This appendix explores the relationship between Yalcin (2007) andl Veltman’s (1996) semantics for epistemic modals. A modification of Yalcin’s semantics from Klinedinst and Rothschild (2012) is used as well as the standard modification of Veltman’s semantics discussed.

##### Basic assumptions

We assume again a language $$\mathcal{L}$$ syntax of propositional logic with modal operators for both semantics and a standard set of worlds. We write $${[\hspace{-.02in}[{a}]\hspace{-.02in}]}$$ for the worlds in which the atomic formula $$a$$ is true (what we wrote as $$I(a)$$ earlier). For the dynamic system we give a semantics that assigns to sentence an update on an arbitrary context $$c$$ (the update of $$\phi$$ on $$c$$ is written $$c[\phi]$$. For the static semantics we define a denotation function $${[\hspace{-.02in}[{\cdot}]\hspace{-.02in}]}$$ that gives relative to a world $$w$$ and a context $$s$$ (again a set of worlds) a truth-value (i.e. $${[\hspace{-.02in}[{\cdot}]\hspace{-.02in}]}: \mathcal L \times W \times \mathcal P (W) \to \{0,1\}$$). We then prove that both systems induce the same ‘updates’.

##### Modified Veltman semantics

$$c[a] = c \cap {[\hspace{-.02in}[{a}]\hspace{-.02in}]}$$

$$c[\lnot \phi] = c \backslash c[\phi]$$

$$c[\phi \land \psi] = (c[\phi])[\psi]$$

$$c[\phi \lor \psi] = c[\phi] \cup (c[\lnot \phi])[\psi]$$

$$c[\Box \phi] = c$$, if $$c[\phi] = c$$, otherwise $$\emptyset$$

$$c[\Diamond \phi] = c[\lnot \Box \lnot \phi]$$

##### Modified Yalcin Semantics

$${[\hspace{-.02in}[{A}]\hspace{-.02in}]}^{w, s}$$ is true iff $$w$$ is in $${[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}$$.

$${[\hspace{-.02in}[{\lnot \phi }]\hspace{-.02in}]}^{w,s}$$ is true iff $${[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}^{w,s}$$ is false.

$${[\hspace{-.02in}[{\phi \land \psi}]\hspace{-.02in}]}^{w,s}$$ is true iff $${[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}^{w,s}$$ is true and $${[\hspace{-.02in}[{\psi}]\hspace{-.02in}]}^{w,s{[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}}$$ is true.

$${[\hspace{-.02in}[{\phi \lor \psi}]\hspace{-.02in}]}$$ is true iff $${[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}^{w,s}$$ is true or $${[\hspace{-.02in}[{\psi}]\hspace{-.02in}]}^{w,s{[\hspace{-.02in}[{\lnot \phi}]\hspace{-.02in}]}}$$ is true.

$${[\hspace{-.02in}[{\Box \phi}]\hspace{-.02in}]}^{w,s}$$ is true iff $$\forall w' \in s$$ $${[\hspace{-.02in}[{ \phi}]\hspace{-.02in}]}^{w',s}$$ is true.

$${[\hspace{-.02in}[{\Diamond \phi}]\hspace{-.02in}]}^{w,s}$$ is true iff $${[\hspace{-.02in}[{\Box \lnot \phi}]\hspace{-.02in}]}$$ is false.

where $$c{[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}$$, the update of $$c$$ by $$\phi$$, is defined as follows:

$$c{[\hspace{-.02in}[{\phi}]\hspace{-.02in}]} = \{w \in c: {[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}^{w,c} \textrm{ is true}\}$$

##### Equivalence

The following limited equivalence statement may be made between the two semantics:

For any sentence $$\phi$$ any context $$c$$, and any world $$w$$ in $$c$$, $$w \in c[\phi]$$ iff $$w \in c{[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}$$.

Proof is by induction on formula complexity.

Base case

if $$\phi$$ is simple then the question is just whether $$w$$ is in its denotations for both semantics, so the equivalence is trivial.

