Showing papers in "Analysis in 2008"

Superplurals in English

Abstract: It is now widely believed among philosophers and logicians that ordinary English contains plural terms that may refer to several things at once. But are there terms that stand to ordinary plural terms the way ordinary plural terms stand to singular terms? Let's call such terms superplural. A superplural term would thus, loosely speaking, refer to several “pluralities” at once. It is reasonably straightforward to devise a formal logic of superplural terms, superplural predicates, and even superplural quantifiers (Rayo 2006). But does this formal logic reflect any features of natural languages? In particular, does ordinary English contain superplural terms and predicates? The purpose of this note is to address these questions. We examine some earlier arguments for the existence of superplural expressions in English, and find these arguments to be either inconclusive or not sufficiently far-reaching. Then we present some better examples of the superplural in ordinary English.

Conversational implicature and the cancellability test

Abstract: Matthew Weiner has recently argued in this journal that not all conversa tional implicatures are explicitly cancellable and he has inferred from this that Grice's familiar 'cancellability test does not help determine when an implicature is present' (2006: 129). Bearing in mind that the cancellability test has traditionally been considered the most reliable and effective crite rion for distinguishing conversational implicatures from other linguistic phenomena - such as conventional implicature, semantic entailment and semantic presupposition - Weiner's conclusion is of great importance for much of contemporary linguistics and philosophy: as a brief look at the literature demonstrates, Grice's cancellability test has been brought to bear not only in linguistics and the philosophy of language but also in areas as diverse as ethics, epistemology, metaphysics and the philosophy of mind.1 Thus, considering the pervasiveness of the cancellability test in philosophi cal debates and its resultant importance for philosophical methodology, Weiner's arguments surely deserve a more thorough examination than they have received thus far. Let us begin the discussion of Weiner's views by taking a closer look at Grice's cancellability test. Here is a quotation from Grice: [A] putative conversational implicature that p is explicitly cancellable if, to the form of words the utterance of which putatively implicates that p, it is admissible to add but not p, or J do not mean to imply that p, and it is contextually cancellable if one can find situations in which the utterance of the form of words would simply not carry the implicature. (Grice 1975: 44)

Against composition as identity

Abstract: Some philosophers, such as Peter van Inwagen (1994), have claimed that the view that composition is identity is incoherent. Van Inwagen cites the apparent ungrammaticality of sentences like ‘the six plots are the farmer’s field’ as evidence for his view. Perhaps van Inwagen is right, but I needn’t settle this question here. I will argue against the view that composition is identity, whatever that view amounts to, in the following way. First, I will elucidate a principle called ‘the Plural Duplication Principle’ [PDP]. Any acceptable way of making sense of the slogan that composition is identity— i.e., any way that properly conforms to the intuitions that lead one to utter this slogan— must validate PDP. Second, I argue that PDP is false. So any acceptable way of making sense of the slogan that composition is identity is false. The slogan that composition is identity will be refuted prior to being properly formulated. Following David Lewis (1986: 59-63), let us say that x and y are duplicates just in case there is a 1-1 correspondence between their parts that preserves perfectly natural properties and relations. Suppose that A is identical with B. Then any duplicate of A must also be a duplicate of B. This follows via Leibniz’s Law: if some duplicate of A were not

‘Ought’ implies ‘can’ and the derivation of the Principle of Alternate Possibilities

Abstract: Fine, K. 1975. Vagueness, truth and logic. Synthese 30: 265-300. Fodor, J. and E. Lepore. 1996. What cannot be evaluated cannot be evaluated, and it cannot be supervalued either. Journal of Philosophy 93: 516-35. Kamp, H. 1975. Two theories about adjectives. In Formal Semantics of Natural Lan guage, ed. E. L. Keenan, 123-55. Cambridge: Cambridge University Press. Keefe, R. 2000a. Supervaluationism and validity. Philosophical Topics 28: 93-105. Keefe, R. 2000b. Theories of Vagueness. Cambridge: Cambridge University Press. Lewis, D. 1975. Language and languages. In Minnesota Studies in the Philosophy of Science vol. VII, ed. K. Gunderson, 3-35. Minneapolis: University of Minnesota Press. Morreau, M. 1999. Supervaluation can leave truth-value gaps after all. Journal of Philosophy 96: 148-56. Varzi, A. 2007. Supervaluationism and its logics. Mind 116: 633-76. Williamson, T. 1994. Vagueness. London: Routledge.

Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach

Abstract: We consider a hypersurface in Rn parametrized by a diffeomorphism φo of the unit sphere in Rn into Rn , and we take a point w in the domain I[φo] enclosed by the image of φo, and we consider the ‘hole’ I[w + ξ] enclosed by the image of the hypersurface w + ξ , where ξ is a diffeomorphism as φo with 0 ∈ I[ξ] and is a small positive real parameter. Then we consider the Dirichlet problem for the Laplace equation in the perforated domain I[φo] with the hole I[w + ξ] removed and show real analytic continuation properties of the solution u and of the corresponding energy integral as functionals of the sextuple of w, , ξ , φo, and of the Dirichlet data in the interior and exterior boundaries of the perforated domain, which we think of as a point in an appropriate Banach space, around a degenerate sextuple with = 0.

How to solve the problem of mixed conjunctions

Abstract: Ayer, A. J. 1946. Language, Truth and Logic. London: Gollancz. Eklund, M. 2007. Fictionalism. In The Stanford Encyclopedia of Philosophy. Geach P. 1958. Imperative and deontic logic. Analysis 18: 49-56. Geach, P. 1960. Ascriptivism. The Philosophical Review 69: 221-25. Geach, P. 1965. Assertion. The Philosophical Review 74: 449-65. Kalderon, M. E. 2005. Moral Fictionalism. Oxford: Oxford University Press. Rawls, J. 1971. Theory of Justice. Cambridge Mass.: Belknap Press. Ross, D. 1939. Foundations of Ethics. Oxford: Oxford University Press. Scanlon, T. 1998. What We Owe to Each Other. Cambridge Mass.: Belknap Press. Searle, J. 1962. Meaning and speech acts. The Philosophical Review 71: 423-32. Searle, J. 1969. Speech Acts, An Essay in the Philosophy of Language. Cambridge: Cambridge University Press. Stevenson, C. 1944. Ethics and Language. New Haven: Yale University Press.

Time scale embedding theorem and coercivity of quadratic functionals

Abstract: V tomto clanku studujeme vztah mezi koercivitou a pozitivitou kvadratickeho funkcionalu J na casove skale (time scale), který může být druhou variaci nelinearniho variacniho problemu (P) na casových skalach. Pro připad obecných smisených okrajových podminek ukazujeme, že funkcional J je koercivni, pravě když je pozitivni a plati přislusna zesilena Legendreova podminka modifikovana pro casove skaly. Abychom toto dokazali, odvodili jsme větu o vnořeni (větu o spojite zavislosti na pocatecnich podminkach a parametrech) pro dynamicke rovnice na casových skalach a tuto větu jsme aplikovali na Riccatiho rovnici přislusejici kvadratickemu funkcionalu J. Nasledně jsme pak obdrželi postacujici podminky optimality pro nelinearni problem (P), ktere jsou formulovane pomoci pozitivity funkcionalu J nebo pomoci Riccatiho rovnice na casove skale. Tento výsledek je nový dokonce pro specialni připad spojiteho casu pro obecne okrajove podminky.

Profiling interest relativity

Abstract: Boolos, G. 1999. To be is to be a value of a variable (or some values of some variables). In Logic Logic, and Logic, ed. R. Jeffrey, 54–72. Cambridge, Mass.: Harvard University Press. Dummett, M. 1991. Frege: Philosophy of Mathematics. Cambridge, Mass.: Harvard University Press. Glanzberg, M. 2004. Quantification and realism. Philosophy and Phenomenological Research, 69: 541–72. Glanzberg, M. 2006. Context and unrestricted quantification. In Absolute Generality, eds. A. Rayo and G. Uzquiano, 45–74. Oxford: Oxford University Press. Linnebo, O. 2006. Sets, properties, and unrestricted quantification. In Absolute Generality, ed. A. Rayo and G. Uzquiano, 149–78. Oxford: Oxford University Press. Parsons, C. 2006. The problem of absolute generality. In Absolute Generality, ed. A. Rayo and G. Uzquiano, 203–19. Putnam, H. 1980. Models and reality. The Journal of Symbolic Logic, 45: 464–82. Rayo, A. 2003. When does ‘everything’ mean everything? Analysis, 63: 100–106. Williamson, T. 2003. Everything. Philosophical Perspectives 17: 415–65.

