Masahiro Yamada

This is material I have used for teaching introductory level formal logic at Pomona College. The coverage is fairly standard--propositional calculus and basic first-order logic--but includes a little bit of more advanced material that are often omitted, in particular proof of soundness and completeness of propositional calculus. The proof system is a natural deduction style sequent calculus that Gerhard Gentzen presents in his "Die Wiederspruchsfreiheit der reinen Zahlentheorie" (1936). Exercise material and a proof checker are available online.

This paper considers fair betting odds for certain bets that might be placed in the situation discussed in the so-called Sleeping Beauty Problem. This paper examines what Thirders, Halfers, and Double Halfers must say about the odds as determined by various decision theoretic approaches and argues that Thirders and Halfers have difficulties formulating plausible and coherent positions concerning the relevant betting odds. Double Halfers do not face this problem and that is an important consideration in favor of Double Halfers.

There are three main points of the paper. 1. There are straightforward ways of manipulating expected gains and losses that result in a divergence between fair betting odds and credence. Such manipulations are familiar from tools of finance. One can easily see that the Sleeping Beauty case is structured in such a way as to result in a divergence between fair betting odds and credence. 2. The inspection of credences and betting odds in certain betting situations shows that the two main extant positions, Elga's and Lewis's, are both mistaken. 3. My proposal may seem to require a revision of probability theory but it does not. The Sleeping Beauty case merely calls attention to a constraint on partitioning of possibility space that is usually satisfied as a matter of course but can in fact be violated.

Some arguments beg the question. Question-begging arguments are bad arguments and cannot increase the level of justification one has for the conclusion. Question-begging arguments, unlike some other bad arguments, need not suffer the problem of having unjustified premises. Even if the premises are justified and even if the premises entail the conclusion, a question-begging argument fails to have any force when it comes to increasing one's justification for the conclusion. For example, many regard Moore's famous response to skepticism as a question-begging argument: 1. I have hands . . . via perception 2. If I have hands, then I am not a handless brain in a vat 3. I am not a handless brain in a vat . . . from 1 and 2, modus ponensWhat about this piece of reasoning makes many feel uneasy? Nonskeptics will agree that Moore is justified in believing . They will also agree that he is justified in believing . In fact, under normal circumstances there will be no dispute that Moore knows the premises. The premises entail the conclusion and Moore knows this, too. In other words, the argument is sound, and is known to be sound. Yet many feel the argument begs the question. Why?

One prominent feature of belief is that a belief cannot be formed at will. This paper argues that the best explanation of this fact is that belief formation is a process that takes aim at the truth. Taking aim at the truth is to be understood as causal responsiveness of the processes constituting belief formation to what facilitates achieving true beliefs. The requirement for this responsiveness precludes the possibility of belief formation responding to intentions in a way that would count as forming a belief at will