2.2 Objetivos de la Investigación
3.2.2 Arquitectura Cliente Servidor
Now if we apply the above distinctions to our considerations regarding the explanatory power of the two set theoretic reductions, we should see that neither proposed reduction is apt to be con- sidered a predictive explanation. Both proposed reductions are deemed sufficient to establish the existence of numbers with their “normal” properties, so the falsity of either one will not ensure the non-existence of numbers, thereby contradicting the third condition in the definition of a predictive explanation. Similarly, while our belief in the existence of numbers might be thought sufficient to induce belief in one or other of the set theoretic reductions, the other direction fails. The belief in the existence of a set theoretic construction, is not alone sufficient to generate belief in the exis- tence of numbers, without first presuming a belief in the fact that the numbers are emergent from a particular set theoretic construction. Seeing that the latter move is question begging ensures that the first condition in the definition of predictive explanation cannot be met.
34Boutilier in [16] articulates this idea with reference to the revision of our theory with φ. The details are not
terrifically important but a nice summary and discussion of AGM belief revision theories can be found in [93].
So now let’s consider whether either set theoretic reduction can be thought of as a hypothetical non-predictive explanation. Indeed they can, assume that Platonism in mathematics is false, and then either set theoretic reduction is an apt non-predictive explanation for the existence of numbers since numbers will be seen to have a constructive existence. Both constructions are possible since they both validate all the “normal” properties of the number sequence. Although strictly speaking, we’re not even providing a normality claim; rather for each reduction we are claiming that the appropriate set theoretic construction serves as evidence for the existence of numbers. So instead of a default rule φ ψ we should really state that there is evidentiary term e, which encodes
φ, i.e. eφ, and eφ: ψ. But, in any case, such non-predictive explanations cannot be expected to
rule out all alternatives. The abductive problem dissolves quickly in this case precisely because the explanatory task has been achieved as well it currently can. Non-predictive explanations do not allow us to achieve a unique inference to the best explanation. In retrospect this should have been obvious, the process of inference to the best explanation, could never uniquely determine a mathematical identity, by its very nature. It is simply not the task of a hypothetical non-predictive explanation to definitively establish the one and only cause of the event (or phenomenon) for which we sought an explanation! To seek a predictive explanation in the philosophy of mathematics is to misconstrue the discipline entirely. This thought might be extended to the conclusion that philosophical explanations lack any kind of knowledge (or unique belief) inducing property at all.36
4.5
Generalising the Picture
In this chapter we have tried to demonstrate the importance of being able to factor for structural information in cases of abductive reasoning. We began with a presentation of Gettier’s argument to show that underdetermination problems can prompt an abductive problem which is used to un- dermine the notions of knowledge and non-question begging belief. We then proceeded to consider a case study in underdetermination in the philosophy of mathematics.
The idea was to show that if we considered the appropriate metaphysics then we could re- examine the underdetermination problem in the hope that we could resolve our abduction problem one way or the other. It became steadily clear that instead of resolving the problem we could at best dissolve the problem. This subtle distinction was motivated by examining the kind of dependency relationship which obtained between complex constructions in mathematics. On examination, it turned out that the grounding relationship often supposed to underlie metaphysical dependency of the relevant kind, was either (1) straightforwardly inapplicable to the debate or (2) ultimately inapplicable, because fundamental issues of mathematical ontology remain unresolved.
We suggested that these kinds of consideration should be expressible at the epistemic level of explanation. To do so we found the need to distinguish between types of explanation so that an explanatory relation in the philosophy of mathematics could be seen as domain-appropriate. It is too easy to consider explanation as a simple relation between premises and conclusions. The crucial observation here is that explanation is not a relation, it is a process! A process which is subtly different in each domain of reasoning.
