SEMINAR - Finished

We will continue to read and discuss the third chapter of Luca Incurvati's book Conceptions of Sets and the Foundations of Mathematics, Cambridge University Press, 2020.

Anti-exceptionalism is the view that logic is, among other things, responsive to a posteriori evidence, and that disputes about logic should take adequacy to the data into account. This brings three questions to the fore: i) what counts as evidence in logic? ii) how to determine when evidence favours a particular system of logic, and iii) what kind of logic we use when in such disputes? In this paper, we argue that problem iii), when added to the thesis that natural language does have a specific logic, makes the whole project of logical theory choice untenable. The reason, we shall argue, is that the logic we are supposed to have vitiates any characterization of evidence, making a change of logic unlikely. We do that by studying a case where it is said that evidences favour paraconsistent logic over classical logic, and a case where it is argued that evidences favour a change in the nature of the truth-values (dialetheism). In both cases, the very idea that we have a logic operating in natural language prevent any possibility of change. We then go on and suggest, in broad lines, a version of anti-exceptionalism without the natural language logic hypothesis.


Keywords: logical abductivism; logical theory choice; evidence; paraconsistency; logic of natural language.

In a series of papers [4, 5, 6] on semantic information, Floridi has argued in favor of veridicality thesis (VT). According to this idea, any epistemically-oriented concept of information must have truth as one of its necessary conditions. That is, for Floridi, if a message x carries information of the kind “that is capable of yielding knowledge” (In Dretske’s words, [2, p. 45]), then x carries something that is true. Two challenges have been raised against VT. First, some philosophers have argued that VT wrongly collapses misinformation with non-informativeness [3, 9]. Secondly, it is not clear whether VT can give an adequate account of the information used in hypothetical reasoning and knowledge. In particular, think about information produced by logical and geometrical diagrams. Such representation systems are surely informative and yield knowledge even if they most of the time represent non-actualized, hypothetical scenarios. Focusing on the latter challenge, in this paper I propose an alternative definition of an epistemically-oriented concept of information. Instead of VT, based on semantic inferentialism [1, 8], I propose a definition of information in terms of what can be called the inferentiality thesis: if a message x carries epistemically-oriented information about an event p, then x reveals the inferential articulation of p. In other words, according to this inferentialist reading, information about p yields knowledge on what follows from and what implies p. Finally, I show that this inferentialist definition gives a nice account of the information used in hypothetical, diagrammatic reasoning.

References

[1] Robert Brandom. Making it Explicit: Reasoning, Representing, and Discursive Commitment. Harvard university press, 1998.

[2] Fred Dretske. Knowledge and the Flow of Information. CSLI Publications, 1981.

[3] James H Fetzer. Information: Does it have to be true? Minds and Machines, 14(2):223–229, 2004.

[4] Luciano Floridi. Outline of a theory of strongly semantic information. Minds and Machines, 14(2):197–221, 2004.

[5] Luciano Floridi. Is semantic information meaningful data? Philosophy and Phenomenological Research, 70(2):351–370, 2005.

[6] Luciano Floridi. In defence of the veridical nature of semantic information. European Journal of Analytic Philosophy, 3(1):31–41, 2007.

[7] Nancy J Nersessian and Miles MacLeod. Models and simulations. In Springer Handbook of Model-Based Science, pages 119–132. Springer, 2017.

[8] Jaroslav Peregrin. Inferentialism: why rules matter. Springer, 2014.

[9] Andrea Scarantino and Gualtiero Piccinini. Information without truth. Metaphilosophy, 41(3):313–330, 2010.

[10] Sun-Joo Shin. The Logical Status of Diagrams. Cambridge University Press, 1994.

[11] Sun-Joo Shin, Oliver Lemon, and John Mumma. Diagrams. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, winter 2018 edition, 2018.

We will continue to read and discuss the second chapter of Luca Incurvati's book Conceptions of Sets and the Foundations of Mathematics, Cambridge University Press, 2020.

In this talk we will explore the current debate regarding justification in the context of non-classical set theories. We will present the main arguments of Incurvati (2020) against paraconsistent set theories based on the naive conception of set, which we will present in the form of two maxims. We conclude that his arguments seems convincing and that we can extend his criticizm to paraconsistent set theories based on the iterative conception of set. In particular we show that ANY lattice-valued model which satisfies minimal properties (such as Extensionality and MP)  fail to satisfy  these maxims.  (!)

So we are either left with the option to explore alternative model constructions for set theory or we need to rethink justification! We go for the second option and introduce a new account of justification based on the ideal element theory of Hilbert.   We call this the ideal account of justification and argue that this account behaves much better with respect to paraconsistent set theory. Moreover, we claim that this account should provide a solid justification for paraconsistent set theories based on the iterative conception of set. We conclude with philosophical remarks about the future love childs of Hilbert and Da Costa.