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 ⊩}𝒞_{𝐴 for all 𝑖 ∈ 𝑊}𝔄_{. }

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, (𝑟_{𝑎}, 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 **⊨_{𝔄}𝒞 ⊆ ℘(𝐹𝑜𝑟𝒞_{) × 𝐹𝑜𝑟}𝒞_{, and given a CS+ model 𝔄 = (𝑊, 𝒞, 𝑟, 𝜌) write }

• ⊨_{𝔄}𝒞 𝐴 iff 𝔄 ⊩𝒞 _{𝐴 }

• Σ ⊨_{𝔄}𝒞 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 ∈ 𝐹𝑜𝑟**𝒞_{ 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 ℒ*𝒞
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] ⊆ ℱ_{𝑊}.

*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: