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 Semantics of Wisdom in the Philosophy of Tang Junyi: Between Transformative Knowledge and Transcendental Reflexivity

In this article, I offer a provisional analysis of the philosophical semantics of “wisdom” in the thought of the New Confucian thinker Tang Junyi. I begin by providing some pointers concerning the concept of wisdom in general and situating the discourse on wisdom in comparative philosophy in the context of the later Foucault’s and Pierre Hadot’s historical investigations into ancient Graeco-Roman philosophy as a mode of spiritual self-cultivation and self-transformation. In the remainder of the paper, I try to describe and think through what Foucault identifies as a “Cartesian moment,” in which self-knowledge becomes the ultimate precondition for the ethico-spiritual project of “caring for the self,” in Tang’s approach of wisdom. In the course of my argument, I outline the complex relation between his vision of a renewed Confucian mode of religious practice on the one hand and his philosophical presuppositions concerning the transcendental status of subjectivity and the reflexivity of consciousness on the other.     

Hintikka’s Independence-Friendly Logic Meets Nelson’s Realizability

Inspired by Hintikka’s ideas on constructivism, we are going to ‘effectivize’ the game-theoretic semantics (abbreviated GTS) for independence-friendly first-order logic (IF-FOL), but in a somewhat different way than he did in the monograph ‘The Principles of Mathematics Revisited’. First we show that Nelson’s realizability interpretation—which extends the famous Kleene’s realizability interpretation by adding ‘strong negation’—restricted to the implication-free first-order formulas can be viewed as an effective version of GTS for FOL. Then we propose a realizability interpretation for IF-FOL, inspired by the so-called ‘trump semantics’ which was discovered by Hodges, and show that this trump realizability interpretation can be viewed as an effective version of GTS for IF-FOL. Finally we prove that the trump realizability interpretation for IF-FOL appropriately generalises Nelson’s restricted realizability interpretation for the implication-free first-order formulas.     

Accidentally True Beliefs and the Williamsonian Mental State of Knowing

In this paper, I will explore some philosophical implications of Williamson’s thesis that knowing is a state of mind (KSM). Using the fake barn case, I will introduce a way to evaluate Williamson’s KSM thesis and determine whether the Williamsonian mental state of knowing can be plausibly distinguished from certain other similar but epistemologically distinctive states of mind (i.e., accidentally true beliefs). Then, some tentative externalist accounts of the supposed differences between the Williamsonian mental state of knowing and accidentally true beliefs will be critically assessed, implying that the evaluated traditional versions of externalism in semantics and epistemology do not fit well with Williamson’s KSM thesis. Ultimately, I suggest that the extended-mind or extended-knower approach may be more promising, which indicates that active externalism would be called for by Williamson’s KSM thesis.     

Propositions as games as types

Without violating the spirit of Game-Theoretical semantics, its results can be re-worked in Martin-Löf's Constructive Type Theory by interpreting games as types of Myself's winning strategies. The philosophical ideas behind Game-Theoretical Semantics in fact highly recommend restricting strategies to effective ones, which is the only controversial step in our interpretation. What is gained, then, is a direct connection between linguistic semantics and computer programming.The idea of re-working the results of Game-Theoretical Semantics in Martin-Löf's Type Theory dates back to a seminar on constructive logic led by Jan von Plato in the Department of Philosophy, University of Helsinki, since Spring 1986. I have gained a lot from discussions in the seminar and personally with Jan von Plato. The essential content of this paper has also been presented in the Departments of Mathematics and Philosophy, University of Stockholm, in seminars led by Per Martin-Löf and Dag Prawitz, respectively, and in this case also I have enjoyed personal conversation with the seminar leaders. Other persons I wish to thank are Jaakko Hintikka and Göran Sundholm.     

From IF to BI

We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges. We show how Hodges’ semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O’Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural rôle, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural rôle.     

