Online research in philosophy

ABSTRACT: Stewart Shapiro recently argued that there is no higher-order vagueness. More specifically, his thesis is: (ST) ‘So-called second-order vagueness in ‘F’ is nothing but first-order vagueness in the phrase ‘competent speaker of English’ or ‘competent user of “F”’. Shapiro bases (ST) on a description of the phenomenon of higher-order vagueness and two accounts of ‘borderline case’ and provides several arguments in its support. We present the phenomenon (as Shapiro describes it) and the accounts; then discuss Shapiro’s arguments, arguing that none is compelling. Lastly, we introduce the account of vagueness Shapiro would have obtained had he retained compositionality and show that it entails true higher-order vagueness.

Evidentialism is the thesis that a person is justified in believing a proposition iff the person's evidence on balance supports that proposition. In discussing epistemological issues associated with disagreements among epistemic peers, some philosophers have endorsed principles that seem to run contrary to evidentialism, specifying how one should revise one's beliefs in light of disagreement. In this paper, I examine the connection between evidentialism and these principles. I argue that the puzzles about disagreement provide no reason to abandon evidentialism and that there are no true general principles about justified responses to disagreement other than the general evidentialist principle. I then argue that the puzzles about disagreement are primarily puzzles about the evidential impact of higher-order evidence – evidence about the significance or existence of ordinary, or first-order, evidence. I conclude by arguing that such higher-order evidence can often have a profound effect on the justification of first-order beliefs.

Is higher-order vagueness a real phenomenon? Dominic Hyde (1994) claims that it is, and that it is part and parcel of vagueness itself. According to Hyde, any genuinely vague predicate must also be higher-order vague. His argument for this view is unsound, however. The purpose of this note is to expose the fallacy, and to make some related observations on the vague, the higher-order vague, and the vaguely vague.

Higher Order theories of consciousness have their fair share of sympathisers, but the arguments mustered in their support are—to my mind—unduly persuasive. My aim in this paper is to show that Higher Order theories cannot accommodate the possibility of misrepresentation without either falling into contradiction, or collapsing into a First-Order theory. If this diagnosis is on the right track, then Higher Order theories—at least in the specific versions here considered—fail to give an account of what they set out to explain: what is distinctive of ‘conscious’ phenomena.

Thus, despite the di culty of higher-order automated theorem proving, which has to deal with problems like the undecidability of higher-order uni - cation (HOU) and the need for primitive substitution, there are proof problems which lie beyond the capabilities of rst-order theorem provers, but instead can be solved easily by an higher-order theorem prover (HOATP) like Leo. This is due to the expressiveness of higher-order Logic and, in the special case of Leo, due to an appropriate handling of the extensionality principles (functional extensionality and extensionality on truth values).

It is generally supposed that borderline cases account for the tolerance of vague terms, yet cannot themselves be sharply bounded, leading to infinite levels of higher order vagueness. This higher order vagueness subverts any formal effort to make language precise. However, it is possible to show that tolerance must diminish at higher orders. The attempt to derive it from indiscriminability founders on a simple empirical test, and we learn instead that there is no limit to how small higher order tolerance may become. That means there is no limit to how precisely we may draw the boundaries of borderline cases, thus forestalling any requirement for higher order vagueness.

We investigate several approaches to resolution based automated theoremproving in classical higher-order logic (based on Church's simply typed-calculus) and discuss their requirements with respect to Henkincompleteness and full extensionality. In particular we focus on Andrews'higher-order resolution (Andrews 1971), Huet's constrained resolution (Huet1972), higher-order E-resolution, and extensional higher-order resolution(Benzmüller and Kohlhase 1997). With the help of examples we illustratethe parallels and differences of the extensionality treatment of these approachesand demonstrate that extensional higher-order resolution is the sole approach thatcan completely avoid additional extensionality axioms.

Why is it that philosophy seems unable to obtain the kinds of agreement regularly achieved by mathematics and the natural sciences? The experimental philosophy movement emphasizes conflicting intuitions as a potential source of philosophical disagreement. This essay draws attention to another, complementary source: the logical imperfection of natural languages. Unlike logic as it is formalized in symbolic notation, the rules governing the correct use of terms in a natural language can be indeterminate, underdetermined, and inconsistent. Though most philosophers recognize the logical imperfection of natural languages in the abstract, everyday philosophical discussion is often conducted as though the argumentative moves agreed upon as legitimate, applied to the raw materials of the discussion, should lead toward agreement. Yet the logical imperfection of natural languages makes it possible for disputants to agree upon natural language premises, adhere to the usual, uncontroversial rules for drawing inferences from those premises, and yet arrive at conflicting conclusions. The essay illustrates this claim through an analogy based on Daniel Dennett’s warning to prospective philosophy graduate students in “Higher-order Truths About Chmess.”.

Consider this chess puzzle. White to checkmate in two. It appeared recently in the Boston Globe, and what startled me about it was that I had thought it had been proven that you can’t checkmate with a lone knight (and a king, of course). This is a counterexample, a strange circumstance that can arise in a legal game of chess. This fact is a higher-order truth of chess, namely that the “proof” that you can never checkmate with a lone knight and king is unsound. Now let’s consider chmess (I made up the term); it’s the game that you get by allowing the king to move two spaces, not one, in any direction. I made it up and I don’t know if anybody has ever played it. I have no idea whether it’s worth playing. Probably it isn’t, but it doesn’t matter; it’s not even worth our attention long enough to figure out. But a moment’s reflection reveals that there are exactly as many higher-order a priori truths of chmess as there are of chess, namely, an infinity of them. And no doubt they would be roughly as difficult to discover and to prove as the higherorder truths of chess. There are people who make a living working out the truths of chess and certainly it’s been a big avocation for many other people. But I doubt if anybody yet has spent more than 5 minutes trying to work out the a priori truths, and the higher-order truths, of chmess.