Hintikka’s Logical Revolution

Hintikka thinks that second-order logic is not pure logic, and because of Gödel’s incompleteness theorems, he suggests that we should liberate ourselves from the mistaken idea that first-order logic is the foundational logic of mathematics. With this background he introduces his independence friendly logic (IFL). In this paper, I argue that approaches taking Hintikka’s IFL as a foundational logic of mathematics face serious challenges. First, the quantifiers in Hintikka’s IFL are not distinguishable from Linström’s general quantifiers, which means that the quantifiers in IFL involve higher order entities. Second, if we take Wright’s interpretation of quantifiers or if we take Hale’s criterion for the identity of concepts, Quine’s thesis that second-order logic is set theory will be rejected. Third, Hintikka’s definition of truth itself cannot be expressed in the extension of language of IFL. Since second-order logic can do what IFL does, the significance of IFL for the foundations of mathematics is weakened.     

The Open-Endedness of the Set Concept and the Semantics of Set Theory

Some philosophers have argued that the open-endedness of the set concept has revisionary consequences for the semantics and logic of set theory. I consider (several variants of) an argument for this claim, premissed on the view that quantification in mathematics cannot outrun our conceptual abilities. The argument urges a non-standard semantics for set theory that allegedly sanctions a non-classical logic. I show that the views about quantification the argument relies on turn out to sanction a classical semantics and logic after all. More generally, this article constitutes a case study in whether the need to account for conceptual progress can ever motivate a revision of semantics or logic. I end by expressing skepticism about the prospects of a so-called non-proof-based justification for this kind of revisionism about set theory.     

Is there a phenomenological argument for higher-order representationalism?

In his 2009 article “Self-Representationalism and Phenomenology,” Uriah Kriegel argues for self-representationalism about phenomenal consciousness primarily on phenomenological grounds. Kriegel’s argument can naturally be cast more broadly as an argument for higher-order representationalism. I examine this broadened version of Kriegel’s argument in detail and show that it is unsuccessful for two reasons. First, Kriegel’s argument (in its strongest form) relies on an inference to the best explanation from the claim that all experiences of normal adult human beings are accompanied by peripheral awareness of those very experiences to the claim that all experiences are accompanied by peripheral awareness of those very experiences. This inference is inadequately defended, for the explanandum may also be given a straightforward evolutionary explanation. Second, contra Kriegel, I argue that phenomenological investigation does not support the thesis that we are always peripherally aware of our experiences. Instead, it delivers no verdict on this thesis. Kriegel’s phenomenological mistake may be explained via a highly diluted version of the famous transparency thesis about experience.     

Actuality, Necessity, and Logical Truth

The traditional view that all logical truths are metaphysically necessary has come under attack in recent years. The contrary claim is prominent in David Kaplan’s work on demonstratives, and Edward Zalta has argued that logical truths that are not necessary appear in modal languages supplemented only with some device for making reference to the actual world (and thus independently of whether demonstratives like ‘I’, ‘here’, and ‘now’ are present). If this latter claim can be sustained, it strikes close to the heart of the traditional view. I begin this paper by discussing and refuting Zalta’s argument in the context of a language for propositional modal logic with an actuality connective (section 1). This involves showing that his argument in favor of real world validity his preferred explication of logical truth, is fallacious. Next (section 2) I argue for an alternative explication of logical truth called general validity. Since the rule of necessitation preserves general validity, the argument of section 2 provides a reason for affirming the traditional view. Finally (section 3) I show that the intuitive idea behind the discredited notion of real world validity finds legitimate expression in an object language connective for deep necessity. Earlier versions of this paper were read at the universities of Graz, Maribor, and Salzburg, and at a workshop on the philosophy of logic at the National Autonomous University of Mexico (UNAM) in Mexico City. My thanks to those present at these events for many helpful suggestions. Thanks are also due to an anonymous referee for Philosophical Studies.     

Revisiting Quine on Truth by Convention

In “Truth by Convention” W.V. Quine gave an influential argument against logical conventionalism. Even today his argument is often taken to decisively refute logical conventionalism. Here I break Quine’s arguments into two—(i) the super-task argument and (ii) the regress argument—and argue that while these arguments together refute implausible explicit versions of conventionalism, they cannot be successfully mounted against a more plausible implicit version of conventionalism. Unlike some of his modern followers, Quine himself recognized this, but argued that implicit conventionalism was explanatorily idle. Against this I show that pace Quine’s claim that implicit conventionalism has no content beyond the claim that logic is firmly accepted, implicit rules of inference can be used to distinguish the firmly accepted from the conventional. As part of my case, I argue that positing syntactic rules of inference as part of our linguistic competence follows from the same methodology that leads contemporary linguists and cognitive scientists to posit rules of phonology, morphology, and grammar. The upshot of my discussion is a diagnosis of the fallacy in Quine’s master critique of logical conventionalism and a re-opening of possibilities for an attractive conventionalist theory of logic.     