HOW TO ANALYZE IMMEDIATE EXPERIENCE:

Abstract: This article discusses Jaakko Hintikka's interpretation of the aims and method of Husserl's phenomenology. I argue that Hintikka misrepresents Husserl's phenomenology on certain crucial points. More specifically, Hintikka misconstrues Husserl's notion of “immediate experience” and consequently fails to grasp the functions of the central methodological tools known as the “epoché” and the “phenomenological reduction.” The result is that the conception of phenomenology he attributes to Husserl is very far from realizing the philosophical potential of Husserl's position. Hence if we want a fruitful rapprochement between analytical philosophy and Continental phenomenology of the kind that is Hintikka's ultimate aim, then Hintikka's account of Husserl needs correcting on a number of crucial points.     

The Agentive Modalities

A number of philosophical projects require a proper understanding of the modal aspects of agency, or of what I call ‘the agentive modalities.’ I propose a general account of the agentive modalities, one which takes as its primitive the decision‐theoretic notion of an option. I relate this account to the standard semantics for ‘can’ and to the viability of some positions in the free will debates.     

Revising Carnap’s Semantic Conception of Modality

I provide a tableau system and completeness proof for a revised version of Carnap??s semantics for quantified modal logic. For Carnap, a sentence is possible if it is true in some first order model. However, in a similar fashion to second order logic, no sound and complete proof theory can be provided for this semantics. This factor contributed to the ultimate disappearance of Carnapian modal logic from contemporary philosophical discussion. The proof theory I discuss comes close to Carnap??s semantic vision and provides an interesting counterpoint to mainstream approaches to modal logic. Despite its historical origins, my intention is to demonstrate that this approach to modal logic is worthy of contemporary attention and that current debate is the poorer for its absence.     

Compositional Semantics and Normative ‘Ought’

According to the paradigm view in linguistics and philosophical semantics, it is lexical semantics (LS) plus the principle of compositionality (PC) that allows us to compute the meaning of an arbitrary sentence. The job of LS is to assign meaning to individual expressions, whereas PC says how to combine these individual meanings into larger ones. In this paper I argue that the pair LS?+?PC fails to account for the discourse-relevant meaning of normative ‘ought’. If my hypothesis is tenable, then the failure of LS?+?CS extends to normative language in general. The reason I offer that this is so is that semantics for normative language is, in an important respect, a substantive semantics (SS). The ‘substantive’ in question means that the meaning of normative vocabulary in use is driven by metanormative views associated with a particular normative concept. SS rejects the model LS?+?CS and replaces it with a discourse-relevant semantics built around an interactional principle that ascribes to a particular surface syntactical form of ‘ought’ sentences a logical form that represents its discourse-salient normative content. In the paper I shall sketch how SS works and why it is worth serious consideration.

     

Game theoretical semantics and entailment

The essence of the meaning of a declarative sentence is given by stating its truth conditions, and consequently semantics, the study of meaning, must include a theory of truth conditions. Such a theory must not only describe accurately the truth conditions of declarative sentences, it must also answer the question of when two sentences have the same truth conditions. The fundamental semantic relation of having the same truth conditions cannot be ignored by any reasonable theory.This paper is an attempt to find a partial account of this relation by using game theoretical semantics as developed by Hintikka and his followers. The account given will establish a connection between this approach to semantics and the theory of firstdegree entailment formulated by Anderson and Belnap.     

Adding a Conditional to Kripke’s Theory of Truth

Kripke’s theory of truth (Kripke, The Journal of Philosophy72(19), 690–716; 1975) has been very successful but shows well-known expressive difficulties; recently, Field has proposed to overcome them by adding a new conditional connective to it. In Field’s theories, desirable conditional and truth-theoretic principles are validated that Kripke’s theory does not yield. Some authors, however, are dissatisfied with certain aspects of Field’s theories, in particular the high complexity. I analyze Field’s models and pin down some reasons for discontent with them, focusing on the meaning of the new conditional and on the status of the principles so successfully recovered. Subsequently, I develop a semantics that improves on Kripke’s theory following Field’s program of adding a conditional to it, using some inductive constructions that include Kripke’s one and feature a strong evaluation for conditionals. The new theory overcomes several problems of Kripke’s one and, although weaker than Field’s proposals, it avoids the difficulties that affect them; at the same time, the new theory turns out to be quite simple. Moreover, the new construction can be used to model various conceptions of what a conditional connective is, in ways that are precluded to both Kripke’s and Field’s theories.     

