The Role of Axioms in Mathematics

To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from a wide variety of philosophical positions. Once the axioms are generally accepted, mathematicians can expend their energies on proving theorems instead of arguing philosophy. Given this account of the role of axioms, I give four criteria that axioms must meet in order to be accepted. Penelope Maddy has proposed a similar view in Naturalism in Mathematics, but she suggests that the philosophical questions bracketed by adopting the axioms can in fact be ignored forever. I contend that these philosophical arguments are in fact important, and should ideally be resolved at some point, but I concede that their resolution is unlikely to affect the ordinary practice of mathematics. However, they may have effects in the margins of mathematics, including with regards to the controversial “large cardinal axioms” Maddy would like to support.

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Bounded Revision: Two-Dimensional Belief Change Between Conservative and Moderate Revision

This paper presents the model of ‘bounded revision’ that is based on two-dimensional revision functions taking as arguments pairs consisting of an input sentence and a reference sentence. The key idea is that the input sentence is accepted as far as (and just a little further than) the reference sentence is ‘cotenable’ with it. Bounded revision satisfies the AGM axioms as well as the Same Beliefs Condition (SBC) saying that the set of beliefs accepted after the revision does not depend on the reference sentence (although the posterior belief state does depend on it). Bounded revision satisfies the Darwiche–Pearl (DP) axioms for iterated belief change. If the reference sentence is fixed to be a tautology or a contradiction, two well-known one-dimensional revision operations result. Bounded revision thus naturally fills the space between conservative revision (also known as natural revision) and moderate revision (also known as lexicographic revision). I compare this approach to the two-dimensional model of ‘revision by comparison’ investigated by Fermé and Rott (Artif Intell 157:5–47, 2004) that satisfies neither the SBC nor the DP axioms. I conclude that two-dimensional revision operations add substantially to the expressive power of qualitative approaches that do not make use of numbers as measures of degrees of belief.     

Leon Trotsky’s Contribution to the Marxist Theory of History

Trotsky’s contribution to historical materialism has been subject to two broadly defined critical assessments. Detractors have tended to dismiss his interpretation of Marxism as a form of productive force determinism, while admirers have tended to defend his Marxism as a voluntarist negation of the same. In this essay I argue that both of these opinions share an equally caricatured interpretation of Second International Marxism against which Trotsky is compared. By contrast, I argue that Trotsky’s Marxism can best be understood as a powerful application and deepening of the strongest elements of Second International methodology to a novel set of problems. Thus, against Trotsky’s admirers, I locate his Marxism as both emerging out of, in addition to breaking with, Second International Marxism; while, against his critics, I argue that it was precisely the strengths of this earlier interpretation of Marxism that informed Trotsky’s powerful contributions to historical materialism: his concept of combined and uneven development and his discussion of the role of individual agents within the Marxist interpretation of history.     

Logic with numbers

Many people regard utility theory as the only rigorous foundation for subjective probability, and even de Finetti thought the betting approach supplemented by Dutch Book arguments only good as an approximation to a utility-theoretic account. I think that there are good reasons to doubt this judgment, and I propose an alternative, in which the probability axioms are consistency constraints on distributions of fair betting quotients. The idea itself is hardly new: it is in de Finetti and also Ramsey. What is new is that it is shown that probabilistic consistency and consequence can be defined in a way formally analogous to the way these notions are defined in deductive (propositional) logic. The result is a free-standing logic which does not pretend to be a theory of rationality and is therefore immune to, among other charges, that of “logical omniscience”.     

A repair of Frege’s theory of thoughts

Frege’s writings contain arguments for the thesis (i) that a thought expressed by a sentence S is a structured object whose composition pictures the composition of S, and for the thesis (ii) that a thought is an unstructured object. I will argue that Frege’s reasons for both (i) and (ii) are strong. Frege’s explanation of the difference in sense between logically equivalent sentences rests on assumption (i), while Frege’s claim that the same thought can be decomposed differently makes (ii) plausible. Thoughts are supposed to do work that requires that they be structured and work that requires that they be unstructured. But this cannot be! While the standard response to this problem is to reject either (i) or (ii), I propose a charitable repair in the spirit of Frege’s theory that accepts both. The key idea can be found in Frege’s Basic Laws of Arithmetic(BL, GGA). Frege argues that the thought expressed by a sentence is determined by the truth-conditions that can be derived from the semantic axioms for the sentence constituents. The fact that the same axiomatic truth-condition can be derived in different ways from different semantic axioms suggests a Fregean solution of the dilemma: A thought is a type that is instantiated by all sequences of senses (decomposed thoughts) that have the same axiomatic truth-conditions. This allows for multiple decomposability of the same thought (for different decomposed thoughts can have the same axiomatic truth-conditions) and for a notion of containment (the decomposed thought contains those senses whose semantic axioms are needed in the derivation of the truth-conditions). My proposal combines the virtues of (i) and (ii) without inheriting their vices.     

A Note on Comparative Probability

A possible event always seems to be more probable than an impossible event. Although this constraint, usually alluded to as regularity, is prima facie very attractive, it cannot hold for standard probabilities. Moreover, in a recent paper Timothy Williamson has challenged even the idea that regularity can be integrated into a comparative conception of probability by showing that the standard comparative axioms conflict with certain cases if regularity is assumed. In this note, we suggest that there is a natural weakening of the standard comparative axioms. It is shown that these axioms are consistent both with the regularity condition and with the essential feature of Williamson’s example.     

On the Possibility of Inference to the Best Explanation

Various proposals have suggested that an adequate explanatory theory should reduce the number or the cardinality of the set of logically independent claims that need be accepted in order to entail a body of data. A (and perhaps the only) well-formed proposal of this kind is William Kneale’s: an explanatory theory should be finitely axiomatizable but it’s set of logical consequences in the data language should not be finitely axiomatizable. Craig and Vaught showed that Kneale theories (almost) always exist for any recursively enumerable but not finitely axiomatizable set of data sentences in a first order language with identity. Kneale’s criterion underdetermines explanation even given all possible data in the data language; gratuitous axioms may be “tacked on.” Define a Kneale theory, T, to be logically minimal if it is deducible from every Kneale theory (in the vocabulary of T) that entails the same statements in the data language as does T. If they exist, minimal Kneale theories are candidates for best explanations: they are “bold” in a sense close to Popper’s; some minimal Kneale theory is true if any Kneale theory is true; the minimal Kneale theory that is data equivalent to any given Kneale theory is unique; and no Kneale theory is more probable than some minimal Kneale theory. I show that under the Craig-Vaught conditions, no minimal Kneale theories exist.     

Humanism and Morality

A theory of morality acceptable to humanists must be one that can be accepted independently of religion. In this paper, I argue that while there is such a theory, it is a non-standard one, and its acceptance would have some far-reaching consequences. As one might expect, the theory is similar to others in various ways. But it is not the same as any of them. Indeed, it is a radically new theory. Like Hume’s ethics, it is founded on our natural sociability, and feelings of empathy for others. Like Aristotle’s theory, it incorporates an ethics of virtue. Like Kant’s theory, it regards the set of moral principles as those appropriate for a socially ideal society. But unlike Kant’s theory, it is essentially utilitarian. I call it ‘social contractual utilitarianism’.     

Might Intentions be the Only Source of Practical Imperatives?

I focus on the broadly instrumentalist view that all genuine practical imperatives are hypothetical imperatives and all genuine practical deliberation is deliberation from existing motivations. After indicating why I see instrumentalism as highly plausible, I argue that the most popular version of instrumentalism, according to which genuine practical imperatives can take desires as their starting point, is problematic. I then provide a limited defense of what I see as a more radical but also more compelling version of instrumentalism. According to the position I defend, genuine practical deliberation and genuine practical imperatives take as their starting point the agent's intentions and only the agent's intentions.Given my loose usage, the Humean position Bernard Williams defends in his seminal article “Internal and External Reasons” (1981) counts as a version of instrumentalism about practical reason, since it incorporates the idea that every genuine practical imperative takes as its starting point some existing motivation(s) of the agent. It deviates from strict instrumentalism in that it leaves room for specificationist reasoning (reasoning aimed at moving from general ends to specific ends) in addition to means-end reasoning. For example, it leaves room for practical reasoning that is focused on “finding constitutive solutions, such as deciding what would make for an entertaining evening, granted that one wants entertainment” or on “thinking how the satisfaction of elements in [one's subjective motivational set] can be combined, e.g. by time-ordering” (Williams, 1981, p. 104).     

