Online research in philosophy

We consider the notion of everyday language. We claim that everyday language is semantically bounded by the properties expressible in the existential fragment of second–order logic. Two arguments for this thesis are formulated. Firstly, we show that so–called Barwise's test of negation normality works properly only when assuming our main thesis. Secondly, we discuss the argument from practical computability for finite universes. Everyday language sentences are directly or indirectly verifiable. We show that in both cases they are bounded by second–order (...) existential properties. Moreover, there are known examples of everyday language sentences which are the most difficult in this class (NPTIME–complete). (shrink)

Among the readings available for NL sentences, those where two or more sets of entities are independent of one another are particularly challenging from both a theoretical and an empirical point of view. Those readings are termed here as ‘Independent Set (IS) readings'. Standard examples of such readings are the well-known Collective and Cumulative Readings. (Robaldo, 2011) proposes a logical framework that can properly represent the meaning of IS readings in terms of a set-Skolemization of the witness sets. One of (...) the main assumptions of Robaldo's logical framework, drawn from (Schwarzschild, 1996), is that pragmatics plays a crucial role in the identification of such witness sets. Those are firstly identified on pragmatic grounds, then logical clauses are asserted on them in order to trigger the appropriate inferences. In this paper, we present the results of an experimental analysis that appears to confirm Robaldo's hypotheses concerning the pragmatic identification of the witness sets. (shrink)

In three experiments, we investigated the computational complexity of German reciprocal sentences with different quantificational antecedents. Building upon the tractable cognition thesis (van Rooij, 2008) and its application to the verification of quantifiers (Szymanik, 2010) we predicted complexity differences among these sentences. Reciprocals with all-antecedents are expected to preferably receive a strong interpretation (Dalrymple et al., 1998), but reciprocals with proportional or numerical quantifier antecedents should be interpreted weakly. Experiment 1, where participants completed pictures according to their preferred interpretation, provides (...) evidence for these predictions. Experiment 2 was a picture verification task. The results show that the strong interpretation was in fact possible for tractable all but one-reciprocals, but not for exactly n. The last experiment manipulated monotonicity of the quantifier antecedents. (shrink)

We study a generalization of the Muddy Children puzzle by allowing public announcements with arbitrary generalized quantifiers. We propose a new concise logical modeling of the puzzle based on the number triangle representation of quantifi ers. Our general aim is to discuss the possibility of epistemic modeling that is cut for specifi c informational dynamics. Moreover, we show that the puzzle is solvable for any number of agents if and only if the quanti fier in the announcement is positively active (...) (satis es a form of variety). (shrink)

The paper presents two case studies of multi-agent information exchange involving generalized quantifiers. We focus on scenarios in which agents successfully converge to knowledge on the basis of the information about the knowledge of others, so-called Muddy Children puzzle and Top Hat puzzle. We investigate the relationship between certain invariance properties of quantifiers and the successful convergence to knowledge in such situations. We generalize the scenarios to account for public announcements with arbitrary quantifiers. We show that the Muddy Children puzzle (...) is solvable for any number of agents if and only if the quantifier in the announcement is positively active (satisfies a version of the variety condition). In order to get the characterization result, we propose a new concise logical modeling of the puzzle based on the number triangle representation of generalized quantifiers. In a similar vein, we also study the Top Hat puzzle. We observe that in this case an announcement needs to satisfy stronger conditions in order to guarantee solvability. Hence, we introduce a new property, called bounded thickness, and show that the solvability of the Top Hat puzzle for arbitrary number of agents is equivalent to the announcement being 1-thick. (shrink)

We study definability of second-order generalized quantifiers. We show that the question whether a second-order generalized quantifier $\sQ_1$ is definable in terms of another quantifier $\sQ_2$, the base logic being monadic second-order logic, reduces to the question if a quantifier $\sQ^{\star}_1$ is definable in $\FO(\sQ^{\star}_2,<,+,\times)$ for certain first-order quantifiers $\sQ^{\star}_1$ and $\sQ^{\star}_2$. We use our characterization to show new definability and non-definability results for second-order generalized quantifiers. In particular, we show that the monadic second-order majority quantifier $\most^1$ is not definable (...) in second-order logic. (shrink)

The paper presents an experimental evidence on differences in the sentence-picture verification under additional memory load between parity and proportional quantifiers. We asked subjects to memorize strings of 4 or 6 digits, then to decide whether a quantifier sentence is true at a given picture, and finally to recall the initially given string of numbers. The results show that: (a) proportional quantifiers are more difficult than parity quantifiers with respect to reaction time and accuracy; (b) maintaining either 4 or 6 (...) elements in the working memory has the same effect on the processing of parity quantifiers; (c) however, in the case of proportional quantifiers subjects performed better in the verification tasks under the 6-digit load condition, and (d) even though the strings of 4 numbers were better recalled by subjects after judging parity there is no difference between quantifiers in the case of the 6-element condition. We briefly outline two alternative explanations for the observed phenomena rooted in the computational model of quantifier verification and the different theories of working memory. (shrink)

