We can define a model for justification logic in analogy with the development of a Kripke struc- ture. The only additional requirement is that we specify an extra constraint so that particular justification terms can feature legitimately in our reasoning at particular worlds. In short, we have to specify at which worlds particular justifications are relevant. To do this we add an evidence functionE which maps states and justifications to sets of propositional formulas.
Definition (An Justification Model) MCS = < W, v, E, V > where W is a set of worlds and
v is a reflexive relation. Strictly speaking E is a function from pairs of justification terms and propositions into the powerset of W. We often sayE(j,φ)⊆W for all available justification terms and propositionsφas restricted by a CS-function.32. Of course V is a valuation function as usual.
The semantics for the boolean connectives is as usual we need only specify the semantics for our modal operator.
Definition (Truth and Satisfaction)
M, w|= j:φ iff (1) ∀w’ (wvw’) M, w’|=φand (2) w ∈ E(j, φ).
Of course, this semantics happily validates the the factivity axiom of justification logic by the reflexivity of v. But we need to add further constraints to E if we are to recover the axioms of application and monotonicity. In particular for monotonicity to be preserved we need to specify that the relevance of some justification to a particular propositional claim is inherited by any com- bination of reasons which includes the initial justification. Similarly, to preserve the justification of implied truths we need to insist that evidentiary relevance is inherited across material implication.
Justification of Implication E(j, φ→ψ)∩ E(r, φ)⊆ E(j·r, ψ)
Monotonicity of Relevance E(j, φ)∪ E(r, φ)⊆ E(j+r, φ)
These and various other closure conditions onE can be imposed so as to capture the intuitive axioms. The virtue of this setting is that it allows us to capture a notion of knowledge previously inexpressible. But in particular, this setting displays the nature in which we might hope to address the grue-problem. Which is to say that the grue-hypothesis presents us with an abduction problem, wherein we are forced to choose between two options when neither is obviously justified by our available information. The grue-reasoning proceeds by appeal to a particular kind of evidence function E∗. The challenge is to say if, and why, the E∗-worlds are less plausible than the other worlds E-worlds. That is to say an abduction problem of the grue-type is essentially a question regarding the plausibility of evidentiary links. As such the real problem amounts to the discovery of when particular information is relevant to the assessment of a particular proposition
31
We will return to this idea below.
32This is a meta-linguistic function which assigns the justification terms to particular propositions. The idea is
that there are only so many grammatically cogent pairs (j;φ) and the evidence functionE must minimally respect the constraints of grammar. We shall discuss this idea further below.
Admissible Justifications
However, before we model the situation, we have to decide which justification claims can be mod- elled given the constraints imposed by the CS-function. That is to say, not all justification claims will be grammatically cogent. You might think that the claim that all emeralds are green, cannot support the further claim that all emeralds are grue i.e. that the notion of dual colour instanti- ation is incoherent, or at least incomprehensible given the strictures of contextually appropriate utterance. To meet this restraint Artemov33defines a meta-linguistic function called the constant- specification. The idea is that every agent has limited access to justificatory information, so we define a partial function CS: LJ {j | j is a justification} 7→ LJ {φ|φis a proposition }. We can think of this function as a resource distribution for an agent which underwrites the cogency of the claim that j: φ for each proposition he takes to be justified. It is a way in which to add contextual parameters to each model. As seen above Artemov does not make the CS-function explicit in the language, but chooses instead to add the information directly to the model by of an Evidence assignment.34
Definition (The full CS function) The full CS-function determines a set of formulas j1
:Ax1....jn : Axn for all instances of each axiom in our logic. CS is injective if there is at most
one justification for each instance of every axiom. Every proof in J naturally generates the CS function corresponding to a justification for each step in the proof by theInternalisationrule.
To validate basic logical reasoning we need to specify CS in such a way that for every instance of the axiom-schemas in J we have an appropriate distribution of justification term to validate our logical reasoning.35 Interestingly the model encodes this information as factual information, that is to say a justification claim is true at world, just when j is actually a justification for φ. This is a hard-line view of the justification relation as one which arguably tracks a relation of dependence between the justification and the claim. For instance, the justification of our logical axioms is prompted because we may observe the relation of entailment between our premises and our conclusions. This feature of justification is worth bearing in mind if we recall the relation between our best information and the grue-hypothesis. Supposing that there are such fine-grained relations of dependence, the discovery of such a relation would allow us to confirm or deny the grue-hypothesis. The model above hard-codes the existence of these dependence relations on, I think, the very reasonable assumption that they do in fact exist.36
So far in the justification logic setting we have only described a model appropriately assigned to be the mental state of some particular agent. As such all the justificatory reasoning is, so to speak, of a subjective nature. We have had the opportunity to generalise the setting to multiple agents, and therefore supply widely agreeable justification relations. An open question is whether these relations hard code objective or subjective species of justification. If the former, then we are faced with the burden of defending the institution of these relationships amongst our beliefs and expectations. If the latter then we face a smaller burden in so far as we need only explain our adoption of certain inference rules, or novel information. In the multi-agent case we might insist on the defining the semantics of j:φ by quantifying over all the evidence functions. Enforcing the view that a justification exists in a multi-agent setting just when it is held by all the agents therein.
33
[5]
34In [7] the authors encode this information explicitly in the language by defining a primitive expression j
≫φ
which works to much the same effect, but is defined at both the syntactic-meta level and the semantic level
35
A logic is calledaxiomatically appropriatejust when each instance of its axiom schemas are justified.