$$\mathrm {ZF}$$ ZF Between Classicality and Non-classicality

Studia Logica - We present a generalization of the algebra-valued models of $$\mathrm {ZF}$$ where the axioms of set theory are not necessarily mapped to the top element of an algebra, but may get...     

A Virtue-Based Defense of Mathematical Apriorism

Mathematical apriorists usually defend their view by contending that axioms are knowable a priori, and that the rules of inference in mathematics preserve this apriority for derived statements—so that by following the proof of a statement, we can trace the apriority being inherited. The empiricist Philip Kitcher attacked this claim by arguing there is no satisfactory theory that explains how mathematical axioms could be known a priori. I propose that in analyzing Ernest Sosa’s model of intuition as an intellectual virtue, we can construct an “intuition–virtue” that could supply the missing explanation for the apriority of axioms. I first argue that this intuition–virtue qualifies as an a priori warrant according to Kitcher’s account, and then show that it could produce beliefs about mathematical axioms independent of experience. If my argument stands, this paper could provide insight on how virtue epistemology could help defend mathematical apriorism on a larger scale.     

Phenomenological immanence,normativity, and semantic externalism

This paper argues that transcendental phenomenology (here represented by Edmund Husserl) can accommodate the main thesis of semantic externalism, namely, that intentional content is not simply a matter of what is ‘in the head,’ but depends on how the world is. I first introduce the semantic problem as an issue of how linguistic tokens or mental states can have ‘content’—that is, how they can set up conditions of satisfaction or be responsive to norms such that they can succeed or fail at referring. The standard representationalist view—which thinks of the problem in first-person terms—is contrasted with Brandom’s pragmatic inferentialist approach, which adopts a third-person stance. The rest of the paper defends a phenomenological version of the representationalist position (seeking to preserve its first-person stance) but offers a conception of representation that does not identify it with an entity ‘in the head.’ The standard view of Husserl as a Cartesian internalist is undermined by rejecting its fundamental assumption—that Husserl’s concept of the ‘noema’ is a mental entity—and by defending a concept of ‘phenomenological immanence’ that has a normative, rather than a psychological, structure. Finally, it is argued that phenomenological immanence cannot be identified with ‘consciousness’ in Husserl’s sense, though consciousness is a necessary condition for it.     

What is the axiomatic method?

The modern notion of the axiomatic method developed as a part of the conceptualization of mathematics starting in the nineteenth century. The basic idea of the method is the capture of a class of structures as the models of an axiomatic system. The mathematical study of such classes of structures is not exhausted by the derivation of theorems from the axioms but includes normally the metatheory of the axiom system. This conception of axiomatization satisfies the crucial requirement that the derivation of theorems from axioms does not produce new information in the usual sense of the term called depth information. It can produce new information in a different sense of information called surface information. It is argued in this paper that the derivation should be based on a model-theoretical relation of logical consequence rather than derivability by means of mechanical (recursive) rules. Likewise completeness must be understood by reference to a model-theoretical consequence relation. A correctly understood notion of axiomatization does not apply to purely logical theories. In the latter the only relevant kind of axiomatization amounts to recursive enumeration of logical truths. First-order “axiomatic” set theories are not genuine axiomatizations. The main reason is that their models are structures of particulars, not of sets. Axiomatization cannot usually be motivated epistemologically, but it is related to the idea of explanation.     

