The authors outline a theory of conditionals of the form If A then C and If A then possibly C. The 2 sorts of conditional have separate core meanings that refer to sets of possibilities. Knowledge, pragmatics, and semantics can modulate these meanings. Modulation can add information about temporal and other relations between antecedent and consequent. It can also prevent the construction of possibilities to yield 10 distinct sets of possibilities to which conditionals can refer. The mental representation of a conditional normally makes explicit only the possibilities in which its antecedent is true, yielding other possibilities implicitly. Reasoners tend to focus on the explicit possibilities. The theory predicts the major phenomena of understanding and reasoning with conditionals. You reason about conditional relations because much of your knowledge is conditional. If you get caught speeding, then you pay a fine. If you have an operation, then you need time to recuperate. If you have money in the bank, then you can cash a check. Conditional reasoning is a central part of thinking, yet people do not always reason correctly. The lawyer Jan Schlictmann in a celebrated trial (see Harr, 1995, pp. 361–362) elicited the following information from an expert witness about the source of a chemical pollutant trichloroethylene (TCE):

Thomassen proved that there is no degree of strong connectivity which guarantees a cycle through two given vertices in a digraph (Combinatorica 11 (1991) 393-395). In this paper we consider a large family of digraphs, including symmetric digraphs (i.e. digraphs obtained from undirected graphs by replacing each edge by a directed cycle of length two), semicomplete bipartite digraphs, locally semicomplete digraphs and all digraphs that can be obtained from acyclic digraphs and those mentioned above, by repeated substitutions of digraphs from one of these classes for vertices. We prove that for every natural number k, every k-strong digraph D from the family above is k-cyclic, i.e. for every set X of k vertices of D, there exists a cycle of D containing all the vertices of X . In particular, this implies that every k-strong quasi-transitive digraph is k-cyclic. We prove that if X is a set of vertices in a k-strong digraph D such that the maximum size of an independent set in the digrap...

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If people are presented with a conditional sentence and a corresponding categorical premise, they easily make modus ponens inferences. But several studies have shown that this valid inference of classical logic can be suppressed by adding another conditional sentence containing an additional necessary condition. Although most authors accounting for this suppression of inferences have stressed the importance of interpretative processes, it has not been investigated sufficiently how subjects really interpret such conditional sentences in practice. Thus, this dissertation examines people’s interpretations in interviews, utilising statements similar to those used in previous studies. The results indicate that there is a wider range of possible interpretations than assumed in accounts of the suppression effect in the literature, and that subjects do not always adopt the interpretations expected by experimenters. Moreover, they show that there is often no clear-cut distinction between interpretation and reasoning. This study investigates the influence of other factors like participants’ understanding of the task, character or the use of world knowledge. It also examines

The conditions for thinking a certain thought must be presupposed in the thinking. 1 Externalism about some mental property, M, is the thesis that whether a person (or other physical being) has M depends, not only on conditions inside the person’s skin, but also on the person’s environment and the way that the person is embedded in that environment. The dependence here is supposed to be conceptual rather than causal; it is the kind of dependence that can be revealed by philosophical theorising. This is an armchair methodology; so, if philosophical theorising yields knowledge, then it is a kind of armchair knowledge. Its status as knowledge does not depend on our conducting any detailed empirical investigation of the world around us. The puzzle for discussion in this paper arises when the possibility of armchair knowledge of an externalist dependence thesis about mental property M is put together with a thesis of first-person authority for that same mental property. For the purposes of generating our puzzle, we do not need to be concerned about a precise formulation of the notion of first-person authority. All that we need to suppose is that we each have a distinctively first-personal way of knowing that we ourselves have property M, when we do have it, without needing to conduct any detailed empirical investigation of the environment and our relation to it. This distinctively first-personal knowledge, self-knowledge, is another kind of armchair knowledge. The puzzle that arises from combining externalism and self-knowledge is clearly visible when we consider the epistemic status of the premises and conclusion of arguments of the following form (Ext): Ext(1) I am thinking that p. Ext(2) If I am thinking that p then E(me). Therefore: Ext(3) E(me). (‘E(x) ’ is some statement about x’s environmental embedding.) The occurrence of the particular mental verb ‘think ’ in Ext(1) is not vital; we could just as well have ‘I believe that p’. All that is vital is that the mental property of thinking or believing that p should meet two conditions. First, it should be a property that is subject to first-person authority. Second, it should be a property for which an externalist dependence thesis