Human intentional communication is marked by its flexibility and context sensitivity. Hypothesized brain mechanisms can provide convincing and complete explanations of the human capacity for intentional communication only insofar as they can match the computational power required for displaying that capacity. It is thus of importance for cognitive neuroscience to know how computationally complex intentional communication actually is. Though the subject of considerable debate, the computational complexity of communication remains so far unknown. In this paper we defend the position that (...) the computational complexity of communication is not a constant, as some views of communication seem to hold, but rather a function of situational factors. We present a methodology for studying and characterizing the computational complexity of communication under different situational constraints. We illustrate our methodology for a model of the problems solved by receivers and senders during a communicative exchange. This approach opens the way to a principled identification of putative model parameters that control cognitive processes supporting intentional communication. (shrink)

We compared the processing of natural language quantifiers in a group of patients with schizophrenia and a healthy control group. In both groups, the difficulty of the quantifiers was consistent with computational predictions, and patients with schizophrenia took more time to solve the problems. However, they were significantly less accurate only with proportional quantifiers, like more than half. This can be explained by noting that, according to the complexity perspective, only proportional quantifiers require working memory engagement.

This paper surveys applications of logical methods in the cognitive sciences. Special attention is paid to non-monotonic logics and complexity theory. We argue that these particular tools have been useful in clarifying the debate between symbolic and connectionist models of cognition.

We prove that PTIME generalized quantifiers are closed under Boolean operations, iteration, cumulation and resumption. -/- .

We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multi-quantifier sentences. First, we show that the standard constructions that turn simple determiners into complex quantifiers, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provide an insight into branching operation yielding intractable natural language multi-quantifier expressions. Next, we focus on a linguistic case study. We use computational complexity results to investigate semantic (...) distinctions between quantified reciprocal sentences. We show a computational dichotomy<br>between different readings of reciprocity. Finally, we go more into philosophical speculation on meaning, ambiguity and computational complexity. In particular, we investigate a possibility to<br>revise the Strong Meaning Hypothesis with complexity aspects to better account for meaning shifts in the domain of multi-quantifier sentences. The paper not only contributes to the field of the formal<br>semantics but also illustrates how the tools of computational complexity theory might be successfully used in linguistics and philosophy with an eye towards cognitive science. (shrink)

The paper presents a study examining the role of working<br>memory in quantifier verification. We created situations similar to the<br>span task to compare numerical quantifiers of low and high rank, parity<br>quantifiers and proportional quantifiers. The results enrich and support<br>the data obtained previously in and predictions drawn from a computational<br>model.

We discuss the thesis formulated by Hintikka (1973) that certain natural language sentences require non-linear quantification to express their meaning. We investigate sentences with combinations of quantifiers similar to Hintikka's examples and propose a novel alternative reading expressible by linear formulae. This interpretation is based on linguistic and logical observations. We report on our experiments showing that people tend to interpret sentences similar to Hintikka sentence in a way consistent with our interpretation.

In the dissertation we study the complexity of generalized quantifiers in natural language. Our perspective is interdisciplinary: we combine philosophical insights with theoretical computer science, experimental cognitive science and linguistic theories. -/- In Chapter 1 we argue for identifying a part of meaning, the so-called referential meaning (model-checking), with algorithms. Moreover, we discuss the influence of computational complexity theory on cognitive tasks. We give some arguments to treat as cognitively tractable only those problems which can be computed in polynomial time. (...) Additionally, we suggest that plausible semantic theories of the everyday fragment of natural language can be formulated in the existential fragment of second-order logic. -/- In Chapter 2 we give an overview of the basic notions of generalized quantifier theory, computability theory, and descriptive complexity theory. -/- In Chapter 3 we prove that PTIME quantifiers are closed under iteration, cumulation and resumption. Next, we discuss the NP-completeness of branching quantifiers. Finally, we show that some Ramsey quantifiers define NP-complete classes of finite models while others stay in PTIME. We also give a sufficient condition for a Ramsey quantifier to be computable in polynomial time. -/- In Chapter 4 we investigate the computational complexity of polyadic lifts expressing various readings of reciprocal sentences with quantified antecedents. We show a dichotomy between these readings: the strong reciprocal reading can create NP-complete constructions, while the weak and the intermediate reciprocal readings do not. Additionally, we argue that this difference should be acknowledged in the Strong Meaning hypothesis. -/- In Chapter 5 we study the definability and complexity of the type-shifting approach to collective quantification in natural language. We show that under reasonable complexity assumptions it is not general enough to cover the semantics of all collective quantifiers in natural language. The type-shifting approach cannot lead outside second-order logic and arguably some collective quantifiers are not expressible in second-order logic. As a result, we argue that algebraic (many-sorted) formalisms dealing with collectivity are more plausible than the type-shifting approach. Moreover, we suggest that some collective quantifiers might not be realized in everyday language due to their high computational complexity. Additionally, we introduce the so-called second-order generalized quantifiers to the study of collective semantics. -/- In Chapter 6 we study the statement known as Hintikka's thesis: that the semantics of sentences like ``Most boys and most girls hate each other'' is not expressible by linear formulae and one needs to use branching quantification. We discuss possible readings of such sentences and come to the conclusion that they are expressible by linear formulae, as opposed to what Hintikka states. Next, we propose empirical evidence confirming our theoretical predictions that these sentences are sometimes interpreted by people as having the conjunctional reading. -/- In Chapter 7 we discuss a computational semantics for monadic quantifiers in natural language. We recall that it can be expressed in terms of finite-state and push-down automata. Then we present and criticize the neurological research building on this model. The discussion leads to a new experimental set-up which provides empirical evidence confirming the complexity predictions of the computational model. We show that the differences in reaction time needed for comprehension of sentences with monadic quantifiers are consistent with the complexity differences predicted by the model. -/- In Chapter 8 we discuss some general open questions and possible directions for future research, e.g., using different measures of complexity, involving game-theory and so on. -/- In general, our research explores, from different perspectives, the advantages of identifying meaning with algorithms and applying computational complexity analysis to semantic issues. It shows the fruitfulness of such an abstract computational approach for linguistics and cognitive science. (shrink)

