Reflections
5. Meta Considerations
There are a number of questions we want, or need, to address at this point. (i) What is the advantage of formalizing philosophical texts? (ii) What is the point in doing epistemology, formally? (iii) Why formalizeOC, in particular?
The inspiration of this research came largely from Achourioti and van Lambalgen [2011, p. 254], where the authors aim “to examine from the point of view of mathematical logic, Kant’s formal logic and its relation to what Kant called ‘transcendental logic’.” What would possess them to do such a thing? What are the merits of such projects quite generally?
To say that the Critique of pure reason is notoriously dif- ficult is an understatement. [. . .] Perhaps a mathematical formalisation, however incomplete, can shed some light on [the Critique’s] concepts and their relations. [. . .] there is hardly a better inducement to modesty than trying to come to grips with the complexities ofCPR, not to men- tion the secondary literature. But the formalisation may provide a starting point.
Likewise, we tried to examine from the point of view of formal epistemol- ogy the epistemological theory put forward inOC. However, our aim was not necessarily to clarify. That is not to say that we have not made an ef- fort to do so along the way. We have seen in chapter 1 the very different readings OCadmits, e.g. differing opinions on the relation between cer- tainty and belief or knowledge. And even though they “reflect current com- peting interpretations of On Certainty” it is not clear whether it “might be that, in spite of appearances, the responses are not that divergent” [Moyal- Sharrock and Brenner 2007, p. 3]. A formal framework, which we have by no means achieved here, can serve as a common ground for such debates.
We have shown that OCdoes admit a formal treatment — or at least aspects of it do. Moreover, the way we can treat it formally is not using
revolutionary techniques, but rather standard techniques of formal episte- mology, i.e. probability theory and modal logic. How they are combined may be new, that is, do without an epistemic accessibility relation and de- fine belief, knowledge and doubt wholly on a probabilistic structure. But new is not bad and certainly approaches like ours are gaining support due to the work of Baltag [unpublished] and Leitgeb [2014].
Our main aim in this thesis, however, was to merge two strands of re- search: interpretations ofOCand formal epistemology. For one, we wanted to show experts working onOCthat it is not a hopeless enterprise to try and formalize Wittgenstein’s writing. Drawing attention to the betting in- terpretation of probability, I believe, we have already opened up numerous ways to formally graspOC. Recall Ramsey [1931, p. 15] arguing that “the kind of measurement of belief with which probability is concerned is [. . .] a measurement of beliefquabasis of action.”1 “[B]eliefquabasis of action” does sound very much in line withOCand, as we have seen, the (so-called) downsides (e.g. bets on events that, if they occurred, would change the value of the pay-off) of this interpretation of probability turn out to be up- sides when applied to Wittgenstein. We can argue that thus merging the betting interpretation with a Wittgensteinian view on certainties gives a more robust interpretation of probability. Moreover, it can be argued that it affirms that the thoughts presented in OC, when given a mathematical footing, draw a consistent picture of the situation.
This brings us to the other way of looking at this thesis. We can look at probability from a Wittgensteinian point of view, after (formally) interpret- ing Wittgenstein in terms of probability. Not only can we account for the odd consequence of the betting interpretation where “placing the bet may change the world, and hence your opinions” [H´ajek 2012, 3.3.2] by simply pointing out that the prior probabilities make up an agent’s life — “[m]y
lifeconsists in my being content to accept many things” [OC, 344]. But this also gives us a Wittgensteinian answer to theproblem of priors:
This weakness of the probability axioms generates the fa- mous problem of the priors, the problem of saying where initial probabilities come from. Are they always based on evidence previously collected? If so, how does scientific inquiry get started? If instead theyre not based on previ- ous evidence but area priori, what principles govern this
a priorireasoning? [Weisberg 2015, 1.4]
The Wittgensteinian answer would here be, I believe, that prior proba- bilities are neither based on evidence nor a priori2 but rather a matter of convention or cultural background. The prior probability simply deter- mines the language-game and science is but one of many games that can be played.
1On a biographical note, let us not forget that Ramsey, by Wittgenstein’s own admis-
sion [cf. PI, preface], has influenced Wittgenstein’s turn from the Tractatus to thePI, and thus has been an influence on Wittgenstein’s later work which includesOC.
2That is, if we read ‘a priori’ in the classical, Kantian way. If we read ‘a priori’ simply as
meaning ‘prior to experience’, then indeed the probabilistic priors, i.e. certainties conceived of as beliefs, area priori.
You must bear in mind that the language-game is so to say something unpredictable. I mean: it is not based on grounds. It is not reasonable (or unreasonable).
It is there — like our life. [OC, 559]
It should be duly noted that we might not be content to accept the Wittgen- steinian answer as convincing and launch a critique ofOCfor its failure to solve the problem of priors — or a similar problem stated in non-proba- bilistic terms. For example, we could argue that if the language-game is not based onanygrounds, then it is also not based on culture, nor on com- munity. But then how come that we inheritthis background? Should it be possible that a child grows up, not sharing any certainties of its parents, family, or community — perhaps not even understanding their language — because the language-game is unpredictable and ungrounded?
Moreover, this (as well as the story of Moore and the king) give us a way of looking at conditioning a prior on evidence with probability 0. This usually causes technical problems and is circumvented by conventions (e.g. thatµ(H|E) =0 ifµ(E) =0) [cf. Baltag and Smets 2008b; van Benthemet al.
2009]. In our case, leaving such probabilities undefined is quite acceptable. Indeed, from an OCpoint of view, I believe, letting logic or mathematics dictate what is to happen in such cases is nonsense. A convention may, of course, be agreed on but this convention need not be the same for every agent or context (recall the context of trusting ones senses).
Quite generally, formalizing an epistemological text can lend weight to formal epistemological theories. In our case this would mean that if we have convincingly argued in section 3, the (probabilistic) stability theory of knowledge as constructed by Baltag [unpublished] has gained weight and it, in turn, lends weight to the stability theory of belief as found in Leitgeb [2014].