A Logical Analysis of Monty Hall and Sleeping Beauty

Hintikka and Sandu’s independence-friendly (IF) logic is a conservative extension of first-order logic that allows one to consider semantic games with imperfect information. In the present article, we first show how several variants of the Monty Hall problem can be modeled as semantic games for IF sentences. In the process, we extend IF logic to include semantic games with chance moves and dub this extension stochastic IF logic. Finally, we use stochastic IF logic to analyze the Sleeping Beauty problem, leading to the conclusion that the thirders are correct while identifying the main error in the halfers’ argument.     

More on Putnam’s models: a reply to Belloti

In an earlier paper, I claimed that one version of Putnam's model-theoretic argument against realism turned on a subtle, but philosophically significant, mathematical mistake. Recently, Luca Bellotti has criticized my argument for this claim. This paper responds to Bellotti's criticisms.     

What's Wrong With Indispensability?

For many philosophers not automatically inclined to Platonism, the indispensability argument for the existence of mathematical objectshas provided the best (and perhaps only) evidence for mathematicalrealism. Recently, however, this argument has been subject to attack, most notably by Penelope Maddy (1992, 1997),on the grounds that its conclusions do not sit well with mathematical practice. I offer a diagnosis of what has gone wrong with the indispensability argument (I claim that mathematics is indispensable in the wrong way), and, taking my cue from Mark Colyvan's (1998) attempt to provide a Quinean account of unapplied mathematics as `recreational', suggest that, if one approaches the problem from a Quinean naturalist starting point, one must conclude that all mathematics is recreational in this way.     

Publicly Accessible Intuitions: "Neutral Reasons" and Bioethics

This article examines Leon Kass's contention that a choice forphysician-assisted suicide is "undignified." Although Kass isJewish rather than Christian, he argues for positions that mostChristians share, and he argues for these positions withoutpresupposing the truth of specific religious claims. I arguethat although Kass has some important intuitions, he too readilyassumes that these intuitions will be shared by his audience,and that this assumption diminishes the force of his argument.An examination of the limitations of Kass's argument is helpfulinsofar as it illustrates the real challenge faced by religiousbelievers who wish to defend their beliefs in the "public forum."For it illustrates that what needs to be made "accessible" isthe Judeo-Christian understanding of man and his place in theworld. While I do not wish to claim that this task is impossible,I do think that it is far more difficult than most realize.Like all important tasks, however, unless we wrestle with thedifficulties it raises, our arguments will strike many as unconvincing.     

Is the Splash Red?

Ball (2009) claims that without phenomenal concepts, the knowledge argument fails. In this article, I argue that Ball doesn’t succeed in proving his claim. The reason is that the Marianna case is not a case where the acquisition of the concept required for entertaining a phenomenal belief content Q alone is sufficient for Marianna, given enough physical information about her environment, to infer Q.     

ON NATURALIZING THE EPISTEMOLOGY OF MATHEMATICS

In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.     

On Quine’s Translation Argument

Quine's translation argumnent figures centrally in his views on logic. The goal of this paper is to get clear on that argument. It can be interpreted as an argument to the effect that one should never translate somebody’s speech as going against a law of the translator’s logic. Key to this reading of the translation argument is the premise that one should never translate somebody's speech such that their speech is unintelligible. Ultimately, it is my aim to reject this reading. I argue that only a weaker conclusion—one that says “not most of the time” instead of the stronger “never”—should be attributed to Quine. Accordingly, I propose and defend a weaker version of the first premise that better coheres with the weaker conclusion of the translation argument. Instead of the claim that one should never translate somebody’s speech such that their speech is unintelligible I argue that we should only ascribe to Quine the claim that one should not most of the time translate somebody’s speech in a way that makes it unintelligible. I go on to  sum up the results of my discussion and respond to a criticism of my reading.

     

Logical and Epistemic Foundationalism About Grounding: The Triviality of Facts and Principles

In this paper, I seek to undermine G.A. Cohen’s polemical use of a metaethical claim he makes in his article, ‘Facts and Principles’, by arguing that that use requires an unsustainable equivocation between epistemic and logical grounding. I begin by distinguishing three theses that Cohen has offered during the course of his critique of Rawls and contractualism more generally, the foundationalism about grounding thesis, the justice as non-regulative thesis, and the justice as all-encompassing thesis, and briefly argue that they are analytically independent of each other. I then offer an outline of the foundationalism about grounding thesis, characterising it, as Cohen does, as a demand of logic. That thesis claims that whenever a normative principle is dependent on a fact, it is so dependent in virtue of some other principle. I then argue that although this is true as a matter of logic, it, as Cohen admits, cannot be true of actual justifications, since logic cannot tell us anything about the truth as opposed to the validity of arguments. Facts about a justification cannot then be decisive for whether or not a given argument violates the foundationalism about grounding thesis. As long as, independently of actual justifications, theorists can point to plausible logically grounding principles, as I argue contractualists can, Cohen’s thesis lacks critical bite.     