We study the computational complexity of reciprocal sentences with quantified antecedents. We observe a computational dichotomy between different interpretations of reciprocity, and shed some light on the status of the so-called Strong Meaning Hypothesis.

We examine the verification of simple quantifiers in natural language from a computational model perspective. We refer to previous neuropsychological investigations of the same problem and suggest extending their experimental setting. Moreover, we give some direct empirical evidence linking computational complexity predictions with cognitive reality.
In the empirical study we compare time needed for understanding different types of quantifiers. We show that the computational distinction between quantifiers recognized by finite-automata and push-down automata is psychologically relevant. Our research improves upon hypothesis and (...) explanatory power of recent neuroimaging studies as well as provides
evidence. (shrink)

Szymanik (2007) suggested that the distinction between first-order and higher-order quantifiers does not coincide with the computational resources required to compute the meaning of quantifiers. Cognitive difficulty of quantifier processing might be better assessed on the basis of complexity of the minimal corresponding automata. For example, both logical and numerical quantifiers are first-order. However, computational devices recognizing logical quantifiers have a fixed number of states while the number of states in automata corresponding to numerical quantifiers grows with the rank of (...) the quantifier. This observation partially explains the differences in processing between those two types of quantifiers (Troiani et al. 2009) and links them to the computational model. Taking this perspective, below, we suggest the experimental setting extending the one by McMillan et al. (2005) and Troiani et al. (2009). (shrink)

We compare time needed for understanding different types of quantifiers. We show that the computational distinction between quantifiers recognized by finite-automata and pushdown automata is psychologically relevant. Our research improves upon hypothesis and explanatory power of recent neuroimaging studies as well as provides evidence for the claim that human linguistic abilities are constrained by computational complexity.

We consider collective quantification in natural language. For many years the common strategy in formalizing collective quantification has been to define the meanings of collective determiners, quantifying over collections, using certain type-shifting operations. These type-shifting operations, i.e., lifts, define the collective interpretations of determiners systematically from the standard meanings of quantifiers. All the lifts considered in the literature turn out to be definable in second-order logic. We argue that second-order definable quantifiers are probably not expressive enough to formalize all collective (...) quantification in natural language. (shrink)

We discuss Hintikka’s Thesis [Hintikka 1973] that there exist natural language sentences which require non–linear quantification to express their logical form.

The problem of computational complexity of semantics for some natural language constructions – considered in [M. Mostowski, D. Wojtyniak 2004] – motivates an interest in complexity of Ramsey quantifiers in finite models. In general a sentence with a Ramsey quantifier R of the following form Rx, yH(x, y) is interpreted as ∃A(A is big relatively to the universe ∧A2 ⊆ H). In the paper cited the problem of the complexity of the Hintikka sentence is reduced to the problem of computational (...) complexity of the Ramsey quantifier for which the phrase “A is big relatively to the universe” is interpreted as containing at least one representative of each equivalence class, for some given equvalence relation. In this work we consider quantifiers Rf, for which “A is big relatively to the universe” means “card(A) > f (n), where n is the size of the universe”. Following [Blass, Gurevich 1986] we call R mighty if Rx, yH(x, y) defines N P – complete class of finite models. Similarly we say that Rf is N P –hard if the corresponding class is N P –hard. We prove the following theorems. (shrink)

We discuss McMillan et al. (2005) paper devoted to study brain activity during comprehension of sentences with generalized quantifiers. According to the authors their results verify a particular computational model of natural language quantifier comprehension posited by several linguists and logicians (e. g. see van Benthem, 1986). We challenge this statement by invoking the computational difference between first-order quantifiers and divisibility quantifiers (e. g. see Mostowski, 1998). Moreover, we suggest other studies on quantifier comprehension, which can throw more light on (...) the role of working memory in processing quantifiers. (shrink)