Assertion

Stalnaker represents what an utterance says as a matrix of truth values. A propositional concept takes each possible world, in its role as the context that determines what is said, to the proposition expressed there, so a concept is a character whose contexts are the worlds themselves. Its diagonal proposition, true at a world when what is said there is true there, is the character's diagonal. The dagger expresses the diagonal at every context. A sentence is a priori, true as uttered in every context, when its diagonal is necessary, which is what necessity applied to the dagger says. The dagger evaluates its argument at the context given by the world of evaluation, so it is a two-dimensional operator, one that no operation on propositions can match.

A context is represented by its context set, the worlds compatible with what the speaker presupposes, and the essential effect of an assertion is to eliminate the worlds of the context set incompatible with what is said. Three principles constrain assertion. What is asserted is true in some but not all worlds of the context set. What is said at each world of the context set has a truth value at each of them, so that a semantic presupposition is presupposed by the speaker. And what is said is the same within the context set, which is exactly what makes the essential effect independent of which world is actual. An utterance that violates the third principle is reinterpreted as asserting its diagonal. With rigid designators, That is either Zsa Zsa Gabor or Elizabeth Anscombe expresses a necessary truth or a necessary falsehood at each world, so that only its diagonal can be asserted, and the same reinterpretation makes Sherlock Holmes does not exist true at the actual world alone.

Under the third principle the essential effect is intersection with the diagonal proposition, and under the dagger it always is. Taking assertion as an eliminative and distributive update of context sets, the diagonal is forced: it is the only such update with the essential effect of what is said wherever the actual world is known, so that Stalnaker's reinterpretation is the update by the diagonal.

Implementation notes

• A propositional concept is a Kaplan.Character W W Prop, and its diagonal proposition is Kaplan.Character.diag id. The dagger is the substrate's Kaplan.shiftBy with the context shifted to the world of evaluation. The upside-down dagger of note 6, which projects the diagonal onto the vertical, is Kaplan.Character.dthat of the diagonal.
• The worlds i, j and k of each example are one type, World. Each example's matrix is derived from the facts the paper stipulates for it. Demonstratives, indexicals and terms whose reference is fixed by a description are Kaplan.Character.dthat, and proper names are constant characters, which is the paper's assumption that both are rigid designators.
• A truth-value gap, for the second principle, is the value 0 of a SignType-valued concept, as in Semantics/Presupposition/Basic.lean. A sentence that fails to express a proposition at a world has a row of gaps there.
• The third principle is stated with the clarification of note 13: what is said at two worlds of the context set need coincide only within the context set.
• The paper proposes the essential effect, not an update rule. That the diagonal is the only eliminative and distributive update agreeing with it where the actual world is known is this file's reconstruction, through DynamicSemantics.CCP.isClassical_iff_up_down_eq.
• Locators are to the 1978 printing.

Errata

• On p. 330, Sherlock Holmes does not exist, if the name does not refer, is said to express a necessarily false proposition, by the rule that the sentence is true at the worlds whose domain does not contain the person the name refers to. That rule makes it necessarily true, as the first row of matrix G, true throughout, shows (holmesDoesNotExist_diag). The 1999 reprint in Context and Content reads "necessarily true".

TODO

• Hesperus is identical with Phosphorus, it is now three o'clock and an ophthalmologist is an eye doctor (p. 328) are diagonalized because what the words say varies across the context set. Their concepts need a model of that variation.
• The counterfactual If Aristotle hadn't existed, the history of philosophy would have been very different from the way it was (p. 331) needs no diagonalization, since every context appropriate to it presupposes that Aristotle exists. Stating this needs a conditional over propositional concepts.
• Nondefective contexts, the effect of the fact that an utterance occurred, and rejection, which blocks only the essential effect (pp. 322–324), are not formalized.

References

• Stalnaker 1978, Assertion