Gödel And The Intuition Of Concepts

Gödel has argued that we can cultivate the intuition or ‘perception’ of abstractconcepts in mathematics and logic. Gödel's ideas about the intuition of conceptsare not incidental to his later philosophical thinking but are related to many otherthemes in his work, and especially to his reflections on the incompleteness theorems.I describe how some of Gödel's claims about the intuition of abstract concepts are related to other themes in his philosophy of mathematics. In most of this paper, however,I focus on a central question that has been raised in the literature on Gödel: what kind of account could be given of the intuition of abstract concepts? I sketch an answer to this question that uses some ideas of a philosopher to whom Gödel also turned in this connection: Edmund Husserl. The answer depends on how we understand the conscious directedness toward ‘objects’ and the meaning of the term ‘abstract’ in the context of a theory of the intentionality of cognition.     

A PEIRCEAN RESPONSE TO THE EVOLUTIONARY DEBUNKING OF MORAL KNOWLEDGE

The evolutionary debunking argument advanced by Sharon Street, Michael Ruse, and Richard Joyce employs the logic of Paul Griffiths and John Wilkins to contend that humans cannot have knowledge of moral truths, since the evolutionary process that has produced our basic moral intuitions lacks causal connections to those (putative) truths. Yet this argument is self‐defeating, because its aim is the categorical, normative claim that we should suspend our moral beliefs in light of the discoveries about their non‐truth‐tracking origins, when it is precisely this claim that relies upon the normativity under attack. This article cites Charles S. Peirce (1839–1914) to argue that such self‐defeat can be avoided by expanding upon the basic structure of the argument put forth by Griffiths and Wilkins, provided that one embraces a version of realism that corresponds with Peirce's doctrine of final causation. So construed, final causation reconciles real generals (including real moral values) with natural selection and undergirds further speculation of moral facts within values per se.     

Prior’s defence of Hintikka’s theorem. A discussion of Prior’s ‘The logic of obligation and the obligations of the logician’

In his paper, The logic of obligation and the obligations of the logician, A.N. Prior considers Hintikka??s theorem, according to which a statement cannot be both impossible and permissible. This theorem has been seen as problematic for the very idea of a logic of obligation. However, Prior rejects the view that the logic of obligation cannot be formalised. He sees this resistance against such a view as an important part of what could be called the obligation of the logician. Prior argues that Hintikka??s theorem should not be seen as something paradoxical. On the contrary, it should be seen as a fully acceptable consequence of a basic and reasonable assumption in deontic logic, namely Hintikka??s rule.     

How Mathematical Concepts Get Their Bodies

When the traditional distinction between a mathematical concept and a mathematical intuition is tested against examples taken from the real history of mathematics one can observe the following interesting phenomena. First, there are multiple examples where concepts and intuitions do not well fit together; some of these examples can be described as “poorly conceptualised intuitions” while some others can be described as “poorly intuited concepts”. Second, the historical development of mathematics involves two kinds of corresponding processes: poorly conceptualised intuitions are further conceptualised while poorly intuited concepts are further intuited. In this paper I study this latter process in mathematics during the twentieth century and, more specifically, show the role of set theory and category theory in this process. I use this material for defending the following claims: (1) mathematical intuitions are subject to historical development just like mathematical concepts; (2) mathematical intuitions continue to play their traditional role in today's mathematics and will plausibly do so in the foreseeable future. This second claim implies that the popular view, according to which modern mathematical concepts, unlike their more traditional predecessors, cannot be directly intuited, is not justified.     

Bocheński's Formalization of Summa Theologiae (Ia,75,6) Reconsidered

I investigate Bocheński's first-order logic formalization of the argument for the incorruptibility of the human soul given by Aquinas in Summa Theologiae (Ia,75,6). I suggest a slightly different axiomatization that reflect better Aquinas' informal argument. Along the way, I also fix a mistake in Bocheński's derivation that the human soul is not corruptible per se.     

EXTERNAL REASONS

Abstract: In this article I consider Bernard Williams's argument against the possibility of external reasons for action and his claim that the only reasons for action are therefore internal. Williams's argument appeals to David Hume's claim that reason is the slave of the passions, and to the idea that reasons are capable of motivating the agent who has them. I consider two responses to Williams's argument, by John McDowell and by Stephen Finlay. McDowell claims that even if Hume is right, there might nevertheless be external reasons. Finlay also claims that external reasons exist but, rejecting the connection between reasons and motivation, claims that they don't matter—that is, aren't motivationally significant for the agent whose reasons they are. Although I reject aspects of McDowell's and Finlay's arguments, I argue that external reasons do exist and in particular that any agent has an external reason to satisfy the preconditions of his or her agency.     

How to Be an Alethically Rational Naturalist

Alvin Plantinga has famously argued that naturalism is self-defeating. Plantinga's argument is, at its heart, an argument from analogy. Plantinga presents various epistemic situations and claims of each that (i) a person in such a situation has an undefeated defeater for each of his beliefs, and (ii) a reflective naturalist is in a relevantly similar situation. I present various epistemic situations and claim of each that a person in such a situation does not have an undefeated defeater for each of his beliefs. I further claim that at least some of these situations are more relevantly like the situation faced by the reflective naturalist than any of the situations Plantinga describes. Therefore, Plantinga's argument fails to establish that the reflective naturalist has an undefeated defeater for each of his beliefs and hence fails to establish that naturalism is self-defeating.