Poetry as anti-discourse: formalism,hermeneutics, and the poetics of Paul Celan

I argue from a hermeneutic point of view that formal elements of poetry can only be identified because poetry is based on both the phenomenon and the conception of poetry, both of which precede the attempt to identify formal elements as the defining moment of poetry. Furthermore, I argue with Gadamer that poetry is based on a rupture with and an epoche of our non-poetic use of language in such a way that it liberates “fixed” universal aspects of everyday language, and that through establishing itself in a new, self-referential and monologue unity, it individualizes speech. From the hermeneutic position, poetry is a form of speaking rather than a “fixed” object. As such, I will try to make sense of what Paul Celan said in his famous “Meridian” speech: namely, that the poem is “actualized language, set free under the sign of a radical individuation, which at the same time stays mindful of the limits drawn by language, the possibilities opened by language.”     

Sets and Plural Comprehension

The state of affairs of some things falling under a predicate is supposedly a single entity that collects these things as its constituents. But whether we think of a state of affairs as a fact, a proposition or a possibility, problems will arise if we adopt a plural logic. For plural logic says that any plurality include themselves, so whenever there are some things, the state of affairs of their plural self-inclusion should be a single thing that collects them all. This leads to paradoxes analogous to those that afflict naïve set theory. Here I suggest that they are the very same paradoxes, because sets can be reduced to states of affairs. However, to obtain a consistent theoretical reduction we must restrict the usual axiom scheme of Comprehension for plural logic to ‘stratified’ formulas, to avoid viciously circular definitions. I prove that with this modification to the background plural logic, the theory of states of affairs is consistent; moreover, it yields the axioms of the familiar set theory NFU.     

The Consistency Argument for Ranking Functions

The paper provides an argument for the thesis that an agent’s degrees of disbelief should obey the ranking calculus. This Consistency Argument is based on the Consistency Theorem. The latter says that an agent’s belief set is and will always be consistent and deductively closed iff her degrees of entrenchment satisfy the ranking axioms and are updated according to the ranktheoretic update rules. Special Issue Formal Epistemology I. Edited by Branden Fitelson     

Three contributions to the two-valued propositional calculus

Summary Three chapters contain the results independent of each other. In the first chapter I present a set of axioms for the propositional calculus which are shorter than the ones known so far, in the second one I give a method of defining all ternary connectives, in the third one, I prove that the probability of propositional functions is preserved under reversible substitutions. This paper appeard orginally under the title “Trois contributions an calcul des propositions bivalent” inStudia Societatis Scientiarum Torunensis, Toruń, Polonia, Sectio A, vol. I (1948), pp. 3–15.     

Minimal Belief Change and the Pareto Principle

This paper analyzes the notion of a minimal belief change that incorporates new information. I apply the fundamental decision-theoretic principle of Pareto-optimality to derive a notion of minimal belief change, for two different representations of belief: First, for beliefs represented by a theory – a deductively closed set of sentences or propositions – and second for beliefs represented by an axiomatic base for a theory. Three postulates exactly characterize Pareto-minimal revisions of theories, yielding a weaker set of constraints than the standard AGM postulates. The Levi identity characterizes Pareto-minimal revisions of belief bases: a change of belief base is Pareto-minimal if and only if the change satisfies the Levi identity (for “maxichoice” contraction operators). Thus for belief bases, Pareto-minimality imposes constraints that the AGM postulates do not. The Ramsey test is a well-known way of establishing connections between belief revision postulates and axioms for conditionals (“if p, then q”). Pareto-minimal theory change corresponds exactly to three characteristic axioms of counterfactual systems: a theory revision operator that satisfies the Ramsey test validates these axioms if and only if the revision operator is Pareto-minimal. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Insufficiency of the Dutch Book Argument

It is a common view that the axioms of probability can be derived from the following assumptions: (a) probabilities reflect (rational) degrees of belief, (b) degrees of belief can be measured as betting quotients; and (c) a rational agent must select betting quotients that are coherent. In this paper, I argue that a consideration of reasonable betting behaviour, with respect to the alleged derivation of the first axiom of probability, suggests that (b) and (c) are incorrect. In particular, I show how a rational agent might assign a ‘probability’ of zero to an event which she is sure will occur.