MEANING AND SEMANTIC COMPOSITION
7. THE ‘GLUE’ LANGUAGE: LINEAR LOGIC
We use expressions of linear logic (Girard 1987) to give instructions on how to assemble meanings. Here, we informally describe only the properties of the small fragment of linear logic (the multiplicative fragment) that we will use.7
Intuitively, linear logic is different from classical logic in that premises in a linear logic deduction are treated as resources that must be kept track of, while this is not true in classical logic. Premises in a deduction in classical logic are statements about what is or is not true. In contrast, premises in a linear logic deduction are commodities, occurrences of resources that can be introduced or consumed.
To illustrate this difference, let us assume that we can deduce the statement You will get wet from the premises If it is raining outside, you will get wet and It is raining outside in classical logic:
(46) Classical logic:
If it is raining outside, you will get wet. It is raining outside.
You will get wet.
In classical logic, if a conclusion can be deduced from a set of premises, the same conclusion can still be deduced if additional premises are added:
(47) Classical logic:
If it is raining outside, you will get wet. It often rains in March.
It was raining yesterday. It is raining outside. You will get wet.
In contrast, linear logic does not allow the same conclusion to be deduced when additional premises are introduced. Instead, propositions in linear logic can be thought of as resources, and an economic metaphor is sometimes used.
For instance, we can use the symbol $1 for the proposition that you have a dollar,
applefor the linear logic proposition that if you have $1, you can get an apple, andapplefor the proposition that you have an apple. The following is valid in linear logic:
(48) apple apple This can be read as:
7In this section, we describe only the properties of the linear implication operator
. In Chapter 11 we introduce the multiplicative conjunction operator
, and in Chapter 13 we introduce the of course operator !. Proof rules for our fragment of linear logic are given in the appendix (page 433).
242 9. Meaning and Semantic Composition
(49) If you have$1, you can get an apple. You have$1.
You can get an apple.
Just as in the real world, it is not possible to get two apples with $1, or to still have $1 as well as the apple:
(50) INCORRECT (obtaining two apples with $1):
apple apple apple
INCORRECT (obtaining an apple while keeping $1):
apple apple
More schematically, inferences in linear logic work in the following way:
(51) INCORRECT: We cannot deduce from . A resource cannot be duplicated. INCORRECT: We cannot deduce from . A resource cannot be discarded. INCORRECT: The resource is consumed by to conclude . A resource is consumed by an implication.
INCORRECT: Both and
are consumed in concluding .
A linear implication is also a resource and is consumed in the deduction. CORRECT:
This resource-sensitivity of linear logic allows us to model the meaning contribu- tions of words as semantic resources that must be accounted for. The meaning of a sentence is deduced from the meanings of its component parts; it would be in- correct to deduce the same meaning for the sentence if words or phrases are added or subtracted. Each word or phrase makes a unique contribution that must be re- flected in the final meaning of the sentence, and meanings cannot be arbitrarily duplicated, added, or discarded.
7.1. Semantic Completeness and Coherence
Formally, we say that a meaning derivation for an utterance is semantically complete if a meaning derivation from the premises contributed by the meaning-
The ‘Glue’ Language: Linear Logic 243
bearing items in the sentence produces a meaning for the semantic structure for the utterance that does not contain any unsaturated expressions (that is, in which all of the meaning contribution requirements are satisfied). If no such meaning can be produced, some required material is missing and the utterance is semantically incomplete.
We say that a meaning derivation for an utterance is semantically coherent if the meaning derivation produces a meaning for the utterance with no additional unused premises remaining. If extra resources besides the semantic resource for the utterance remain, the utterance is semantically incoherent.
Semantic completeness and coherence are related in a clear way to the syntactic Completeness and Coherence conditions on f-structures discussed in Chapter 2, Section 3.6. This is as expected, since most syntactic arguments also make a semantic contribution and thus must be accounted for in a meaning derivation; indeed, our logically defined semantic completeness and coherence conditions subsume syntactic Completeness and Coherence in all cases except for pleonastic or semantically empty arguments, which make no semantic contribution and are not accounted for in a semantic derivation. The following sentence is syntactically and semantically incomplete:
(52) *Yawned.
The sentence is syntactically incomplete because the verb yawned requires aSUBJ,
and no subject is present; the sentence is semantically incomplete because the meaning constructor for yawned requires a semantic resource corresponding to its subject, but none can be found. Example (53) is both syntactically and seman- tically incoherent:
(53) *David yawned Chris.
This example is syntactically incoherent due to the presence of anOBJargument,
which yawned does not require. It is semantically incoherent because the meaning constructor for yawned requires only aSUBJresource, and in a meaning deduction for these premises the semantic resource for Chris remains unused.
Semantic and syntactic completeness differ for arguments that make no seman- tic contribution:
(54) *Rained.
The verb rained requires aSUBJ, but there is noSUBJin example (54); therefore, the sentence is syntactically incomplete. The semantic completeness condition is not violated, however, because theSUBJof rained is not required to make a
semantic contribution.
Another difference between syntactic and semantic coherence involves mod- ifying adjuncts: a semantic deduction in which the meaning contribution of a
244 9. Meaning and Semantic Composition
modifier is not incorporated is semantically incoherent, since all meanings must be taken into account. That is, the semantic coherence condition prevents us from assigning an unmodified meaning to a sentence with a modifier:
(55) David ran quickly. cannot mean:run
David
The modifier quickly is not constrained by the syntactic Completeness and Co- herence conditions, which apply only to governable grammatical functions. Se- mantically, however, its meaning contribution must be taken into account, and the deduction is semantically incoherent if the modifier meaning does not appear. 7.2. Glue Deductions and Meaning
Glue semantic deductions have an interesting property: as shown by Dalrymple et al. (1999a), whether or not a glue deduction is possible depends only on the linear logic glue formulas on the right-hand side of the meaning constructor, never on the meanings involved in the deduction. This means that we can think of the meaning deduction process purely in terms of the linear logic deduction over semantic structures; on the basis of the resulting deduction, we can determine the meaning of the resulting constituents by function abstraction and application.
For example, we can present deductions in an abbreviated form like (56), which is the same as the deduction in (37) of this chapter except that meaning terms have been omitted. On the basis of this deduction, we can determine the meaning corresponding to the semantic structure
by function application, following the function application rule presented in (35) of this chapter.
(56)
TheSUBJsemantic structure
is present.
If we find a resource for theSUBJsemantic structure
, we can produce a resource for the semantic structure for the full sentence
.
We have produced a semantic structure for the full sentence
.
In fact, as discussed by Dalrymple et al. (1999a), this aspect of glue semantic deductions is strongly similar to Categorial Grammar (Oehrle et al. 1988; Moort- gat 1988, 1996; Morrill 1994; Steedman 1996). Linguistic analysis in Categorial Grammar is a deductive process, in which the syntactic structure and the meaning of a sentence are obtained by a logical deduction from premises contributed by its words. The Lambek calculus (Lambek 1958), the logical system commonly used
Quantification 245
in syntactic analysis in categorial frameworks, is actually a fragment of noncom- mutative multiplicative linear logic and so is very close to the linear logic glue language.
Probably the most important difference between the categorial approach and the glue approach is in the syntactic primitives that are relevant for semantic composition. In categorial grammar, a predicate combines with its arguments on the basis of relations defined on the surface string, like to-the-left-of and to- the-right-of; in the glue approach, in contrast, semantic deductions are guided by f-structural relations likeSUBJ,COMP, andADJ. This frees the glue approach from concerns with crosslinguistically variable constituent structure relations and allows semantic composition to proceed according to the more abstract syntactic organization of f-structure.