This talk will be an introduction to the ultraproduct construction and the model theoretic notion of saturation, which are two of the themes in Maryanthe Malliaris’s Annual Logic Lecture next week. My goal is to introduce these concepts with some examples and motivation to give anyone interested some familiarity with the key characters in Maryanthe’s story before she arrives. Maryanthe will define these concepts in her talk and will not presuppose any material from my talk.
The framework of strict-tolerant consequence championed by Cobreros, Egré, Ripley and van Rooij provides a novel setting that permits one to have a transparent truth predicate without abandoning classical logic. The semantics for this notion of consequence, due to van Rooij, employs the three-valued strong Kleene matrices. A second framework that has received renewed attention is the collection of weak Kleene matrices, which have frequently appeared in the context of so-called logics of nonsense, making the pairing of these topics a natural avenue for investigation. In this talk, I’ll discuss strict-tolerant consequence on the weak Kleene matrices, its corresponding proof theory, and its interpretation. I’ll also discuss how the resulting notion of consequence bears on several matters in philosophical logic, including the content-theoretic interpretation of bounds consequence, the semantic properties of paradoxical sentences in the Principia Mathematica, and debates concerning the logical analysis of category mistakes.
When speakers utter conflicting moral sentences (“X is wrong”/“X is not wrong”), it seems clear that they disagree. It has often been suggested that the fact that the speakers disagree gives us evidence for a claim about the semantics of the sentences they are uttering. Specifically, it has been suggested that the existence of the disagreement gives us reason to infer that there must be an incompatibility between the contents of these sentences (i.e., that it has to be the case that at least one of them is incorrect). This inference then plays a key role in a now-standard argument against certain theories in moral semantics. In this paper, we introduce new evidence that bears on this debate. We show that there are moral conflict cases in which people are inclined to say both (a) that the two speakers disagree and (b) that it is not the case at least one of them must be saying something incorrect. We then explore how we might understand such disagreements. As a proof of concept, we sketch an account of the concept of disagreement and an independently motivated theory of moral semantics which, together, explain the possibility of such cases.
The recent literature maintains that the behavior of modal expressions motivates a non-truth-conditional account of their meaning, and non-classical account of their underlying logic. The key aspect of interpretation of modal claims is the characteristic dynamic effect they have on the context, and the corresponding dynamic notion of validity captures their seemingly non-classical behavior. While prima facie supported by the puzzling behavior of modals in discourse, I argue that this approach is empirically inadequate. Instead I develop and argue for a dynamic theory of context-change which assigns standard truth-conditional meaning to modal utterances, and a corresponding dynamic notion of validity which preserves classical logic.
Logical pluralism is commonly described as the view that there is more than one correct logic. It has been claimed that, in order for that view to be interesting, there has to be at least a potential for rivalry between the correct logics (e.g., in Field 2009, Priest 2006, Read 2006, Russell 2008). In this talk, I explore how the relevant notions of rivalry and correctness could be combined when relying on a semantic conception of rivalry in terms of disagreement. I first give a brief intuitive characterization of the sort of rivalry in question before reviewing some standard proposals on how to capture it. I argue that none of those proposals aligns well with pluralism. More recent proposals (Shapiro 2014) to adopt the semantic framework used in the debate on context-dependence and disagreement in the philosophy of language seem to do a better job, but ultimately, it remains doubtful whether the semantics of disagreement is able to capture a notion of rivalry suitable for pluralists.
A historical survey of mathematical practice in support of a pragmatic inductive philosophy of mathematics.
I argue for a distinction between eventive and evidential speech reports. In eventive speech reports the at-issue contribution is the introduction of a speech event with certain properties. Typical examples include direct and free indirect speech. In evidential speech reports, by contrast, the fact that something was said is not at issue, but serves to provide evidence for the reported content. Typical examples include Quechua reportative evidential morphology, Dutch reportative modals, or German reportive subjunctive. Following up on an observation by Von Stechow & Zimmermann (2005:fn.16), I argue that English indirect discourse is ambiguous. In the current framework this means it allows both an eventive reading, where a reported speech act is at issue, and an evidential reading, where it is backgrounded.
The semantics of degree constructions has motivated the implementation of a MAX operator, a function from a set of degrees to its maximal member (von Stechow 1985, Rullmann 1995, a.o.). This operator is unsatisfying: it’s arbitrary (cf. MIN), and therefore not explanatory. There have thus been several calls to reduce MAX to a more pragmatic principle of maximal informativity (Dayal 1996, Beck & Rullmann 1999, Fox & Hackl 2007, von Fintel et al. 2014).
Intriguing differences between before and after have caused some to posit an EARLIEST operator in the temporal domain (Beaver & Condoravdi 2003, Condoravdi 2010). This operator is unsatisfying for similar reasons (cf. LATEST), and some have suggested it, too, can be redefined in terms of informativity (Rett 2015). However, recent cross-linguistic evidence (reported here) complicates the reduction of EARLIEST to `maximize informativity’: while counterparts of `before’ and `after’ across languages share many foundational semantic properties, they appear to differ in a principled way in how certain before constructions are interpreted. I discuss this and other related observations with respect to the future of a domain-general `maximize informativity’ program.
The following is an example of a counterfactual conditional in English:
(1) If I had thrown a six, I would have won the game.
One normally infers from (1) that the antecedent is counterfactual (i.e. I did not throw a six; I write this as CF-p), and that the consequent is counterfactual (i.e. I did not win the game; written CF-q). Whereas most previous literature focuses on the counterfactuality of the antecedent exclusively (perhaps assuming that an analysis of CF-p extends to CF-q), this work provides an analysis for how the counterfactual inference of the consequent (CF-q) is generated, and explains its empirical distribution.
I identify several contexts in which the CF-q inference gets cancelled. In some of these, cancellation is the result of the presence of a specific lexical item (such as “also”). In other cases, it is the intonation contour of the conditional that leads to cancellation. By analyzing the topic-focus structure of conditionals, I argue that the various contexts in which CF-q gets cancelled have a pragmatic property in common: they are multiple cause contexts. This means that they make more than one cause for the same consequent salient.
The next step in my analysis is adopting an idea going back to Karttunen (1971), who suggests thatconditional perfection (the pragmatic strengthening of conditionals to biconditionals) is a necessary ingredient for the CF-q inference to arise. The key prediction, which has not been explored before, is that if for some reason conditional perfection is not triggered, the CF-q inference is not drawn. I derive the independent result that multiple cause contexts do not trigger conditional perfection. This provides the desired explanation of why in multiple cause contexts the CF-q inference is not drawn. This analysis opens a new way to investigate counterfactuality, namely by using tools from the study of discourse (questions and answers, topic and focus, exhaustivity). Finally, I sketch some directions of future work on how using causal networks to represent multiple causation can be applied to the pragmatics of counterfactual conditionals.
One traditional role of a logic of a logic of entailment is as a set of closure principles for theories. Looking at logics in this way, and as theories themselves, can be very interesting. A logic determines a closure operator (in Tarski’s sense) on sets of formulas. The theories generated by a closure operator themselves (sometimes) determine closure operators. Looking at the space of theories generated by a “master theory”, and the interaction of the closure operators that they determine, I motivate a variety of different logical systems.