Proceso Unificado de Modelado (RUP)

Recall our logic KT4, we argued above that such a logic could not be said to mimic the appropriate properties of belief, since KT4 ` Bφ → φ. Our job now is to show how to recover a notion of justified belief by deriving model of KT4 from a justification model of JT4. First we note that there is a soundness and completeness result for the logic KT4. The proof is pretty standard 38, so we reproduce the proof in the appendix only to show the connection between justification logics and Doxastic or Epistemic logic. What follows will hold for both Doxastic and Epistemic frames. Furthermore we shall show that there is a finite model of any consistent set of KT4 sentences. But the important result for our current purposes is that we can take any set of justification logic sentences and recover a set of doxastic claims. These can be intuitively thought of as justified beliefs, since they have been recovered from a justification logic setting.

Recovering Belief from Justifications

Having shown that there is a completeness theorem for the standard modal logics of belief, and analogously knowledge. It remains to show that each valid theorem of our doxastic logic can also be understood as a result of the appropriate justification logic. The thought here is that each sentence of our doxastic logic has a hidden structure which has not been revealed. To show this we define a translation function t: LJT47→ LKT4

t(p) =p t(¬φ) =¬(t(φ)) t(φ∨ψ) =t(φ)∨t(ψ) t(j:φ) =B(t(φ)) 37_{In [56]} 38

This mapping is called theforgetful functor.39 Melvin Fitting40discusses its usage with respect to the notions of explicit and implicit knowledge. Where justification logic can be seen encoding a very explicit notion of knowledge, and direct epistemic logic suffices only to articulate claims for which we have an unexamined reason to believe true.

Proposition (Preservation) The forgetful functor maps theorems of JT to theorems of KT. Likewise it maps theorems of JT4 to theorems of KT4 and theorems of KT45 to theorems of JT45.41

The result is straightforward but it highlights a neat connection between our beliefs and our justifications. Furthermore, it allows for a tolerable notion of truth-entailing beliefs. This goes a long way to rehabilitating some of mentioned failures of models which only allow for the expression of primitive beliefs.

Recovering Justifications from Beliefs

You might now wonder whether given a set of beliefs we can discover a justification for those beliefs. Indeed we can. We shall prove this result in the appendix, but the crucial notions are as follows:

Definition (Realisation) If we letφ be a formula in our doxastic logic. A realisation ofφ is a formula in the appropriate justification logic, that results from applying a realisation function to every subformula Bψ ∈ φ. A realisation is normal if negative occurrences of B are replaced with distinct justification variables (which are always part of the language of our justification logic regardless).

In short, a CS function can be found to ensure the success of each realisation. The technicalities of this definition only become important when proving the next theorem.

Theorem (The Realisation Theorem) Ifφis a theorem of one of KT, KT4 or KT45, there is some normal realisation of φthat is a theorem of JT, JT4, or JT45 respectively.

This latter result was proven syntactically by Artemov and semantically by Fitting.42 By way of example, we say that (j:φ→!j:j:φ) = r(Bφ→BBφ). Where r is a realisation function that takes as input validity in the doxastic epistemic logic, and returns a validity in our justification logic. Another example of a KT4-theorem where (x:φ∨y:ψ)→ (j·!x + e·!y):(x:φ∨y:ψ)) = r(Bφ∨Bψ) →

B(Bφ∨Bψ)). These examples were taken from Artemov, the idea in the latter realisation is that the constants j and e are from the CS-function so that they stand as justification for the major premises, while for the classical axioms x:φ →(x:φ∨y:ψ). So while the theorem receives a more complicated validation in the justification logic setting, we can nevertheless recover a valid theorem of our doxastic logic by means of a (yet to be) specified method of translation. The realisation theorem states that we can work a normalised “translation” in the opposite direction of the forgetful

39_{Bryan Renne uses this translation to prove the consistency of the basic theories of justification logic. [17]pg122}

Theorem 3.14. This goes towards proving a completeness result for each justification logic via a canonical model construction. See his Theorem 3.22 for details. Alternatively see Aretemov’s overview paper [?]

40

In [30]

41_{Proof The proof proceeds in two steps (i) we fix a formulaφ valid in (for instance) JT4, and proceed to show}

that for any subformulas ofφthe translation function works, ultimately applying the translation function toφitself i.e. t(φ)↔t(ψ) for allψ∈φ. Now we show that for any valid formulaφin JT4, we can find a model of KT4 which validates t(φ). The proof is by contraposition, suppose that MK, w 2KT4 t(φ) for some w∈W, we need to show

that MJ similarly invalidatesφ. So we define an evidence functionE such that by addingE to MK creates a model

MJ =<W, R,E V>where we preserve the truths at every world v∈W. It is now a simple matter to prove that

MK, v|=KT4 t(τ) iff MJ, v|=τ. The proof is by induction as expected and it ensures (given our assumption) that

MJ w2J T4φas desireda 42

functor. In a sense we discover the greater structure underlying each belief. 43 We have given an example of one normal realisation to give a flavour of the idea, but the more significant thought is that all basic modal logics can now be thought of as short hand for a much more involved process of justification and emergent justified belief.

Justification Relations: Perfectly Relevant?

To summarise the issues so far; we have examined two formal settings which attempt to cash out the notions of belief or knowledge. Both the quantitative Bayesian, and the qualitative Epistemic Doxastic logic settings failed to adequately represent the notions of belief and knowledge in so far as both treated belief as a unexamined primitive. Slight nuance was added by the inclusion of somewhat natural axioms but ultimately both approaches suffered, since they made no mention of the source of our beliefs or the manner in which we come to believe certain claims over others.

This deficit can be addressed in a purely formal setting by adding (as in justification logic) an evidence function E, but the really interesting notions in epistemology involves explaining how such an evidence function arises. These evidence relations encode the relevance of our premises to certain conclusions, so it is vital to suggest a manner in which we can discover how and why certain conclusions are prompted by particular kinds of evidence, and not others. In the next chapter we attempt to provide a theory as to (a) how such connections can be discovered and (b) why such connections perform a vital role in epistemology.

3.4

Conclusion

The models of belief and knowledge elaborated in this chapter point to a common problem. We have not distinguished adequately between the source of our beliefs and knowledge. We have argued that the notion of belief (knowledge) and its attainment is better understood if we explicitly factor for the source our (belief) knowledge. The importance of our current suggestion relates more to the notion that the structure of our information state is inadequately represented if it leaves out a mechanism for distinguishing between the justification of our information, and (perhaps more importantly) a priority ordering on the species of justification deployed. These are the kind of considerations that underlie any solution to an underdetermination problem.

Chapter 4

True: Structural Commitments

Quine taught us some time ago (though not with these words), the metaphysician’s task of describing the structure of logical space - the space of all possible worlds - is not so easily separated from the scientist’s task of locating the actual world in it.1 - Robert Stalnaker

4.1

Introduction: Species of Evidence

In this chapter we shall attempt to unite the foregoing considerations and address the problem of underdetermination which afflicts traditional models of rational belief. First we shall set the scene and demonstrate the pervasive effect of underdetermination problems by discussing abduction and the construction of Gettier cases. We will then elaborate informally a means to develop solutions to abductive problems. The core idea is to distinguish between justificatory claims and the source of our justification. The observation that there are multiple kinds of justification relation defined to be dependent on different sources of evidence prompts the idea that no formal epistemology of justification should be elaborated with an attendant formal metaphysics of dependence. The rest of the chapter will showcase an attempt to apply this method to Benacceraf’s famous multiple reduction argument in the philosophy of mathematics.