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Semantic Indexicality
Semantic Indexicality shows how a simple syntax can be combined with a propositional language at the level of logical analysis. It is the adoption of such a base language which has not been attempted before, and it is this which constitutes the originality of the book. Cresswell's simple and direct style makes this book accessible to a wider audience than the somewhat specialized subject matter might initially suggest.
230 pages, Hardcover
First published February 29, 1996
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September 27, 2024
Cressewell's strategy is to coin a formal semantics for indexicality, using chiefly to operators, Abs and Arg, to (respectively) fix and extract indices of evaluation.
Cresswell introduces a language L, in which every interpretation consists of a triple < W, D, V >.
• W is the class of all world-time pairs,
• D is the class of ‘things’ broadly construed as everything L quantifies over
• V is a function that takes expressions to their meaning
The meaning of every sentence, in turn, is a set of triples (= a proposition P) of the form < w, u, σ > where w ∈ W, u ∈ D and σ is a sequence such that when u = σ(0), the triple reduces to a pair < w, σ > ; σ(n) ∈ D otherwise, meaning that σ is only helpful to deal with n-place predicates (n > 1), i.e. when you need to parse who Fs who, what modifies what; finally, a codes the value of a sentence of L iff a ∈ P. If we take the example of the predicate admires, V(admires) would be a set of triples such that
< w, u, σ > ∈ a iff σ(1) admires σ(2) at w,
where σ(n) codes the value of admires for a member of D. Cresswell writes that the u of the P set of triples marks the ‘evaluation individual’, which indicates what is currently being abstracted on (9), yielding an unambiguous interpretation of, for example, ‘Everyone admires someone’. This means that a quantifier can be defined in L as an operator that ‘abstracts’ on the second term of the triple. Cresswell introduces a family of abstraction operators Absn, the semantics of which goes as follows:
V(Absn) is the function ω such that for a ∈ P, any < w, u, σ > ∈ ω (a) iff, if σ(n) = u, < w, u, σ > ∈ a
Hence, Absn admires is true of a given u relative to w and σ, iff u admires σ(n+1). This abstraction operator can help represent the difference between logical forms with quantifiers in two orders:
(1) ∀x∃y x admires y
(2) ∃y∀x x admires y
Give us, respectively, the fact that we’re either talking about
(1) Everyone Abs1 someone Abs2 admires (=> all the people that at least some one admires)
Or about
(2) Someone Abs2 everyone Abs1 admires (=> the one person that is admired by all the people)
Cresswell adds that we can think of Abs1 as a nominative case marker, and Abs2 as an accusative case marker (11). However, in order to account for passivization, we need a means of changing the order of the arguments of admires, so that (1) can be expanded to
(3) ∀x∃y x admires y and y admires x
What happens then is the index is ‘permuted’ (15).
The problem with all this is that Abs operators only bind variables one by one, and Cressewell claims that, in donkey anaphora cases, ‘we want operators which simultaneously bind a number of variables’ (17). But the first thing that Cresswell notes is that Geach’s (1962) donkey sentence, universally quantified
Every farmer who owns a donkey beats it
Can be existentially quantified by adding a temporal circumstancial, functioning as a Lewisian adverb of quantification:
A farmer who owns a donkey always beats it
Later, Cresswell introduces a shift operator (ch 3) that changes the default index of evaluation, so that ‘Dan’s mother’ could be used to refer to a mother other than Dan’s own biological one, like the mother he picked for a ‘mother of the year’ contest. Here, the index shift is equivalent to coining an ad hoc elaborately indexed Poss operator encoding the kind of relation between Dan and ‘his’ mother.
On QDR, Cresswell (ch 6) holds that, in ‘Everybody is present’, the semantics of ‘everybody’ tells us it universally quantifies over the intended set, but only the pragmatics tells us what set that is. More accurately, I think, the semantics of ‘everybody’ doesn’t quantify over the largest set (everybody in the world), as if by default (as Stanley & Szabo 2000 would say), but it quantifies over a domain of sets, which pragmatic theory qualifies as, according to Cresswell’s theory for instance, possibly intended sets. Pragmatics, then, broadly construed, gives us more than Cresswell thinks, namely 1) what that set is (an interpretation: like the set of first-year students in university U), and 2) what its domain is (a theory: like the Gricean theory that the domain encompasses possibly intended sets).
