Autumn 2020 Ling 5201 Syntax I 2: Syntax as deduction Robert Levine - - PowerPoint PPT Presentation

Autumn 2020 Ling 5201 Syntax I 2: Syntax as deduction

Robert Levine

Ohio State University levine.1@osu.edu

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Trees as proof histories?

◮ What is happening at each step in the tree in (1)?

(1)

S NP\S NP Bill (NP\S)/NP criticized NP Mary

◮ At the lowest point in the tree,

◮ a symbol of the form B/A (B = NP\S, A = NP) appears on the left

• f a symbol of the form A

◮ with the node dominating them labeled B.

◮ ‘Give me an A to my right and I’ll give you back a B’ meets ‘A’. . . ◮ with the result ‘B’.

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Proof histories, cont’d.

◮ At the next level up,

◮ a symbol of the form A\B (A = NP, B = S) appears on the right of a

symbol of the form A

◮ with the node dominating them labeled B.

◮ ‘Give me an A to my left and I’ll give you back a B’ meets ‘A’. . . ◮ with the result ‘B’.

The essential story

◮ a syntactic type defined as a guarantee to combine with a category of a different

type so that a particular category is the result. . .

◮ combines with that indicated type. . . ◮ . . . and the promised category does indeed result. ◮ This way of presenting how the combination of elements illustrated in (1) has a

strict analogue in elementary logic,

◮ and points the way to a very different interpretation of what kind of object (1)

actually is.

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(1): a reminder

(1)

S NP\S NP Bill (NP\S)/NP criticized NP Mary

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Syntax as deduction

◮ We can take (1) not to be a representation of the internal structure of the sentence, in the way that the PS trees we’ve seen are understood, ◮ but as a sequence of proof steps, where each node label working up from the bottom of the tree is a valid deduction, ◮ validated by the ancient rule Modus Ponens (lit. ‘the way to put [things] together’): (2) φ ⊃ ψ φ ψ ◮ In logical notation, φ ⊃ ψ asserts that the truth of one proposition (φ, the antecedent) implicates—is a guarantor of the truth of—a second proposition ψ (the consequent). ◮ So what (2) tells you is that when one true proposition validates the truth of a second, and you have a proof that the first one is true, you can infallibly conclude that the second

• ne is true as well.

◮ But we’re not talking about truth here. . . ◮ . . . we’re talking about valid descriptions of the syntactic types which correspond to the combination of other syntactic types. ◮ So what we have is not Modus Ponens in some version of propositional logic, ◮ but rather a strict analogue of Modus Ponens in a logic of syntactic types, ◮ where both / and \ correspond to the logical implication connective ⊃.

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A (very) simple type logic

◮ So now: given that we are considering a logical analogue to (some

subportion of) propositional logic, what is the analogue of (2)? (3) / Elim(ination) \ Elim(ination) X Y/X Y X\Y X Y

◮ Slashed categories combine with the element they’re ‘slashed for’ to

yield the promised syntactic type

◮ which is the type on the side from which the slash is falling away.

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But don’t we need more information than just the syntactic type?

◮ Something is definitely missing from the rules in (3), however.

◮ In particular: what’s missing from (3) so far as the tree we’ve already

set up, repeated here, is concerned?

(1)

S NP\S NP Bill (NP\S)/NP criticized NP Mary

◮ I’ve been talking about a slashed category ‘looking to the left’ or ‘looking to the

right’ for the right kind of category to combine with.

◮ But the rules given so far say nothing about the the actual words in the sentence.

◮ There is nothing in the rules to indicate about how the combination of the word in

(1) is pronounced.

◮ and there is nothing about the semantic result of combining the linguistic signs

described by these types, i.e., what the meaning of the combination of words from the the different constituents of the sentence is.

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A not-so-simple type logic

◮ A logic—any logic—is a system for drawing certain conclusions from

given premises, valid in every case.

◮ We’re used to thinking of logics as guides to reasoning to true

conclusions from true premises.

◮ But a logic isn’t necessarily about truth. ◮ What is critical is that we have a set of statements defining, from

specified hypotheses, the inevitable outcome of those hypotheses.

◮ Our logic is a logic of syntactic types, as propositional logic is a logic

• f propositions.

◮ But in propositional logic, the proof terms are simple formulæ, ◮ whereas our type logic, to be useful, must be a logic not only of types

(‘formulæ’) but of combinations of information—of sounds with sounds, and meanings with meanings–which make up the linguistic signs belonging to those types.

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Type logic, cont’d.

◮ Specifications of such information—commonly referred to in type logic

as labels—have their own (possibly quite complex) organization.

◮ They ‘go along for the ride’, so to speak, in the course of a deduction, ◮ so that the inference of a particular syntactic type on the basis of

prosodically and semantically labeled premises will simultaneously yield the prosodic and semantic labels for that deduced type.

◮ Or, in terms more familiar to the linguist, given a set of signs each of

which has a specified syntactic type,

◮ the prosody and meaning of the combination will be completely

determined.

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Rules of the labeled deduction system

◮ What do we want our rules to do? ◮ On the prosodic side,

◮ we want the combination of syntactic types to order the pronunciation

• f the word sequences corresponding to the combined types as dictated

by the direction of the slash,

◮ so that if Y/X combines with X, the result is a category Y whose

pronunciation consists of the prosodic string labeling Y/X followed by the string labeling X,

◮ and likewise for X\Y.

criticized; (NP\S)/NP bill; NP criticized • bill; NP\S criticized • bill; NP\S mary; NP mary • criticized • bill; S

◮ where a • b connects two phonological strings a,b by the concatenation

• perator.

◮ Now all we need is a set of rules that validate these proofs. . .

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