider (S).
(S) The president was a crook.
This sentence is ambiguous. It might be taken to claim of the present president that he (Obama at the time this was written) used to be a crook. On the other hand, it might be read ‘At some time in the past the president (at that time) was a crook’. On this last reading, we are saying that the whole sentence (or dictum, in Latin) ‘the president is a crook’ was true at some past time. This is the de dicto reading of (S). Here both ‘the president’ and ‘is a crook’ are read in the past tense. We can represent this interpretation of (S) by applying the past tense operator P to the sentence ‘the president is a crook’, so that both ‘the president’ and ‘is a crook’ lie in its scope.
P(the president is a crook) de dicto reading of (S)
On the first reading of (S), we are saying a certain thing (in Latin, res) has a past tense property: of having been a crook. This is the de re reading of (S). Here we read ‘the president’ in the present tense, and ‘is a crook’ in the past tense. We can represent this reading by restricting the scope of the past tense operator P to the predicate ‘is a crook’.
the president P(is a crook) de re reading of (S)
The distinction between these two readings of (S) is a crucial one, for given that Obama never was a crook, and that Nixon was, the de dicto version of (S) is true, while the de re version is false.
The aim of this study was twofold: we investigated (a) the effect of two types of captioned video (i.e., on-screen text in the same language as the video) on listening comprehension; (b) L2 learners’ perception of the usefulness of captions while watching L2 video. The participants, 226 university-level students from a Flemish university, watched three short French clips in one of three conditions: the control group watched the clips without captions (N = 70), the second group had fully captioned clips (N = 81), the third group had keyword captioned clips (N = 75). After each clip, all participants took a listening comprehension test, which consisted of global and detailed questions. To answer the detailed questions, participants had access to an audio passage of the corresponding clip. At the end of the experiment, participants completed a questionnaire and open-ended survey questions about their perception of captions. Our findings revealed that the full captioning group outperformed both the no captioning and the keyword captioning group on the global comprehension questions. However, no difference was found between the keyword captioning and the no captioning group. Results of the detailed comprehension questions (with audio) revealed no differences between the three conditions. A content-analysis approach to the questionnaire indicated that learners’ perceived need for full captions is strong. Participants consider captions useful for speech decoding and meaning-making processes. Surprisingly, keyword captions were considered highly distracting. These findings suggest that full rather than keyword captioning should be considered when proposing video-based listening comprehension activities to L2 learners.
This study concerned multiple exposures to English before writing and aimed to explore the possibility of an increase in free active vocabulary with a focus on latent productive vocabulary beyond the first 2,000 most frequent words. The researcher incorporated online video into her college freshman composition class and examined its effects on non-basic vocabulary use. To activate previously known vocabulary, a variety of audiovisual modes before writing were applied to four groups alternately: (1) video with captions, (2) video without captions, (3) silent video with captions, and (4) video with screen off (soundtrack only). The results show that the writing involving non-captioned videos contained a higher percentage of advanced vocabulary than that with the other three conditions (specifically, 12.45% versus 11.33% with captioned videos, 5.2% with silent but captioned videos and 8.63% with audio only). Drawing upon the dual-coding theory, this study also points out some pedagogical implications for a video-based writing course.
Evaluative techniques offer a tremendous potential for online controller design. However, when the optimization space is large and the performance metric is noisy, the overall adaptation process becomes extremely time consuming. Distributing the adaptation process reduces the required time and increases robustness to failure of individual agents. In this paper, we analyze the role of the four algorithmic parameters that determine the total evaluation time in a distributed implementation of a Particle Swarm Optimization (PSO) algorithm. For an obstacle avoidance case study using up to eight robots, we explore in simulation the lower boundaries of these parameters and propose a set of empirical guidelines for choosing their values. We then apply these guidelines to a real robot implementation and show that it is feasible to optimize 24 control parameters per robot within 2 h, a limited amount of time determined by the robots' battery life. We also show that a hybrid simulate-and-transfer approach coupled with a noise-resistant PSO algorithm can be used to further reduce experimental time as compared to a pure real-robot implementation.
For many EFL learners, listening poses a grave challenge. The difficulty in segmenting a stream of speech and limited capacity in short-term memory are common weaknesses for language learners. Specifically, reduced forms, which frequently appear in authentic informal conversations, compound the challenges in listening comprehension. Numerous interventions have been implemented to assist EFL language learners, and of these, the application of captions has been found highly effective in promoting learning. Few studies have examined how different modes of captions may enhance listening comprehension. This study proposes three modes of captions: full, keyword-only, and annotated keyword captions and investigates their contribution to the learning of reduced forms and overall listening comprehension. Forty-four EFL university students participated in the study and were randomly assigned to one of the three groups. The results revealed that all three groups exhibited improvement on the pre-test while the annotated keyword caption group exhibited the best performance with the highest mean score. Comparing performances between groups, the annotated keyword caption group also emulated both the full caption and the keyword-only caption groups, particularly in the ability to recognize reduced forms. The study sheds light on the potential of annotated keyword captions in enhancing reduced forms learning and overall listening comprehension.