Domains and anaphora: I note that an UC can serve as a linguistic antecedent in an anaphora.
The last few chapters went a bit over my head, but some of the examples discussed were thought-provoking.
Cresswell introduces a language L, in which every interpretation consists of a triple < W, D, V >.
• W is the class of all world-time pairs,
• D is the class of ‘things’ broadly construed as everything L quantifies over
• V is a function that takes expressions to their meaning
The meaning of every sentence, in turn, is a set of triples (= a proposition P) of the form < w, u, σ > where w ∈ W, u ∈ D and σ is a sequence such that when u = σ(0), the triple reduces to a pair < w, σ > ; σ(n) ∈ D otherwise, meaning that σ is only helpful to deal with n-place predicates (n > 1), i.e. when you need to parse who Fs who, what modifies what; finally, a codes the value of a sentence of L iff a ∈ P. If we take the example of the predicate admires, V(admires) would be a set of triples such that
< w, u, σ > ∈ a iff σ(1) admires σ(2) at w,
where σ(n) codes the value of admires for a member of D. Cresswell writes that the u of the P set of triples marks the ‘evaluation individual’, which indicates what is currently being abstracted on (9), yielding an unambiguous interpretation of, for example, ‘Everyone admires someone’. This means that a quantifier can be defined in L as an operator that ‘abstracts’ on the second term of the triple. Cresswell introduces a family of abstraction operators Absn, the semantics of which goes as follows:
V(Absn) is the function ω such that for a ∈ P, any < w, u, σ > ∈ ω (a) iff, if σ(n) = u, < w, u, σ > ∈ a
Hence, Absn admires is true of a given u relative to w and σ, iff u admires σ(n+1). This abstraction operator can help represent the difference between logical forms with quantifiers in two orders:
(1) ∀x∃y x admires y
(2) ∃y∀x x admires y
Give us, respectively, the fact that we’re either talking about
(1) Everyone Abs1 someone Abs2 admires (=> all the people that at least some one admires)
Or about
(2) Someone Abs2 everyone Abs1 admires (=> the one person that is admired by all the people)
Cresswell adds that we can think of Abs1 as a nominative case marker, and Abs2 as an accusative case marker (11). However, in order to account for passivization, we need a means of changing the order of the arguments of admires, so that (1) can be expanded to
(3) ∀x∃y x admires y and y admires x
What happens then is the index is ‘permuted’ (15).
The problem with all this is that Abs operators only bind variables one by one, and Cressewell claims that, in donkey anaphora cases, ‘we want operators which simultaneously bind a number of variables’ (17). But the first thing that Cresswell notes is that Geach’s (1962) donkey sentence, universally quantified
Every farmer who owns a donkey beats it
Can be existentially quantified by adding a temporal circumstancial, functioning as a Lewisian adverb of quantification:
A farmer who owns a donkey always beats it
Later, Cresswell introduces a shift operator (ch 3) that changes the default index of evaluation, so that ‘Dan’s mother’ could be used to refer to a mother other than Dan’s own biological one, like the mother he picked for a ‘mother of the year’ contest. Here, the index shift is equivalent to coining an ad hoc elaborately indexed Poss operator encoding the kind of relation between Dan and ‘his’ mother.
On QDR, Cresswell (ch 6) holds that, in ‘Everybody is present’, the semantics of ‘everybody’ tells us it universally quantifies over the intended set, but only the pragmatics tells us what set that is. More accurately, I think, the semantics of ‘everybody’ doesn’t quantify over the largest set (everybody in the world), as if by default (as Stanley & Szabo 2000 would say), but it quantifies over a domain of sets, which pragmatic theory qualifies as, according to Cresswell’s theory for instance, possibly intended sets. Pragmatics, then, broadly construed, gives us more than Cresswell thinks, namely 1) what that set is (an interpretation: like the set of first-year students in university U), and 2) what its domain is (a theory: like the Gricean theory that the domain encompasses possibly intended sets).
Domains and anaphora: I note that an UC can serve as a linguistic antecedent in an anaphora.
The last few chapters went a bit over my head, but some of the examples discussed were thought-provoking.
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