The Dirichlet problem for graphs of prescribed anisotropic mean curvature in ℝn+1

Abstract: We consider the Dirichlet problem for graphs of prescribed mean curvature in ℝn+1 where the prescribed mean curvature function H=H(X,N) may depend on the point X in space and on the normal N of the graph as well. In some special cases this Dirichlet problem arises as the Euler equation of a generalised nonparametric area functional.

On a side effect of solving Fitch's paradox by typing knowledge

Abstract: a sufficiently robust metaphysics of propositions to justify accepting (1) while rejecting (3). More subtle is to reject the classical picture of truth and modality which is used in establishing (5) and (6). If you take that line you need to appreciate just how much else you are rejecting. If you accept the argument you must either reject (1) or accept (2). One reason for the intuition behind (1) might be that the denial of a proposition ought to be equivalent to the affirmation of some other proposition and that the content of what is denied determines uniquely the content of what would have to be affirmed.3 If you reject (1) you will need to explain why that intuition is misguided. The other response is to accept (2) as, for instance, Stalnaker (2003: 36) does. Philosophers worry about (2) because they think that it trivializes logical inference or logical knowledge. The problem of accounting for such knowledge is indeed a vexing one,4 but what the present note is intended to show is that there are hidden costs in supposing that nothing more than a simple rejection of (2) is all it takes to address it.

Definite descriptions are ambiguous

Abstract: 1. The problem Keith Donnellan (1966) contrasted two uses of definite descriptions, the referential and the attributive. In using a definite description referentially the speaker communicates content about a particular object in mind, whereas in using the same description attributively the speaker communicates content about whatever object uniquely satisfies the description. Assuming that definite descriptions have a quantificational attributive meaning, the main problem raised by Donnellan’s contrast between uses is whether descriptions also have a referential meaning. If they do, it is plausible to think that the definite article is ambiguous between a referential and an attributive meaning. In what follows, I will call this thesis `Ambiguity’. For ease of exposition, I will take it as a thesis about English. The most influential arguments against Ambiguity invoke the independently motivated Gricean distinction between what a speaker means and what he or she says (Grice 1989). According to these arguments, we do not need to postulate a referential meaning for definite descriptions to account for referential uses. We can account for such uses in terms of what a speaker means but does not literally say. Thus, in using a definite description referentially the speaker means or communicates content about a particular object in mind, but what the speaker literally says is determined by the description’s quantificational attributive meaning. We are then told that on the grounds of parsimony this account of referential uses is superior to Ambiguity (see Grice 1969; Kripke 1977; Bach 1981; Neale 1990). I think all such arguments against Ambiguity have been seriously weakened in the face of the following points: (a) definite descriptions are regularly used referentially, and this shows that there is no prima facie reason to deny that referential uses are literal; actually, this usage regularity strongly suggests that referential uses are literal, just as literal as attributive uses (Devitt 1997; Reimer 1998; compare Neale 2004); (b) complex demonstratives and referentially used definite descriptions are used similarly in a wide variety of situations, and this strongly suggests that both expressions have a similar referential meaning (Devitt 2004) (e.g. `That/the concert last night was great, wasn’t it?’); (c) referentially used `incomplete’ descriptions may be used to express truths even when speakers cannot provide completions for them, and this also strongly suggests that descriptions have a referential meaning (e.g. `the tall kid who used to sit in the front row in first grade was born in Rio de Janeiro’) (Wilson 1991; Devitt 2004; compare Wettstein 1981; Schiffer 2005). Thus, not only do I think that the Gricean arguments above do not succeed in undermining Ambiguity. I also think there is a strong case for Ambiguity.

