𝑊 × 𝐹𝑜𝑟𝒞. We read 𝑖 ⊩𝒞 𝐴 as ‘𝐴 is true at 𝑖’.
When we want to explicitly refer to truth at a world in a particular CS+ model 𝔄, we shall employ the following notation: 𝔄, 𝑖 ⊩𝒞 𝐴 and 𝔄, 𝑖 ⊩𝒞Σ, as we have done for ⊩.
Given a CS+ model (𝑊, 𝒞, 𝑟, 𝜌) any 𝑖 ∈ 𝑊, and any 𝑐 ∈ 𝒞 define ⊩𝒞 as follows:
(1) 𝑖 ⊩𝒞 𝑝 iff 𝑝𝜌𝑖1 (2) 𝑖 ⊩𝒞 ~𝐴 iff not 𝑖 ⊩𝒞 𝐴 (3) 𝑖 ⊩𝒞 𝐴 ∧ 𝐵 iff 𝑖 ⊩𝒞 𝐴 and 𝑖 ⊩𝒞 𝐵 (4) 𝑖 ⊩𝒞 𝐴 ∨ 𝐵 iff 𝑖 ⊩𝒞 𝐴 or 𝑖 ⊩𝒞 𝐵 (5) 𝑖 ⊩𝒞 𝐴 ⊃ 𝐵 iff 𝑖 ⊩𝒞 ~𝐴 or 𝑖 ⊩𝒞 𝐵 (6) 𝑖 ⊩𝒞 □𝐴 iff ∀𝑗 ∈ 𝑊: 𝑗 ⊩𝒞 𝐴. (7) 𝑖 ⊩𝒞 ◊𝐴 iff ∃𝑗 ∈ 𝑊: 𝑗 ⊩𝒞 𝐴. (8) 𝑖 ⊩𝒞 𝐴 > 𝑐 𝐵 iff (𝑟𝑐, 𝜌), 𝑖 ⊩ 𝐴 >𝑐𝐵
What’s going on in (8)? The truth conditions for a formula (𝐴 >𝑐 𝐵), i.e. with an indexed
connective as the main connective, in a CS+ model are defined in terms of truth conditions for the corresponding non-indexed formula (𝐴 > 𝐵) in a CS model – a model based on the ordering frame that is the image of c under 𝑟 i.e. the ordering frame that is said to represent the context corresponding to the index of the indexed formula. This is how we formally capture the idea of indexed formulae being evaluated in contexts (represented by ordering frames) corresponding to the context index. Note that reference to a CS model is not required in any of the other clauses (1)-(8), since there are no formulae with an indexed connective as the main connective contained in the definienda of clauses (1)-(7).161
161 An alternative, and semantically equivalent formulation of the truth conditions for the contextualized language would be to have 𝑟 assigning comparative similarity assignments to worlds directly, relative to some context, i.e. to have 𝑟: 𝑊 × 𝒞 ⟶ ℘(𝑊) × ℘(𝑊 × 𝑊) be the function such that 𝑟(𝑤, 𝑐) is a comparative similarity assignment to world 𝑤, in context 𝑐. If we recall, this approach is closely aligned with Nolan’s suggestion, highlighted at the end of chapter 1. But I chose not to go this way, since we can accommodate the contextual variability in CS+ models by recycling the formalism already present in CS ordering frames. Recall from the definition of CS ordering frames, that we already have defined a function ≲: 𝑊 ⟶ ℘(𝑊) × ℘(𝑊 × 𝑊), which uniquely characterizes each ordering frame and whose image, for each world, consists of comparative similarity assignments being defined on ℘(𝑊) × ℘(𝑊 × 𝑊).
• As in the case of CS models, let’s introduce the following notation for convenience: 𝑖 ⊩𝒞 Σ iff 𝑖 ⊩𝒞 𝐴 for all 𝐴 ∈ Σ
• Also denote with 𝔄 ⊩𝒞 𝐴 when 𝔄, 𝑖 ⊩𝒞𝐴 for all 𝑖 ∈ 𝑊𝔄.
Note that it follows from the above definition that formulae whose index set ranges over more than one index may be evaluated on more than one CS model, e.g.
(𝑊, 𝒞, 𝑟, 𝜌), 𝑖 ⊩𝒞 (𝐴 >
𝑎𝐵) ∨ (𝐶 >𝑏 𝐷)
iff (𝑊, 𝒞, 𝑟, 𝜌), 𝑖 ⊩𝒞 (𝐴 >
𝑎𝐵) or (𝑊, 𝒞, 𝑟, 𝜌), 𝑖 ⊩𝒞 (𝐵 >𝑏𝐶)
iff (𝑟𝑎, 𝜌), 𝑖 ⊩ 𝐴 >𝑎𝐵 or (𝑟𝑏, 𝜌), 𝑖 ⊩ 𝐶 >𝑏𝐷
That is, (𝑟𝑎, 𝜌) and (𝑟𝑎, 𝜌) are CS models, by definition, and they need not be the same.
Just as we have relativized formula validity to a model 𝔄 ⊩𝒞 𝐴 it will be of use to define valid inference relativized to a model.
Definition 4.4.13.3: Let ⊨𝔄𝒞 ⊆ ℘(𝐹𝑜𝑟𝒞) × 𝐹𝑜𝑟𝒞, and given a CS+ model 𝔄 = (𝑊, 𝒞, 𝑟, 𝜌) write
• ⊨𝔄𝒞 𝐴 iff 𝔄 ⊩𝒞 𝐴
• Σ ⊨𝔄𝒞 𝐴 iff for all 𝑖 ∈ 𝑊: if 𝔄, 𝑖 ⊩𝒞 Σ, then 𝔄, 𝑖 ⊩𝒞𝐴.