We study a discrete time self-interacting random process on graphs, which we call greedy random walk. The walker is located initially at some vertex. As time evolves, each vertex maintains the set of adjacent edges touching it that have not yet been crossed by the walker. At each step, the walker, being at some vertex, picks an adjacent edge among the edges that have not traversed thus far according to some (deterministic or randomized) rule. If all the adjacent edges have already been traversed, then an adjacent edge is chosen uniformly at random. After picking an edge the walker jumps along it to the neighbouring vertex. We show that the expected edge cover time of the greedy random walk is linear in the number of edges for certain natural families of graphs. Examples of such graphs include the complete graph, even degree expanders of logarithmic girth, and the hypercube graph. We also show that GRW is transient in $\mathbb{Z}^d$ for all d ≥ 3.
This paper carries out stochastic comparisons of series and parallel systems with independent and heterogeneous components in the sense of the hazard rate order, the reversed hazard rate order, and the likelihood ratio order. The main results extend and strengthen the corresponding ones by Misra and Misra [18] and by Ding, Zhang, and Zhao [8]. Meanwhile, the results on the hazard rate order of parallel systems and the reversed hazard order of series systems serve as nice supplements to Theorem 16.B.1 of Boland and Proschan [4] and Theorem 3.2 of Nanda and Shaked [20], respectively.
We consider the problem of a firm facing failures with weak forewarning signals. In the base model that we study, the firm watches for signals of a random arrival of a disruptive innovation and continuously updates the posterior probability that a disruptive innovation has already happened. A disruptive innovation is marked by a rapid increase in the growth rate of the market for a new technology, and it is followed by a random arrival of catastrophic failure of the firm. The firm can invest capital to adopt the innovation to prevent failure. The optimal policy is to adopt it when the posterior probability exceeds an optimally chosen threshold. We investigate the probability of failure under the optimal policy when the cost of failure is large and the arrival rate of disruptive innovation is low. The probability of failure is close to one if the arrival rate is extremely low while it is close to zero if the arrival rate is moderate. We also consider an extension of the base model to incorporate recurrence of disruptive innovation; when the arrival rate is moderate, the optimal threshold and the failure probability can be significantly larger than those of the base model.
In Dynamic Programing, mixed strategies consist of randomizing the choice of actions. In some problems, such as portfolio management, it makes sense to diversify actions rather than choosing among them purely or randomly. Optimal betting in casinos and roulette by a gambler with fixed goal was studied by Dubins and Savage [9] and their school without the element of diversification (betting simultaneously on different holes of the roulette), once it was proved (Smith's theorem - Smith [16], Dubins [8] and Gilat and Weiss [10]) that diversification does not increase the probability of reaching the goal. We question the scope of this finding, that was based on the assumption that the holes on which gamblers can bet are disjoint, such as 1 and BLACK in regular roulette. A counter example is provided in which holes are nested, such as 1 and RED. Thus, it may be rational for gamblers with a fixed goal to place chips on more than one hole at the table.
This note is related to a joint work with Michèle Cohen on the preference for safety in the Choquet Expected Utility model.
In this paper we show how the permutation Monte Carlo method, originally developed for reliability networks, can be successfully adapted for stochastic flow networks, and in particular for estimation of the probability that the maximal flow in such a network is above some fixed level, called the threshold. A stochastic flow network is defined as one, where the edges are subject to random failures. A failed edge is assumed to be erased (broken) and, thus, not able to deliver any flow. We consider two models; one where the edges fail with the same failure probability and another where they fail with different failure probabilities. For each model we construct a different algorithm for estimation of the desired probability; in the former case it is based on the well known notion of the D-spectrum and in the later one—on the permutational Monte Carlo. We discuss the convergence properties of our estimators and present supportive numerical results.
The so called “Israeli Queue” is a single server polling system with batch service of an unlimited size, where the next queue to be visited is the one in which the first customer in line has been waiting for the longest time. The case with finite number of queues (groups) was introduced by Boxma, Van der Wal and Yechiali [3]. In this paper we extend the model to the case with a (possibly) infinite number of queues. We analyze the M/M/1, M/M/c, and M/M/1/N—type queues, as well as a priority model with (at most) M high-priority classes and a single lower priority class. In all models we present an extensive probabilistic analysis and calculate key performance measures.
Many modern networks grow from blocks. We study the probabilistic behavior of parameters of a blocks tree, which models several kinds of networks. It grows from building blocks that are themselves rooted trees. We investigate the number of leaves, depth of nodes, total path length, and height of such trees. We use methods from the theory of Pólya urns and martingales.