The Versatility of Universality in Principia Mathematica

In this article, I examine the ramified-type theory set out in the first edition of Russell and Whitehead's Principia Mathematica. My starting point is the ‘no loss of generality’ problem: Russell, in the Introduction (Russell, B. and Whitehead, A. N. 1910. Principia Mathematica, Volume I, 1st ed., Cambridge: Cambridge University Press, pp. 53–54), says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, hence a problem. The purpose of this article is to clarify Russell's claim and to solve the ‘no loss of generality’ problem. I first remark that the hierarchy of propositional functions calls for a fine-grained conception of ramified types as propositional forms (‘ramif-types’). Then, comparing different important interpretations of Principia’s theory of types, I consider the question as to whether Principia allows for non-predicative propositional functions and variables thereof. I explain how the distinction between the formal system of the theory, on the one hand, and its realizations in different epistemic universes, on the other hand, makes it possible to give us a more satisfactory answer to that question than those given by previous commentators, and, as a consequence, to solve the ‘no loss of generality’ problem. The solution consists in a substitutional semantics for non-predicative variables and non-predicative complex terms, based on an epistemic understanding of the order component of ramified types. The rest of the article then develops that epistemic understanding, adding an original epistemic model theory to the formal system of types. This shows that the universality sought by Russell for logic does not preclude semantical considerations, contrary to what van Heijenoort and Hintikka have claimed.     

Counterfactual epistemic scenarios

In two-dimensional semantics in the tradition of Davies and Humberstone, whether a singular term receives an epistemically shifted reading in the scope of a modal operator depends on whether the world considered as actual is shifted. This means that epistemically shifted readings should be available only in environments where an explicit contrast between the actual world and some counterfactual worlds cannot be made. In this paper, I argue that this is incorrect. Whether a singular term receives an epistemically shifted reading is independent of whether the world treated as actual is shifted. This, I argue, undermines the two-dimensionalist account of epistemic shift. I then turn to the question how a positive view should handle these two phenomena separately. I argue for treating singular terms with a version of counterpart theory in which the difference between epistemically shifted and other readings is determined in the context of utterance.     

Some Notes on Boolos’ Semantics: Genesis,Ontological Quests and Model-Theoretic Equivalence to Standard Semantics

The main aim of this work is to evaluate whether Boolos’ semantics for second-order languages is model-theoretically equivalent to standard model-theoretic semantics. Such an equivalence result is, actually, directly proved in the “Appendix”. I argue that Boolos’ intent in developing such a semantics is not to avoid set-theoretic notions in favor of pluralities. It is, rather, to prevent that predicates, in the sense of functions, refer to classes of classes. Boolos’ formal semantics differs from a semantics of pluralities for Boolos’ plural reading of second-order quantifiers, for the notion of plurality is much more general, not only of that set, but also of class. In fact, by showing that a plurality is equivalent to sub-sets of a power set, the notion of plurality comes to suffer a loss of generality. Despite of this equivalence result, I maintain that Boolos’ formal semantics does not committ (directly) second-order languages (theories) to second-order entities (and to set theory), contrary to standard semantics. Further, such an equivalence result provides a rationale for many criticisms to Boolos’ formal semantics, in particular those by Resnik and Parsons against its alleged ontological innocence and on its Platonistic presupposition. The key set-theoretic notion involved in the equivalence proof is that of many-valued function. But, first, I will provide a clarification of the philosophical context and theoretical grounds of the genesis of Boolos’ formal semantics.     

A logic for epistemic two-dimensional semantics

Epistemic two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. While this theory is usually presented in an informal manner, I take some steps in formalizing it in this paper. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that captures the relevant ideas of epistemic two-dimensional semantics. I also describe some properties of the logic that are interesting from a philosophical perspective, and apply it to the so-called nesting problem.     

Meredith, Prior, and the History of Possible Worlds Semantics

This paper charts some early history of the possible worlds semantics for modal logic, starting with the pioneering work of Prior and Meredith. The contributions of Geach, Hintikka, Kanger, Kripke, Montague, and Smiley are also discussed.