Since each context set 𝒞 gives rise to a distinct language ℒ𝒞, and consequently a distinct set of wffs 𝐹𝑜𝑟𝒞, we need a semantic consequence relation for each language. The definition below is of semantic consequence for each ℒ𝒞. In most cases however, I’ll omit the
superscript 𝒞 unless the discussion will hinge on some specific property of the context set.
Definition 4.4.14: Given a set 𝒞 let ⊨𝐂𝐒+𝒞 ⊆ ℘(𝐹𝑜𝑟𝒞) × 𝐹𝑜𝑟𝒞, and define:
Σ ⊨𝐂𝐒+𝒞 𝐴 iff for all CS+ models 𝔄 and 𝑖 ∈ 𝑊: if 𝔄, 𝑖 ⊩𝒞 𝐵 for all 𝐵 ∈ Σ, then 𝔄, 𝑖 ⊩𝒞 𝐴.
We say an inference from Σ to 𝐴 is CS+ valid iff Σ ⊨𝐂𝐒+𝒞 𝐴. That is, valid inference is defined as truth preservation at all worlds in all CS+ models. A formula 𝐴 ∈ 𝐹𝑜𝑟𝒞 is said to be CS+
valid iff ∅ ⊨𝐂𝐒+𝒞 𝐴. Call this logic (schema) CS+.
I use the term ‘logic schema’ since if 𝒞 ≠ 𝒞′, in particular if |𝒞| ≠ |𝒞′|, then ⊨𝐂𝐒+𝒞 ≠ ⊨ 𝐂𝐒+
𝒞′ by
definition. For example, see Corollaries 4.9.1 and 4.9.2.
Note that it is immediate from the above definitions that ⊨𝐂𝐒+𝒞 ⊆ ⊨𝔄𝒞 , for any CS+ model 𝔄.
With the aid of the notation from Definition 4.4.13.3 we can express CS+ semantic consequence definition more succinctly: Σ ⊨𝐂𝐒+𝒞 𝐴 iff for all CS+ models 𝔄: Σ ⊨𝔄𝒞 𝐴.
Note that since the truth conditions for □ and ◊ formulae are defined in terms of unrestricted quantification over possible worlds, i.e. only >𝑐-formulae truth conditions depend on 𝒞 and r,
the above validity conditions give the modal logic S5 for the basic modal language. This allows us to formulate a more precise statement about a special case, when 𝒞 is empty.
Corollary 4.9.1: If 𝒞 = ∅, then Σ ⊨𝐂𝐒+𝒞 𝐴 iff Σ ⊨𝐒𝟓𝐴.
Proof : This follows immediately from the fact that if 𝒞 = ∅, then by Definition 4.4.1 ℒ𝒞 becomes {~, □,.◊,.∧,.∨, ⊃} ∪ {>𝑐 : 𝑐 ∈ ∅} = {~, □,.◊,.∧,.∨, ⊃}, i.e. the basic modal language. □
There is another special case with interesting properties, when 𝒞 is a singleton, which is expressed in Corollary 4.9.2 at the beginning of the next section, shortly after Theorem 4.9.
The part of the basic modal language is indistinguishable between the two classes of models in the following sense.
Lemma 4.6: For any CS+ model 𝔐 = (𝑊, 𝒞, 𝑟, 𝜌) and any 𝐴 ∈ 𝐹𝑜𝑟̅̅̅̅̅̅̅>𝒞, 𝐹 ∈ ℱ𝑊, 𝑖 ∈ 𝑊:
𝔐, 𝑖 ⊩𝒞 𝐴 iff (𝐹, 𝜌), 𝑖 ⊩ 𝐴
Proof : It suffices to note that elements of 𝐹𝑜𝑟̅̅̅̅̅̅> depend only on 𝑊 and 𝜌, which are the same
for 𝔐 and (𝐹, 𝜌), by definition. □
Theorem 4.7: If Σ ∪ {𝐴} ⊆ 𝐹𝑜𝑟̅̅̅̅̅̅̅>𝒞: then Σ ⊨𝐂𝐒 𝐴 iff Σ ⊨𝐂𝐒+𝐴.
Proof : Immediate from Lemma 4.6. □
Definition 4.4.15: Call frame 𝐻 ∈ 𝐂𝐒 a mutual refinement of frames 𝐹 and 𝐺 iff (𝐹, 𝐻) ∈ ℛ and (𝐺, 𝐻) ∈ ℛ. Note that 𝐻 is a mutual refinement of 𝐹 and 𝐺 iff 𝐻 ∈ ℛ[𝐹] ∩ ℛ[𝐺].162
It will be worthwhile (useful later) emphasizing a relatively obvious, yet important fact.
Lemma 4.8: If (𝑊, ≲) = 𝐹 ∈ 𝐂𝐒, then ℛ[𝐹] ⊆ ℱ𝑊.
Proof : Immediate from definition of ℱ𝑊 and the fact that refinements preserve domains. □
4.4.4 Results
Much of Lewis’ analysis is preserved on this account. This occurs when the premises and conclusion of an inference are confined to a single context. This makes sense intuitively, and the semantics manages to align with our intuition in this regard.
Theorem 4.9: For all Σ ∪ {𝐴} ⊆ 𝐹𝑜𝑟𝒞: