This talk will introduce Hoare logic, one of the most (if the the most) influential works in program verification. I will draw parallels between propositional and predicate logic and present a rudimentary way (the way it was done in 1960s) for verifying programs.
The idea of a conclusion following formally from a set of premises is central to our conception of logic today: logical consequence is often taken for the subject matter of logic, and ‘formal consequence’ and ‘logical consequence’ are routinely taken as synonyms. It would come as a surprise, then, to know that consequences generally and formal consequence in particular did not always hold this place of prominence: the first treatises specifically devoted to the study of consequences did not appear until the beginning of the fourteenth century – over 1600 years after the death of Aristotle; and the notion of formal consequence didn’t begin to take a shape resembling its modern successor until nearly a half century after these treatises appeared.
Prior to the later nineteenth century, the main developments of the concept of formal consequence as we know it occur in the following stages:
All but the first and last of these developments occur in a period of intense logical activity stretching from about 1285 to 1341, just over half a century. This talk provides an overview of the developments of this period. I begin with an outline of the topical tradition on which earlier treatises on consequences depended. Next, I detail the account of John Buridan, whose distinction between formal and material consequences is more familiar than others, both because it has been better researched and because of its basic similarity to modern model-theoretic accounts. From here, I move backward to the previous developments Buridan’s account relies on and engages with, particularly those of William of Ockham and Walter Burley.
This talk provides a formal pragmatic analysis of (im)precision which accounts for its essential properties, but also for Lewis’s (1979) observation of asymmetry in how standards of precision may shift in a given discourse: Only up, not down. I propose that shifts of the kind observed and discussed by Lewis are in fact cases of underlying disagreement about the standard of precision, which is only revealed when one interlocutor uses an expression which signals their adherence to a higher standard than the one adhered to be the other interlocutor(s). I show that a modest formal pragmatic analysis along the lines of game-theoretic approaches by Franke (2009), Jaeger (2012) and others can easily capture the natural asymmetry in standard-signaling that gives rise to Lewis’s observation, so long as such an account is enriched with a notion of relevance. If there’s time, I’ll also discuss how this approach can be helpful for understanding a longstanding puzzle in the semantics of counterfactuals, namely the issue of Sobel Sequences and their reversals.
The talk will be about a question that is generally taken to be settled: are there different sizes of infinity? That is, are there infinite sets of different sizes? This is one of the most natural questions that one can ask about the infinite. But it is of course generally taken to be settled by results, such as Cantor’s theorem, to the effect that there are infinite sets without bijections between them. These results (which I of course do not dispute) answer the question given an almost universally accepted principle relating size to the existence of functions. The principle is: for any sets A and B, if A is the same size as B, then there is a bijection from A to B. The aim of the talk, however, is to argue that the question is in fact wide open: to argue that we are not in a position to know the answer, because we are not in a position to know this principle. I will do this by looking at what seem to be the strongest reasons for thinking that we are in a position to know the principle, and arguing against them.
There are several options for logical pluralism on the table. Rudolf Carnap puts forward a position in which the logical connectives never mean the same thing in distinct logics, and JC Beall & Greg Restall put forward a position in which they always do. However, neither of them are capable of accounting for all of our intuitions. In particular, neither Carnap nor Beall & Restall can make sense of the fact that it seems there are some contexts in which distinct logics seem to have the same logical terminology, and some context in which distinct logics have distinct logical terminology. In this paper, I will present a view (in line with that of Stewart Shapiro’s) which can account for both of these intuitions.
In this talk, I will frame a number of definitions of algorithmic randomness as instances of what I refer to as the logical approach to randomness. In order to better understand this logical approach, I will contrast it with one of the standard approaches to defining randomness in classical mathematics, which I call the valuative approach to randomness. I will focus specifically on two potential problems faced by the logical approach that threaten to trivialize this approach to defining randomness. I will address these two potential problems, arguing that the logical approach fills an important role, namely, that of supplementing the valuative approach by yielding additional information about classically random objects, which is unearthed when we bring the tools of mathematical logic to bear on the study of randomness.
Detachment via Modus Ponens (factual detachment) faces well-known problems for conditionals with deontic modals. One class of conditionals where the validity of factual detachment has been contested are those conditioning on an agent’s preferences, known in the linguistic literature as anankastic conditionals and in the philosophical literature as hypothetical imperatives. This talk presents a semantics for such conditionals which validates detachment and then examines why detachment appears to fail when it does.
We claim that an endorsement component enters the interpretation of the modal in the conclusion, but, crucially, not that of the conditional premise. In the problematic cases, this endorsement component is at odds with what can be reasonably assumed about normal speakers, hence the unease assenting to the conclusion even when assenting to both premises. We argue that the source of the endorsement component is pragmatic, and that, therefore, these cases do not provide a reason to adopt a semantics for this type of conditional that invalidates Modus Ponens.
Orthodox approaches to formal semantics for modal operator languages, following Kripke and Kaplan, supplement the familiar compositional semantics embodied in the recursive definition of truth-at-a-world by a “postsemantics” that consists in identifying truth simpliciter with truth at the actual world. I will propose an alternative, Tarski-inspired approach that makes do without any appeal to a postsemantics. This Tarskian approach, I will argue, is superior to the orthodoxy on both empirical and methodological grounds, and has some remarkable philosophical consequences.
Any semantics for modals and conditionals must prove its worth by resolving some standard problem cases. I start with a solution to Frank-Zvolensky conditionals. Consider the following:
(1) If John speeds, he should speed.
(2) If the Dalai Lama is mad, then he should be mad.
These conditionals should not be predicted as theorems of any natural language semantics for conditional constructions. A solution to this problem extends the Kratzerian restrictor analysis with a consistent modal base expansion and contraction operation. Following this, some past-shifted conditional constructions will be considered. Past-shifted conditionals, such as the Morgenbesser conditional:
(3) If you had bet heads, you would have won.
Are shown to be resolvable by connecting the modal base, in a Kratzerian framework, and the Stalnakerian common ground. This connection is shown to generate truth conditional differences exactly where they should be predicted. Along the way, the standard Kratzerian ordering source will be generalized to accommodate an arbitrary number of independent ordering sources, factoring in the possibility of inconsistencies.