Stefan Wintein

How to be fairer

We confront the philosophical literature on fair division problems with axiomatic and game-theoretic work in economics. Firstly, we show that the proportionality method advocated in Curtis (in Analysis 74:417–57, 2014) is not implied by a general principle of fairness, and that the proportional rule cannot be explicated axiomatically from that very principle. Secondly, we suggest that Broome’s (in Proc Aristot Soc 91:87–101, 1990) notion of claims is too restrictive and that game-theoretic approaches can rectify this shortcoming. More generally, we argue that axiomatic and game-theoretic work in economics is an indispensable ingredient of any theorizing about fair division problems and allocative justice.

A Gentzen Calculus for Nothing but the Truth

In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic (ETL), an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof.

Assertoric Semantics and the Computational Power of Self-Referential Truth

There is no consensus as to whether a Liar sentence is meaningful or not. Still, a widespread conviction with respect to Liar sentences (and other ungrounded sentences) is that, whether or not they are meaningful, they are useless. The philosophical contribution of this paper is to put this conviction into question. Using the framework of assertoric semantics, which is a semantic valuation method for languages of self-referential truth that has been developed by the author, we show that certain computational problems, called query structures, can be solved more efficiently by an agent who has self-referential resources (amongst which are Liar sentences) than by an agent who has only classical resources; we establish the computational power of self-referential truth. The paper concludes with some thoughts on the implications of the established result for deflationary accounts of truth.

From Bi-facial Truth to Bi-facial Proofs

In their recent paper Bi-facial truth: a case for generalized truth values Zaitsev and Shramko [7] distinguish between an ontological and an epistemic interpretation of classical truth values. By taking the Cartesian product of the two disjoint sets of values thus obtained, they arrive at four generalized truth values and consider two “semi-classical negations” on them. The resulting semantics is used to define three novel logics which are closely related to Belnap’s well-known four valued logic. A syntactic characterization of these logics is left for further work. In this paper, based on our previous work on a functionally complete extension of Belnap’s logic, we present a sound and complete tableau calculus for these logics. It crucially exploits the Cartesian nature of the four values, which is reflected in the fact that each proof consists of two tableaux. The bi-facial notion of truth of Z&S is thus augmented with a bi-facial notion of proof. We also provide translations between the logics for semi-classical negation and classical logic and show that an argument is valid in a logic for semi-classical negation just in case its translation is valid in classical logic.

Dividing the indivisible: Apportionment and philosophical theories of fairness

Philosophical theories of fairness propose to divide a good that several individuals have a claim to in proportion to the strength of their respective claims. We suggest that currently, these theories face a dilemma when dealing with a good that is indivisible. On the one hand, theories of fairness that use weighted lotteries are either of limited applicability or fall prey to an objection by Brad Hooker. On the other hand, accounts that do without weighted lotteries fall prey to three fairness paradoxes. We demonstrate that division methods from apportionment theory, which has hitherto been ignored by philosophical theories of fairness, can be used to provide fair division for indivisible goods without weighted lotteries and without fairness paradoxes.

Analytic Tableaux for all of SIXTEEN 3

In this paper we give an analytic tableau calculus PL16 for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ⊧t, ⊧f, ⊧i, and ⊧ under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the first three entailment relations will in general require developing four tableaux, while proving that they are in the ⊧ relation may require six.

Alternative Ways for Truth to Behave When There’s no Vicious Reference

In a recent paper, Philip Kremer proposes a formal and theory-relative desideratum for theories of truth that is spelled out in terms of the notion of ‘no vicious reference’. Kremer’s Modified Gupta-Belnap Desideratum (MGBD) reads as follows: if theory of truth T dictates that there is no vicious reference in ground model M, then T should dictate that truth behaves like a classical concept in M. In this paper, we suggest an alternative desideratum (AD): if theory of truth T dictates that there is no vicious reference in ground model M, then T should dictate that all T-biconditionals are (strongly) assertible in M. We illustrate that MGBD and AD are not equivalent by means of a Generalized Strong Kleene theory of truth and we argue that AD is preferable over MGBD as a desideratum for theories of truth.

On the Strict–Tolerant Conception of Truth

We discuss four distinct semantic consequence relations which are based on Strong Kleene theories of truth and which generalize the notion of classical consequence to 3-valued logics. Then we set up a uniform signed tableau calculus(the strict–tolerant calculus), which we show to be sound and complete with respect to each of the four semantic consequence relations. The signs employed by our calculus are , , and , which indicate a strict assertion, strict denial, tolerant assertion and tolerant denial respectively. Recently, Ripley applied the strict–tolerant account of assertion and denial (originally developedby Cobreros et al. [2012] to bear on vagueness) to develop a new approach to truth and alethic paradox, which we call the Strict–Tolerant Conceptionof Truth (STCT). The paper aims to contribute to our understanding of STCT in at least three ways. First, by developing the strict–tolerant calculus.Second, by developing a semantic version of the strict–tolerant calculus(assertoric semantics), which informs us about the (strict–tolerant) assertoric possibilities relative to a fixed ground model. Third, by showing that the strict–tolerant calculus and assertoric semantics jointly suggest that STCTs claim that the strict and tolerant can be understood in terms of one another has to be reconsidered.

A Framework for Riddles about Truth that do not involve Self-Reference

In this paper, we present a framework in which we analyze three riddles about truth that are all (originally) due to Smullyan. We start with the riddle of the yes-no brothers and then the somewhat more complicated riddle of the da-ja brothers is studied. Finally, we study the Hardest Logic Puzzle Ever (HLPE). We present the respective riddles as sets of sentences of quotational languages, which are interpreted by sentence-structures. Using a revision-process the consistency of these sets is established. In our formal framework we observe some interesting dissimilarities between HLPE’s available solutions that were hidden due to their previous formulation in natural language. Finally, we discuss more recent solutions to HLPE which, by means of self-referential questions, reduce the number of questions that have to be asked in order to solve HLPE. Although the essence of the paper is to introduce a framework that allows us to formalize riddles about truth that do not involve self-reference, we will also shed some formal light on the self-referential solutions to HLPE.

From closure games to strong Kleene truth

In this paper, we study the method of closure games, a gametheoretic valuation method for languages of self-referential truth developed by the author. We prove two theorems which jointly establish that the method of closure games characterizes all 3and 4-valued Strong Kleene fixed points in a novel and informative manner. Amongst others, we also present closure games which induce the minimal and maximal intrinsic fixed point of the Strong Kleene schema.