In the Twentieth Century, Logic and Philosophy of Language are two of the few areas of philosophy in which philosophers made indisputable progress. For example, even now many of the foremost living ethicists present their theories as somewhat more explicit versions of the ideas of Kant, Mill, or Aristotle. In contrast, it would be patently absurd for a contemporary philosopher of language or logician to think of herself as working in the shadow of any figure who died before the Twentieth Century began. Advances in these disciplines make even the most unaccomplished of its practitioners vastly more sophisticated than Kant. There were previous periods in which the problems of language and logic were studied extensively (e.g. the medieval period). But from the perspective of the progress made in the last 120 years, previous work is at most a source of interesting data or occasional insight. All systematic theorizing about content that meets contemporary standards of rigor has been done subsequently. The advances Philosophy of Language has made in the Twentieth Century are of course the result of the remarkable progress made in logic. Few other philosophical disciplines gained as much from the developments in logic as the Philosophy of Language. In the

This report is to be used as the basis for the oral qualifying examination, which has to take place during the fourth semester of the 4+4 Ph.D. programme at Odense University. In this report I will state the results I have obtained so far. I will start by giving the results I have obtained concerning Semicomplete Multipartite Digraphs, which is the main research area I have been involved in. I will thereafter state results which I have obtained in other areas of graph theory. Finally I will describe future research plans for the last four semesters of my Ph.D. work.

The concept of conditional probability has been employed for hundreds of years. Thomas Bayes used the expression "the probability that [B] on the supposition that [A] " in the statement of a basic law (1763, p. 378). Frank Ramsey, developing the application of probability to uncertain epistemic attitudes, the "logic of partial belief (1926, p. 166), wrote of your "degree of belief in [B] given [A]": there are degrees of belief (degrees of closeness to certainty); and there are degrees of belief-under-a-supposition. Ramsey suggested that the latter illuminate conditional judgements—judgements expressed using "if: we add the antecedent hypothetically to our stock of beliefs, and assess the consequent on that basis (1929, p. 247). The idea is attractive. Will I recover if I have the operation? Trying to answer questions like that can leave you more or less uncertain. "On the supposition that", "given", and "if are prima facie interchangeable. Especially since Ernest Adams's work (1965), many (I think most) philosophers who have a working knowledge of probability theory, a sympathy for its application to uncertain epistemic attitudes, and an interest in the theory of conditionals, have recognized a promising connection here. 1 According to E. J. Lowe (1996) they are rather obviously mistaken: on its "standard definition" conditional probability throws no light on conditionals (pp. 604-6); the same is true on Lowe's preferred definition of conditional probability (pp. 608—9), for it presupposes an understanding of a conditional judgement (p. 609); and anyway, conditional probability is an incorrect measure of the degree to which a conditional judgement is acceptable (pp.

I argue that indicative conditionals are best viewed as having partial truth conditions: “If A, B ” is true if A and B are both true, false if A is true and B is false, and lacks truth value if A is false. The truth conditions are shown to explain a variety of important phenomena regarding indicative conditionals, including Adams ’ Thesis about the assertability conditions of conditionals, and how indicative conditionals embed in more complex constructions. In particular, the truth conditions are shown to provide the semantic basis for characterising several distinct logics of indicative conditionals, of which the logic of assertion is the main focus of the paper. 1

oured graphs and digraphs (see section 4), tournaments (see section 3) and some problems in extremal graph theory (see section 5). I have, together with J. Bang-Jensen and J. Huang, introduced the notion of round graphs (see section 7) and together with G. Gutin extended the most central theorems on tournaments to hypertournaments (see section 6). I have, together with D. Blokh and G. Gutin, also done work on networks (see (11)). Below I will describe some of my papers in more detail. In order to keep this survey short, I will not describe the results in (1,2,3,11,17,29,30). 2. Semicomplete multipartite digraphs The papers (4,5,7,13,15,20,21,22,24,27,28,29,31) address this topic. The problem of finding necessary and sufficient conditions for a semicomplete multipartite digraph (SMD) to be Hamiltonian, seems to be both very interesting and difficult. In [5] Bang-Jensen, Gutin and Huang proved a sufficient condition for a SMD to be Hamiltonian. In (4) I strengthen this