Indexicals, speech acts and pornography

Abstract: In the last twenty years, recorded messages and written notes have become a significant test and an intriguing puzzle for the semantics of indexical expressions (see Smith 1989, Predelli 1996, 1998a, 1998b, 2002, Corazza et al. 2002, Romdenh-Romluc 2002). In particular, the intention-based approach proposed by Stefano Predelli has proven to bear interesting relations to several major questions in philosophy of language. In a recent paper (Saul 2006), Jennifer Saul draws on the literature on indexicals and recorded messages in order to criticize Rae Langton's claim that works of pornography can be understood as illocutionary acts – in particular acts of subordinating women or acts of silencing women. Saul argues that it does not make sense to understand works of pornography as speech acts, because only utterances in contexts can be speech acts. More precisely, works of pornography such as a film may be seen as recordings that can be used in many different contexts – exactly like a written note or an answering machine message. According to Saul, bringing contexts into the picture undermines Langton's radical thesis – which must be reformulated in much weaker terms. In this paper, I accept Saul's claim that only utterances in contexts can be speech acts, and that therefore only works of pornography in contexts may be seen as illocutionary acts of silencing women. I will, nonetheless, show that Saul's reformulation doesn't undermine Langton's thesis. To this aim, I will use the distinction Predelli proposes in order to account for the semantic behaviour of indexical expressions in recorded messages – namely the distinction between context of utterance and context of interpretation.

Entire functions sharing a polynomial with their derivatives and normal families

Abstract: Let ƒ be a nonconstant entire function and Q be a nonconstant polynomial. Let M be a positive number and k ≥ 2 be an integer. We prove that if ƒ and ƒ' share Q CM and |ƒ(k) (z)| ≤ M · (1 + |Q(z)|) whenever ƒ(z) = Q(z), then is constant. Furthermore, we prove the corresponding normality criterion.

A simple solution to the hardest logic puzzle ever

Abstract: 1. The simplest solution to the ‘hard’ puzzle The puzzle. Three gods A, B, and C are called, in some order, ‘True’, ‘False’, and ‘Random’. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for ‘yes’ and ‘no’ are ‘da’ and ‘ja’, in some order. You do not know which word means which. 1

How to talk about unobservables

Abstract: In this journal, Dicken & Lipton [2006] argued, following Musgrave [1985], that a constructive empiricist cannot coherently draw the distinction between observable objects (events, processes, …) and unobservable ones. We argue to the contrary: the distinction can be drawn coherently, but add a qualification to deal with an associated problem concerning the language of science.

Reply to Brueckner

Abstract: Anthony Brueckner (2008) offers a compelling argument for incompati bilism the New Third Argument (here after 'NTA') in response to my criticisms of the Third Argument (Campbell 2007).11 explicate NTA (?2), offer some preliminary considerations (?3), and then close with a discus sion of Brueckner's critique of my Adam example (2007: 109) (?4). In the end I show that, like the Third Argument, NTA does not establish incom patibilism, the view that no one has free will if determinism is true.

Remark on uniqueness of weak solutions to the Navier–Stokes equations

Abstract: Consider the Navier–Stokes equations with the initial data a ∈ L2σ(Rd). Let u and v be two weak solutions with the same initial value a. If u satisfies the usual energy inequality and if where X1(Rd) is the multiplier space (see the definition in the text), then we have u = v.

On rules of inference and the meanings of logical constants

Abstract: In the theory of meaning, it is common to contrast truth-conditional theories of meaning with theories which identify the meaning of an expression with its use. One rather exact version of the somewhat vague use-theoretic picture is the view that the standard rules of inference determine the meanings of logical constants. Often this idea also functions as a paradigm for more general use-theoretic approaches to meaning. In particular, the idea plays a key role in the anti-realist program of Dummett and his followers. In the theory of truth, a key distinction now is made between substantial theories and minimalist or deflationist views. According to the former, truth is a genuine substantial property of the truth-bearers, whereas according to the latter, truth does not have any deeper essence, but all that can be said about truth is contained in T-sentences (sentences having the form: ‘P’ is true if and only if P). There is no necessary analytic connection between the above theories of meaning and truth, but they have nevertheless some connections. Realists often favour some kind of truth-conditional theory of meaning and a substantial theory of truth (in particular, the correspondence theory). Minimalists and deflationists on truth characteristically advocate the use theory of meaning (e.g. Horwich). Semantical anti-realism (e.g. Dummett, Prawitz) forms an interesting middle case: its starting point is the use theory of meaning, but it usually accepts a substantial view on truth, namely that truth is to be equated with verifiability or warranted assertability. When truth is so understood, it is also possible to accept the idea that meaning is closely related to truth-conditions, and hence the conflict between use theories and truth-conditional theories in a sense disappears in this view.

