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  Dynamic Contexts Heim and Dynamic Predicate Logic Veltman's Epistemic Modals Veltman/Yalcin comparison Generalized Quantifiers Quantifiers and Modals

• Veltman/Yalcin comparison
• Daniel Rothschild

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