A ${\Pi }_1^0$ class can be defined as the set of infinite paths through a computable tree. For classes $P$ and $Q$, say that $P$ is Medvedev reducible to $Q$, $P{\le }_{M}Q$, if there is a computably continuous functional mapping $Q$ into $P$. Let ${ℒ}_M$ be the lattice of degrees formed by ${\Pi }_1^0$ subclasses of $2^{\omega }$ under the Medvedev reducibility. In "Non-branching degrees in the Medvedev lattice of ${\Pi }_1^0$ classes," I provided a characterization of nonbranching/branching and a classification of the nonbranching degrees. In this paper, I present a similar classification of the branching degrees. In particular, $P$ is separable if there is a clopen set $C$ such that $P\cap C\ne \varnothing \ne P\cap C^c$ and $P\cap C{\perp }_{M}P\cap C^c$. By the results in the first paper, separability is an invariant of a Medvedev degree and a degree is branching if and only if it contains a separable member. Further define $P$ to be hyperseparable if, for all such $C$, $P\cap C{\perp }_{M}P\cap C^c$ and totally separable if, for all $X,Y\in P$, $X{\perp }_{T}Y$. I will show that totally separable implies hyperseparable implies separable and that the reverse implications do not hold, that is, that these are three distinct types of branching degrees. Along the way I will show some related results and present a combinatorial framework for constructing ${\Pi }_1^0$ classes with priority arguments.

It is argued that the "inner" negation $\sim$ familiar from 3-valued logic can be interpreted as a form of "conditional" negation: $\sim A$ is read '$A$ is false if it has a truth value'. It is argued that this reading squares well with a particular 3-valued interpretation of a conditional that in the literature has been seen as a serious candidate for capturing the truth conditions of the natural language indicative conditional (e.g., "If Jim went to the party he had a good time"). It is shown that the logic induced by the semantics shares many familiar properties with classical negation, but is orthogonal to both intuitionistic and classical negation: it differs from both in validating the inference from $A\to \sim B$ to $\sim \left(A\to B\right)$.

We develop a logical system that captures two different interpretations of what extensive games model, and we apply this to a long-standing debate in game theory between those who defend the claim that common knowledge of rationality leads to backward induction or subgame perfect (Nash) equilibria and those who reject this claim. We show that a defense of the claim à la Aumann (1995) rests on a conception of extensive game playing as a one-shot event in combination with a principle of rationality that is incompatible with it, while a rejection of the claim à la Reny (1988) assumes a temporally extended, many-moment interpretation of extensive games in combination with implausible belief revision policies. In addition, the logical system provides an original inductive and implicit axiomatization of rationality in extensive games based on relations of dominance rather than the usual direct axiomatization of rationality as maximization of expected utility

From classical, Fraïissé-homogeneous, ($\le \omega$)-categorical theories over finite relational languages, we construct intuitionistic theories that are complete, prove negations of classical tautologies, and admit quantifier elimination. We also determine the intuitionistic universal fragments of these theories.

Humberstone asks whether every theorem of BCI provably implies $\phi \to \phi$ for some formula $\phi$. Meyer conjectures that the axiom $\mathbf{B}$ does not imply any such "self-implication." We prove a slightly stronger result, thereby confirming Meyer's conjecture.

We use Shelah's theory of possible cofinalities in order to solve some problems about ultrafilters. Theorem: Suppose that $\lambda$ is a singular cardinal, $\lambda \text{'}<\lambda$, and the ultrafilter $D$ is $\kappa$ -decomposable for all regular cardinals $\kappa$ with $\lambda \text{'}<\kappa <\lambda$. Then $D$ is either $\lambda$-decomposable or ${\lambda }^{+}$-decomposable. Corollary: If $\lambda$ is a singular cardinal, then an ultrafilter is ($\lambda$,$\lambda$)-regular if and only if it is either $\mathrm{cf}\hspace{0.17em}\lambda$-decomposable or ${\lambda }^{+}$-decomposable. We also give applications to topological spaces and to abstract logics.

We start with Fodor's critique of cognitive science in "The mind doesn't work that way: The scope and limits of computational psychology": he argues that much mental activity cannot be handled by the current methods of cognitive science because it is nonmonotonic and, therefore, is global in nature, is not context-free, and is thus not capable of being formalized by a Turing-like mental architecture. We look at the use of nonmonotonic logic in the artificial intelligence community, particularly with the discussion of the so-called frame problem. The mainstream approach to the frame problem is, we argue, probably susceptible to Fodor's critique; however, there is an alternative approach, due to McCain and Turner, which is, when suitably reformulated, not susceptible. In the course of our argument, we give a proof theory for the McCain-Turner system and show that it satisfies cut elimination. We have two substantive conclusions: first, that Fodor's argument depends on assumptions about logical form which not all nonmonotonic theories satisfy and, second, that metatheory plays an important role in the context of evolutionary accounts of rationality.