The Simulation Argument again

Abstract: Nick Bostrom has argued that given some plausible assumptions, we should believe that we are not humans but rather conscious computer simulations of humans (Bostrom 2003). I will offer a new way of thinking about Bostrom's Simulation Argument. Bostrom envisages post-humans who create conscious minds on com puters. The post-humans use their vast computing power to implement programming that ensures that 'the computational processes of a human brain are structurally replicated in suitably fine-grained detail', in a way that 'would suffice for the generation of subjective experiences' (Bostrom 2003: 244). Brian Weatherson calls these non-human, computer-generated minds Sims (Weatherson 2003). Sims have experiences which represent the Sims to be normal, embodied humans living in a Sim-free 21st century world. The post-humans thus run ancestor simulations within the artifi cially created, computer-instantiated minds of the Sims. Here is how Weatherson sketches the Simulation Argument: since (1) 'the percentage of human-like agents that are Sims' (i.e. the number of Sims divided by the number of Sims plus humans) is 'far above 50 per cent', and since (2) 'we don't have any specific evidence that tells on whether we are a Sim or a human', we should believe that we are Sims, not humans (Weatherson 2003: 425). Says Bostrom: 'One's credence in the hypothesis that one is in a simulation should be close to unity', given the way in which the human like Sims vastly outnumber normal humans (Bostrom 2003: 249). It is worth noting that one reason why Bostrom thinks that the number of Sims will vastly outstrip the number of humans is that Sims 'will run their own ancestor-simulations on powerful computers they build in their simu lated universe' (Bostrom 2003: 253). The computers in question will be 'virtual machines' which can be 'stacked': 'it is possible to simulate one machine simulating another machine, and so on, in arbitrarily many steps of iteration' (Bostrom 2003: 253). However, since a Sim cannot really build a computer he merely seems to build one within his simulation it follows that he cannot really create another human-like Sim mind that is instanti ated in the programming of a real computer that he really builds. The Sims cannot really create other Sims. The idea that 'stacked virtual machines' can give rise to more and more conscious Sims seems to be just a confusion.1

Smilansky's alleged refutation of compatibilism

Abstract: Saul Smilanky aims to refute the traditional compatibilist view that ‘determinism does not really change anything, morally’ (2007: 348). His argument runs as follows. Assume determinism plus complete predictability. Then ‘the compatibilist does not have a strong principled objection’ (348) to prepunishing someone who we now know will commit a crime in a week, since they will commit the crime of their own free will, and hence will be morally responsible for it, and hence liable to blame and punishment. There is no need to wait in case they change their mind, since we already know that they will not. But the common-sense view is that prepunishment ‘runs counter to the idea of respect for persons’ (348). So compatibilism is not, in fact, compatible with our ordinary intuitions about punishment. And this means that compatibilism must be false, since it is false that ‘determinism does not really change anything, morally’ (348): it makes a big difference, with respect to the permissibility of prepunishment, whether or not determinism is true. Smilansky’s argument fails on more than one front. Firstly, it is ‘complete predictability’ that is causing all the trouble, and not determinism itself. Imagine that we have libertarian free will, but that we also have access to affordable and reliable time travel. Then we could travel to the future, watch Smith commit a crime, and then come back to the present and prepunish the soon-to-be-guilty party. The mere fact that Smith has not yet committed the crime, and could (in a libertarian sense of ‘could’), at the last moment, decide to desist, makes no difference. For of course we know that Smith will not desist. So there is no need to wait in case he changes his mind. One might be tempted to object that the libertarian time-travel scenario is too far-fetched. But in fact it is not that much more far-fetched than Smilansky’s determinism-plus-perfect-predictability scenario. Perhaps time travel is physically impossible – and it is certainly unlikely to be within the budget of any police force any time soon. But perfect predictability is also well-nigh impossible, even assuming determinism, at least if there are any chaotic systems in the vicinity – and, again, it is certainly unlikely to be within the budget of any police force any time soon. For Smilansky’s argument to succeed, he would thus have to show that foreknowledge is possible assuming determinism but impossible assuming indeterminism – and in some sense of ‘possible’ that is somehow relevant to the question about prepunishment. Since in both cases foreknowledge is

Asymptotic analysis of generalized Hermite polynomials

Abstract: We analyze the polynomials $H_{n}^{r}(x)$ considered by Gould and Hopper, which generalize the classical Hermite polynomials. We present the main properties of $H_{n}^{r}(x)$ and derive asymptotic approximations for large values of $n$ from their differential-difference equation, using a discrete ray method. We give numerical examples showing the accuracy of our formulas.