Induction Step

Suppose for sentences $$\phi$$ and $$\psi$$, for any $$c$$ and $$w \in c$$, that $$w \in c[\phi]$$ iff $$w \in c{[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}$$.

By case:

Negation

$$w$$ in $$c[\lnot\phi]\\ \textrm{iff } w \in c \backslash c[\phi]\\ \textrm{iff } w \not \in c[\phi] \\ \textrm{iff } w \not \in c {[\hspace{-.02in}[{\phi}]\hspace{-.02in}]} \text{ (by induction hypothesis)} \\ \textrm{iff } {[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}^{w,c}\textrm{ is false}\\ \textrm{iff } {[\hspace{-.02in}[{\lnot \phi}]\hspace{-.02in}]}^{w, c}\textrm{ is true}\\ \textrm{iff } w \in c{[\hspace{-.02in}[{\lnot \phi}]\hspace{-.02in}]}$$

Conjunction

$$w \in c[\phi \land \psi] \\ \textrm{iff } w \in (c[\phi])[\psi] \\ \textrm{iff } w \in c[\phi] \text{ and } w\in (c[\phi])[\psi] \text{ (given monotonicity of dynamic updates)}\\ \textrm{iff } w \in c{[\hspace{-.02in}[{\phi}]\hspace{-.02in}]} \textrm{ and } w\in (c{[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}){[\hspace{-.02in}[{\psi}]\hspace{-.02in}]} \text{ (by induction hypothesis)}\\ \textrm{iff } {[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}^{w,c}\textrm{ is true}\textrm{ and } {[\hspace{-.02in}[{\psi}]\hspace{-.02in}]}^{w,c{[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}}\textrm{ is true} \\ \textrm{iff } c{[\hspace{-.02in}[{\phi \land \psi}]\hspace{-.02in}]}^{w,c}\textrm{ is true}\\ \textrm{iff } w \in (c{[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}){[\hspace{-.02in}[{\psi}]\hspace{-.02in}]}$$

Disjunction

Very similar to conjunction, so shortened here:
$$w \in c[\phi \lor\psi] \\ \textrm{iff } w \in c[\phi]\textrm{ or }w \in (c[\lnot \phi])[\psi] \\ \textrm{iff } {[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}^{w,c}\textrm{ is true or } {[\hspace{-.02in}[{\psi}]\hspace{-.02in}]}^{w,c{[\hspace{-.02in}[{\lnot \phi}]\hspace{-.02in}]}}\textrm{ is true} \\ \textrm{iff } w \in c{[\hspace{-.02in}[{\phi \lor \psi}]\hspace{-.02in}]}$$

Necessity Modal

$$w \in c[\Box \phi]\\ \textrm{iff } c[\phi] = c \\ \textrm{iff } \forall w \in c, w \in c[\phi] \\ \textrm{iff } \forall w \in c, w \in c{[\hspace{-.02in}[{\phi}]\hspace{-.02in}]} \\ \textrm{iff } \forall w \in c, {[\hspace{-.02in}[{\phi}]\hspace{-.02in}]}^{w,c}\textrm{ is true}\\ \textrm{iff } {[\hspace{-.02in}[{\Box \phi}]\hspace{-.02in}]}^{w,c}\textrm{ is true}\\ \textrm{iff } w \in c {[\hspace{-.02in}[{\Box \phi}]\hspace{-.02in}]}$$

Possibility Modal

Follows from previous proofs since defined in terms of necessity modal and negation.1

Klinedinst, Nathan, and Daniel Rothschild. 2012. “Connectives Without Truth-Tables.” Natural Language Semantics 20: 137–75.

Veltman, Frank. 1996. “Defaults in Update Semantics.” Journal of Philosophical Logic 25 (3): 221–61. doi:10.1007/BF00248150.

Yalcin, Seth. 2007. “Epistemic Modals.” Mind 116: 983–1026. doi:10.1093/mind/fzm983.

1. Thanks to Matt Mandelkern for a few corrections.