Indefiniteness in semi-intuitionistic set theories: On a conjecture of Feferman

The paper proves a conjecture of Solomon Feferman concerning the indefiniteness of the continuum hypothesis relative to a semi-intuitionistic set theory

The Place of Philosophy in European Culture

In this paper the author investigates the place of philosophy in European culture. Philosophy has taken a considerable time to be recognised, or to recognise itself, as distinct from other disciplines. Although philosophy gave birth to physics and, more recently, to other sciences, it is not seen as a “technical” subject, like mathematics and natural sciences, or even social sciences such as economics. Philosophy is available to the general (educated) public while the technical subjects are not. All educated people know the names of the great Western philosophers. Less people know the names of the great mathematicians (other than those such as Descartes and Leibniz which were, at the same time, philosophers). Therefore, philosophy has not lost its place as part of high culture, as have the natural sciences and mathematics. Philosophy continues to exert a pervasive effect upon European culture in general. However, according to the author, two tasks lie before philosophers; two gulfs are for us to bridge. The fi rst one is the gulf between philosophers of all schools and scientists (particularly physicists); the other one is that between divergent philosophical schools – between analytical philosophy and so called “continental” philosophy. If it solves these problems, philosophy will remain what it has been in the past – a shining component of European culture

Future Contingents and the Logic of Temporal Omniscience

At least since Aristotle’s famous 'sea-battle' passages in On Interpretation 9, some substantial minority of philosophers has been attracted to the doctrine of the open future--the doctrine that future contingent statements are not true. But, prima facie, such views seem inconsistent with the following intuition: if something has happened, then (looking back) it was the case that it would happen. How can it be that, looking forwards, it isn’t true that there will be a sea battle, while also being true that, looking backwards, it was the case that there would be a sea battle? This tension forms, in large part, what might be called the problem of future contingents. A dominant trend in temporal logic and semantic theorizing about future contingents seeks to validate both intuitions. Theorists in this tradition--including some interpretations of Aristotle, but paradigmatically, Thomason (1970), as well as more recent developments in Belnap, et. al (2001) and MacFarlane (2003, 2014)--have argued that the apparent tension between the intuitions is in fact merely apparent. In short, such theorists seek to maintain both of the following two theses: (i) the open future: Future contingents are not true, and (ii) retro-closure: From the fact that something is true, it follows that it was the case that it would be true. It is well-known that reflection on the problem of future contingents has in many ways been inspired by importantly parallel issues regarding divine foreknowledge and indeterminism. In this paper, we take up this perspective, and ask what accepting both the open future and retro-closure predicts about omniscience. When we theorize about a perfect knower, we are theorizing about what an ideal agent ought to believe. Our contention is that there isn’t an acceptable view of ideally rational belief given the assumptions of the open future and retro-closure, and thus this casts doubt on the conjunction of those assumptions

Truthmakers and modality

This paper attempts to locate, within an actualist ontology, truthmakers for modal truths: truths of the form or . In section 1 I motivate the demand for substantial truthmakers for modal truths. In section 2 I criticise Armstrong’s account of truthmakers for modal truths. In section 3 I examine essentialism and defend an account of what makes essentialist attributions true, but I argue that this does not solve the problem of modal truth in general. In section 4 I discuss, and dismiss, a theistic account of the source of modal truth proposed by Alexander Pruss. In section 5 I offer a means of (dis)solving the problem

Conocimiento práctico y conocimiento del lenguaje

Para aclarar qué tipo de conocimiento requiere el lenguaje, se examina el conflicto aparente entre dos aportaciones fregeanas: la prioridad de la función comunicativa del lenguaje y una teoría del significado que parece comprometida con el psicologismo que Frege rechaza. El artículo defiende que el lenguaje es instrumento de comunicación antes que vehículo del pensamiento, examina la teoría fregeana del significado en relación con las nociones de “acto lingüístico”, “frase”, “sentido” y “fuerza”, y sostiene que los hablantes de un idioma requieren tanto una habilidad práctica (saber cómo) como un conocimiento implícito de las reglas del lenguaje.In order to clarify the kind of knowledge that language requires, this paper examines the apparent conflict between two Fregean contributions: first, the priority of the communicative role of language and second, a theory of meaning that seems to embrace the kind of psychologism rejected by Frege. The article defends that language is a tool for communication rather than a vehicle of thought, examines Frege’s theory of meaning in connection with the notions of “linguistic act,” “phrase,” “sense,” and “force,” and holds that language speakers need both practical skill (know-how) and implicit knowledge of linguistic rules

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic, and $\Omega$-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets

Presupposition and Policing in Complex Demonstratives

Philosoph

The scope of Feferman’s semi-intuitionistic set theories and his second conjecture

The paper is concerned with the scope of semi-intuitionistic set theories that relate to various foundational stances. It also provides a proof for a second conjecture of Feferman’s that relates the concepts for which the law of excluded middle obtains to those that are absolute with regard to the relevant test structures, or more precisely of ∆1 complexity. The latter is then used to show that a plethora of statements is indeterminate with respect to various semi-intuitionistic set theories