Intuitionism and Logical Tolerance

New Thinking about Propositions, 2014

Croatian Journal of Philosophy, 2024

In this paper, I deal with recognising an appropriate criterion for distinguishing two competing conceptions of the propositional content among the content realists—the Fregean and the Russellian—especially in connection to some classical proponents of the realist view (Frege, Moore, and Russell). My starting point is a survey characterisation of the two conceptions and the accompanying classifi cation of Russell’s and Moore’s conceptions of the propositional content, which I fi nd problematic on several accounts. I set up a context for my consideration and elaborate on why I fi nd it problematic. My central point is that, given how the classical proponents of propositions understood their respective conceptions, as well as how more recent proponents of propositions (for example, David Kaplan) understood them, one should draw the distinction between the Fregean and the Russellian conception on the grounds of what propositional components do rather than the nature of propositional components (unless, of course, one ultimately reduces the latter to the former).

Dialectica, 2005

In the unpublished work Theory of Knowledge† a complex is assumed to be “anything analyzable, any- thing which has constituents” (p. 79), and analysis is presented as the “discovery of the constituents and the manner of combination of a given complex” (p. 119). The notion of complex is linked in various ways with the notions of relating relation, logical form and proposition, taken as a linguistic expression provided with meaning. This paper mainly focuses on these notions, on their links and, more widely, on the role of logical form, by offering a new way of understanding what Russell was doing in TK as concerns the logical-ontological matter of this manuscript. In particular, a new account of Russell's theory of judgment will be given, by taking a stand with respect to the main accounts already given, and it will be argued for the presence in TK of a notion of type different from the one applied to propositional functions in ML and PM.

41st International Wittgenstein Symposium "Philosophy of Logic and Mathematics", Kirchberg am Wechsel, Austria, August 5-11, 2018

In this paper I argue that bivalent systems in logic and beyond are often too inexact and that bivalence about truth-values and other value spectra will consequently and generally need to be replaced with multivalence. I begin arguing for that conclusion by showing how, contrary to an at least potential misconception and framed in set theoretical terms, the somewhat reconceived law of excluded middle does not entail bivalence but is entirely compatible with multivalence too. From these logico-semantic considerations I then move on to some ontologico-empirical considerations in favor of multivalence and against bivalence that can be traced back to Lukasiewicz or to the Vienna Circle's logical empiricism. I conclude by pointing out that the paradigm shift from bivalence to multivalence has already occurred in the logic-related and so-called new psychology of reasoning and by going through some options of how the traditional values of "true" and "false" and the spectrum of truth-values can be adapted to a framework of 'quantified multivalence.'

Intuitionist logic is often understood as similar to classical logic except that it denies the principle of excluded middle. Extending the classical interpretation of the principle of excluded middle, that a proposition is either true or false, it will be shown that the propositional formula of the principle allows for three interpretations. Intuitionist logic actually employs the extended interpretation of the principle. In this view, intuitionist logic " s rejection of excluded middle is pre-emptory, however, the recognition that mathematical proofs can be true, false as well as uncertain, is well founded. It is suggested that U 4 is better suited than intuitionist logic to mathematical proofs.

Nordic Wittgenstein Review, 2017

It is often held that Wittgenstein had to introduce numbers in elementary propositions due to problems related to the so-called colour-exclusion problem. I argue in this paper that he had other reasons for introducing them, reasons that arise from an investigation of the continuity of visual space and what Wittgenstein refers to as ‘intensional infinity’. In addition, I argue that the introduction of numbers by this route was prior to introducing them via the colourexclusion problem. To conclude, I discuss two problems that Wittgenstein faced in the writings before Some Remarks on Logical Form (1929), problems that are independent of the colour-exclusion problem but dependent on the introduction of numbers in elementary propositions. 1. Numbers in Some Remarks on Logical Form In Some Remarks on Logical Form (RLF) Wittgenstein wrote: I wish to make my first definite remark on the logical analysis of actual phenomena: it is this, that for their representation numbers (rational and irr...

International Journal for the Study of Skepticism Special Issue, 2016

Wittgenstein's notion of 'hinge propositions'-those propositions that stand fast for us and around which all empirical enquiry turns-remains controversial and elusive, and none of the recent attempts to make sense of it strike me as entirely satisfactory. The literature on this topic tends to divide into two camps: either a 'quasi-epistemic' reading is offered that seeks to downplay the radical nature of Wittgenstein's proposal by assimilating his thought to more mainstream epistemological views, or a nonepistemic, 'quasi-pragmatic' conception is adopted that goes too far in the opposite direction by, for example, equating 'hinge propositions' with a type of 'animal' certainty. Neither interpretative strategy, I will argue, is promising for the reason that 'hinges' are best not conceived as certainties (or uncertainties) at all. Rather, what Wittgenstein says in respect to them is that doubt is "logically" excluded, and where there can be no doubt, I contend, there is no such thing as knowledge or certainty either.

Disputatio, 2019

By 1929, after the full acknowledgment of the colour–exclusion problem, Wittgenstein had to admit that material incompatibilities presented in conceptual systems (Satzsysteme) could not be reduced to formal tautologies and contradictions. Wittgenstein then, in his middle period, had to examine the kind of negation which, for instance, colour systems should render, which expose not just one but many or, in some cases, infinite inferentially articulated alternatives. Here, inspired by Brandom’s inferentialism (1994, 2001, 2008), I explore the idea that Wittgenstein, in his middle period, advocated a form of inferentialism based on the inferentially articulated content of propositions in Satzsysteme. At that time, Wittgenstein suggested that every sentence should be logically connected to many others. I call this feature inferential thickness. Therefore, I use Löf’s (2013) normative read of verificationism to explain Middle Wittgenstein’s holist solution to problems concerning the use of negation related to material incompatibilities and determination of propositional sense. I also investigate the distinction between contrariety and contradiction and some possible connections to a mandatory restriction of the principle of excluded middle in Satzsysteme