Compatibilism can resist prepunishment: a reply to Smilansky

Abstract: Bernstein, A. R. and F. Wattenberg. 1969. Non-standard measure theory. In Appli cations of Model Theory of Algebra, Analysis, and Probability, ed. W. A. J. Luxemburg. New York: Holt, Rinehard and Winston. Lewis, D. 1980. A subjectivist's guide to objective probability. In Ifs: Conditionals, Belief, Decision, Chance and Time, ed. W. L. Harper, R. Stalnaker and G. Pearce, 1981: 267-97. Boston: Reidel. Williamson, T. 2007. How probable is an infinite sequence of heads? Analysis 67: 173-80.

Ancestral arithmetic and Isaacson's Thesis

Abstract: First-order Peano Arithmetic (PA) is incomplete. So the question naturally arises: what kinds of sentences belonging to PA's language LA can we actually establish to be true even though they are unprovable in PA? There are two familiar classes of cases. First, there are sentences like the canonical Godel sentence for PA. Second, there are sentences like the arithmetization of Good- stein's Theorem. In the first sort of case, we can come to appreciate the truth of the Godelian undecidable sentences by reflecting on PA's consistency or by coming to accept the instances of the 1 reflection schema for PA. And those routes involve deploying ideas beyond those involved in accepting PA as true. To reason to the truth of the Godel sentence, we need not just to be able to do basic arithmetic, but to be able to reflect on our practice. In the second sort of case, we come to appreciate the truth of the sentences which are undecidable in PA by deploying transfinite induction or other infinitary ideas. So the reasoning again involves ideas which go beyond what's involved in grasping basic arithmetic. Thinking about these sorts of cases suggests a plausible general conjecture. Given the arguments of Daniel Isaacson (1987, 1992), let's call it

More prepunishment for compatibilists: a reply to Beebee

Abstract: plausible that the common-sense view or at least a common-sense view is that such legislation is legitimate because it prevents the relevant inten tions being carried out: that is, because it prevents a crime (murder or a terrorist act) from being committed. If that is right, then these really are cases of prepunishment, as Smilansky defines it. The requirement that there be actual intentions and preparations can be seen merely as a reflection of the fact that these are the routes by which prosecutors come to have reasonable beliefs about what will happen if they do not intervene. (I do not say that they should be so seen; only that such a characterization of this kind of legislation has a perfectly good claim to being a common-sense view.) Smilansky's argument thus fails on two counts. It fails to show that the permissibility of prepunishment runs counter to our ordinary intuitions; and even if it succeeded on this front, it would fail to demonstrate the falsity of compatibilism, because it would still fail to show that determin ism makes a difference to anything, morally speaking.

Why contingent facts cannot necessities make

Abstract: Halbach, V., H. Leitgeb, and P. Welch. 2003. Possible worlds semantics for modal notions conceived as predicates. Journal of Philosophical Logic 32: 179–223. Horsten, L. 1998. A Kripkean approach to unknowability and truth. Notre Dame Journal of Formal Logic 39: 389–405. Horsten, L. and H. Leitgeb. 2001. No future. Journal of Philosophical Logic 30: 259–65. Kaplan, D. and R. Montague. 1960. A paradox regained. Notre Dame Journal of Formal Logic 1: 79–90. Repr. in Montague 1974: 271–85. Koons, R. 1992. Paradoxes of Belief and Strategic Rationality. Cambridge: Cambridge University Press. Linsky, B. 2007. Logical types in arguments about knowability and belief. In New Essays on the Knowability Paradox, ed. J. Salerno. Oxford: Oxford University Press. McGee, V. 1991. Truth, Vagueness, and Paradox: An Essay on the Logic of Truth. Indianapolis and Cambridge: Hackett Publishing. Montague, R. 1963. Syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability. Acta Philosophica Fennica 16: 153–67. Repr. in his Formal Philosophy: Selected Papers of Richard Montague. New Haven and London: Yale University Press. Edited and with an introduction by Richmond H. Thomason, 1974, 286–302. Paseau, A. 2008. Fitch’s argument and typing knowledge. Notre Dame Journal of Formal Logic 49: to appear. Williamson, T. 2000. Knowledge and its Limits. Oxford: Oxford University Press.