Update Semantics

In this section of the thesis we will consider the attempt to modeling natural language indicative conditionals as presented by Update Semantics [44]. In contrast to the accounts discussed above, the update semantic account is dynamic, i.e., it is no longer concerned solely with truth-preservation, but rather it is focused on the notion of information change and update. More specifically, the meaning of a sentence is no longer associated with its truth conditions, but it is an operation on information states, where information states are intuitively what an agent takes to be true. The difference between previous logics and the current logic is well reflected in the slogan of Update Semantics, namely: “You know the meaning of a sentence if you know the change it brings in the information state of anyone who accepts the news conveyed by it” ([44], pp. 1). As will be demonstrated in Section 2.3, Update Semantics gives rise to a system that allows one to account for some of the problems of the horse-shoe analysis of implication. Moreover, it is useful to keep in mind that this system is closely related to Inquisitive Semantics, which, to a certain extent, can be viewed as a static counterpart of this dynamic system. This section is based on [44] and [43].

Definition 36 (Language). Let P be a finite set of propositional letters. We denote by LP the set of formulas built up from letters in P using the connectives ¬,∨,∧,→,_♦.

Definition 37 Let W be the powerset of P. Then: 1) σ is an information state iff σ⊆W

2) For every two states σ, τ, σ+τ :=σ∩τ

Definition 38 The semantics for US is given recursively in the following way: σ[p] =σ∩ {w∈W|p∈w} σ[¬θ] =σ\σ[θ] σ[θ∧ψ] =σ[θ]∩σ[ψ] σ[θ∨ψ] =σ[θ]∪σ[ψ] σ[_♦ θ] =σ if σ[θ]6=∅ σ[_♦ θ] =∅ if σ[θ] =∅ σ[θ→ψ] =σ if σ[θ][ψ] =σ[θ] σ[θ→ψ] =∅ if σ[θ][ψ]6=σ[θ]

Definition 39 (Support) A state σ supports a sentenceθ , σ |=θ, iff σ[θ] =

σ, where σ[θ] is an update of a state σ with information θ.

Definition 40 (Validity) An argument is valid iff updating a state σ with premises ψ1,· · · , ψn, yields an information state in which the conclusionθ is supported, i.e., ψ1,· · ·ψn|=θ iff ∀σ, σ[ψ1]· · ·[ψn]|=θ.

As there are significant differences between Update Semantics and the other systems we discussed, it is important to examine Update Semantics in greater detail. Notice that states σ can be viewed just as sets of possible worlds, whereas for a sentence θ an operation σ[θ] is a result of updating an information state σwith information encoded byθ. According to this frame- work, the support of a sentence θ at a stateσ (i.e., σ[θ] =σ) is equivalent to accepting θ in an information state σ. This can be thought of as reflecting the intuition that, if we already accept a sentenceθ in our information state, then the update of this state with information θ does not change what we take to be true. The update clauses given by the semantics of US define how σ changes when somebody in a state σ accepts a sentenceθ. Hence, for instance, updating an information stateσwith a sentence¬θis equivalent to removing from σ all the possible worlds s.t. θ holds at them. Notice that in this framework, the semantics for all connectives apart from the conditional and ‘_♦’ are dynamic, i.e., after their acceptance conversational participant modifies his information state. As for the conditional and ‘_♦’, these are to be treated as consistency tests: by accepting them, a conversational partic- ipant only verifies if they hold, but does not update his information state. So to speak, the sentence involving might and the conditional correspond to performing a test on σ, rather than introducing some information to our information state.

The main benefit of pursuing update semantics is that it gives an intuitive and desirable account of our natural language as a dynamic process, it is not, however, the only one. Another key benefit of US relates to the Ramsey test5_.

For notice that, in order to verify if the conditional θ →ψ holds, one checks if after updating his information state with θ, ψ holds. This is in line with the Ramsey Test, in which when one verifies the truth of a conditional, he appends θ to the set of his beliefs and checks if it is such that ψ holds in it.

5_{The Ramsey test is one of the first and most influential methods suggested to analyse} conditionals. It defines a procedure for verifying whether an indicative conditional holds. As originally stated by Ramsey in his 1929 footnote: “If two people are arguing ‘If A will C?’ and are both in doubt as to A, they are adding A hypothetically to their stock of knowledge and arguing on this basis about C... We can say they are fixing their degrees of

Indeed, the links between the update framework and the nature of natural language implication are visible in the analysis in Chapter 4.

Update Semantics allows us to account for 5 out of the 16 paradoxical inferences considered. Namely, inferences (8) |= (p → q)∨(q → p), (12) (p∧q)→s|= (p→s)∨(q→s), (13) (p→q)∧(s→t)|= (p→t)∨(s→q), (14) ¬(p → q) |= p and (15) ¬(p → q) |= ¬q. In all of these inferences, it is the modeling of implication as a test on states that can be attributed to be the key semantic feature that allows us to account for them. To explicate this point consider the counter-model for (10):

Let σ = {w1, w2}, |p| = {w1}, |q| = {w2}. Then, it follows that

σ[p] ={w1} 6=σ[p][q] =∅. Hence, by the definition of ‘→’, it follows that

σ[p→q] =∅. Similarly, it follows that σ[q] ={w2} 6=σ[q][p] =∅. Hence, by

the definition of ‘→’, it follows that σ[q → p] = ∅. Thus, it follows by the definition of ‘∨’, that σ[(p → q)∨(q → p)] = ∅ 6= σ. Thus, it follows that

6|= (p→q)∨(q→p), as required.

It is also worth to consider the counter-model for the inference (14): Let σ = {w1, w2}, |p| = {w1} and |q| = {w2}. Then, it follows by the

definition of ‘¬’ and ‘→’ thatσ[¬(p→q)] =σ\σ[p→q] =σ\∅=σ. Notice, however that σ[p] = {w1} 6= σ. Hence, it follows that σ[¬(p → q)] 6|= p.

Thus,¬(p→q)6|=p. N.B. this example also demonstrates that that there is a close relation between ‘_♦’ and ‘→’. This is because when we test if a state

σ supports ¬(p→q), in practice we also verify whether σ is consistent with

p∧ ¬q.

In similar fashion the Update Semantic modeling of implication allows us to account for (12), (13) and (15).

Unfortunately, the semantic features of US do not allow one to account for all the problematic inferences. Importantly, out of the systems considered it is the only system that validates (1) p|=q → p, (2)¬q |= q →p and (8)

p → (q → p). Let us consider the proofs for (1) and (2) to see how the semantic features of US validate these inferences.

Consider a proof by contradiction to see why (1) holds. Letσbe arbitrary. Suppose for contradiction that ∃σ s.t. σ[p] 6|= p → q. Then it follows that

σ[p][q][p]6=σ[p][q] . This is a contradiction, since it follows by the definition of an update with an atomic sentence that for any state σ supporting an atomic sentence p,σ =σ[p]. Thus, it follows that p|=q →p.

Similarly consider the following proof to see why (2) holds. Let σ be arbitrary. Suppose for contradiction that ∃σ s.t. σ[¬p] 6|= σ[p][q]. Then, it follows that σ[¬p][p][q] 6= σ[¬p][p] ?. Notice that it follows by the support definition for atomic sentence and negation thatσ |= [p][¬p] iffσ =∅. Hence, by ?∅ 6=∅[q] =∅ . Thus, it follows that ¬p|=p→q.