An explanation is a process of moving from our shared structural information, and the ob- servation of structural connection between event types, to the inauguration of default rules of expectation, or evidentiary relevance and the conclusion that certain claims justify (or explain)37
36
A similar suggestion was made by Eric Schliesser on the NewApps blog on July 17th 2013 in a post entitled “What is the reach of philosophical argument”
37In lieu of developing a systematic “logic of explanation” we treat justification and explanation as complementary
notions. Whatever claim justifiesψmay be used to explainψ. So if we have j : ψwhere j encodes a formulaχsuch that χjustifiesψ thenχis also an abductive solution to an abductiveψ-problem. Both are answers to the why-ψ
others just when they track a dependency relation we have seen hold in the world. These justifi- cations prompt the formation of beliefs and or knowledge where appropriate. An explanation is successful if upon deployment within in a community, the community converges upon acceptance of the theories in which the appropriate default rules, or evidence functionE feature. The process is not so much the idea of reasoning by appeal to structural information, but by doing so in such a way that others are compelled by your arguments - compelled because they share the same struc- tural information, and see your reasoning to be adequate.
To illustrate the manner in which justification and indeed explanation is a process, we should note the manner in which justification terms or analogously explanatory terms can inherit prop- erties from the dependency relation which they track. For instance, consider a language of ex- planatory groundingLGfrom three tiers of linguistic utterance. For convenience we list them here:
Lpg, Lexp2, Lexp3. The idea is construct our logic of justification from bottom up, by defining justification with respect to our structural information as determined by, for instance, our logic of full ground.
φ::= p | ¬φ|φ∨ψ |j: ψ
φ::= j : ψ|j + r : ψ|j! : j : ψ |(j ·e) : ψ φ::= p |φ-ψ|∆-ψ
Where the language LG is constructed in three stages. We take that Lpg is primitive as it
stems from our implicit logic of grounding. We, then define a meta-syntactic function in analogy with the CS-function of justification logic. The idea is provide an agent with a limited but well motivated set of justificatory claims from which we can motivate belief and knowledge claims. Each justification is based on a result achieved in our metaphysical theory of grounding.
(φi)∈ LG= j:ψ∈ Lexp3 if j :ψ∈ Lexp2
Bool(φi)∈ Lexp3 ifφi is a boolean combination of atomics
p∈ Lexp3 ifφi∈ Lpg and is atomic
j:ψ∈ Lexp2 if `pg χ-ψ and there is codingj:=χ
j+r :ψ∈ Lexp2 if `pg χ·τ -ψand there is a coding j :=χ, r:=τ
j! :j:ψ∈ Lexp2 if `pg τ -χ∧χ-ψ and there is coding j :=χ, !j :=τ j·e:ψ∈ Lexp2 if `pg (χ-τ ∧τ -ψ)where j:= (χ-τ), e:=χ
φi∈ Lpg if>
Crucially, we can see now that the properties of the justification relation e.g. monotonicity etc, are derived from from the underlying relation on which the justification procedure is founded. In our case, the non-monotonicity of the full grounding relation should ensure that the monotonic- ity axiom of justification logic will fail. This syntactic information can be encoded semantically in an evidence function E as appropriate. This picture of ascent from our base ontological and metaphysical theories to our reasoning about justification and belief is the crucial missing compo- nent in formal epistemology as standardly developed. The mistake oft repeated is to think that beliefs and knowledge can be studied in vacuum. The defence for the unreasonable idealisation is that it becomes too messy to incorporate such information. We would like to think that we have shown two things (1) the problems emerging from our failure to incorporate such structural information in our epistemology are much worse than any worry about theoretical nicety, and (2) by viewing justification (and analogously explanation) as a process of tracking rigorously definable dependency relations we need not be too concerned about the incursion of messiness.
In the above construction we have effectively stipulated evidentiary relevance by fiat, but there is another more subtle method which would achieve analogous results. The idea is that justification logic encodes justification terms based on the relation of logical entailment i.e. proof. The Lift- ing Lemma shows that we can construct justification terms which precisely encode the conditions under which the consequences are logically entailed by our premises. Hence, by analogy we might hope to prove a “Lifting Lemma” for each such dependency relation. In other words we might think that once we observe conditions under which our consequences are grounded in our premises, we could determine the type of information required to properly construct justification terms for reasoning about grounding. So there could be a construction which shows for each conclusion, how the conclusion followed from our premises by appeal only to the inference rules of Fine’s logic of pure ground.
Some issues remain. We have been intent to demonstrate the importance of structural infor- mation in both the development of an ontological picture, but also as a constraint on the types of justificatory (or explanatory) procedure we develop. However, we have barely scratched the surface with respect to systematically investigating the interactions of structural information upon our doxastic and epistemic states. Perhaps worse, we have not fully developed a picture of how this works in a multi-agent epistemic setting. This will be crucial if we expect to see explanations as a process of justification which converges to community consensus. For instance, we might need to insist that explanatory solutions can only develop and converge in a community if they share the same basic structural information and assumptions. We do not hope to adequately address either of these important issues within the scope of this thesis. Both raise significant and non-trivial questions which we defer for later work.
In the next chapter we shall focus on examining the relation of causality in an effort to discern its structure and the type of role it has in both ontological and epistemic reasoning. We wish to underline the importance of structural information in rational inference. We also seek to further elaborate the notion of explanation as a process of reasoning from structural information to the appropriate changes of epistemic and doxastic states. As such the next chapter will provide a model that might be emulated in the development of further explanatory notions. Entirely in line with our current suggestions.
Chapter 5
Because: Identified Dependency
On the plain behind him are the wanderers in search of bones...and they move haltingly in the light like mechanisms whose movements are monitored with escapement and pallet so that they appear restrained by a prudence or reflectiveness which has no inner reality, and they cross in their progress one by one that track of holes that runs to the rim of the visible ground and which seems less the pursuit of some continuance than the verification of a principle, a validation of sequence and causality - as if each round and perfect hole owed its existence to the one before it there on that prairie upon which are the bones and the gatherers of bones...He strikes a fire in the hole and draws out his steel. Then they all move on again. - Cormac McCarthy1
5.1
Introduction: Patterns of pattern seeking
Seek and you shall find. This phrase, once a religious benediction, is now an infamous fallacy named optimism. Nevertheless, we shall argue, there are reasons to be optimistic.
In the previous chapter we argued that structural information had a role to play in episte- mology. In particular we argued that certain dependency relations should be included in our explanatory reasoning. We attempted to showcase this by appeal to the relation of grounding. The choice was appropriate at the time, but we would be remiss if we did not mention what many2 feel to be the paradigmatic explanatory relation - causality. We shall argue that the world has a discernible causal structure, that our best theories are those which seek and find such information.
We do not pretend to address every available theory of causality. We deliberately narrow our focus to avoid over-inflating the need for exposition. In particular we treat Judea Pearl’sCausality3
in addition with two papers he wrote with Joseph Halpern on the same topic. Our intent is to view the study of causality as a prelude the study of explanation.4 The primary reason for making this choice is that Pearl and Halpern take seriously the notion of structural information and they are rigorous about distinguishing causal structural information from statistical data. These reasons are sufficient, but we provide another. Pearl explicitly links the recognition of true causal infor- mation with the emergence of a “deep understanding.”5 He relates the observation of structural information with the growth of understanding and the emergence of knowledge and belief. Since this is exactly the course we recommend, we shall proceed to assess his proposal.
We shall break down the discussion into three areas where the work of Pearl and Halpern
1InBlood Meridian the Epilogue. 2
cf. [66]
3 [73] 4
In this choice we follow Pearl and Halpern in [47]
5
can be seen to be innovative. In particular we shall examine (1) their discussion of the graphical representations of dependency information (2) graphical conditions for the identification of causal relationships (3) their semantics for causal statements. All three contribute towards the broad theory of causality and (1) and (2) are utilised to develop a theory of explanation derivative of the predictive power of causal reasoning.