# Logical Form

# LOGICAL FORM

One can use sentences to present arguments, some of which are valid. Sentences are complex linguistic expressions that exhibit grammatical structure. And the grammatical properties of sentences need not be obvious. As discussed in this entry, certain arguments seem to be valid because the relevant premises and conclusions exhibit nonobvious logical structure. But this raises questions about what logical structure is and how it is related to grammatical structure.

## Patterns of Reasoning

An ancient thought is that premises and conclusions have parts and that valid arguments exhibit valid forms, like the following: **Q** if **P** , and **P** ; so **Q** . One can say that the variables (in bold) range over propositions, leaving it open for now what propositions are: sentences of some (perhaps unspoken) language, abstract states of affairs, or whatever. One can also assume that declarative sentences can be used, in contexts, to indicate or express propositions. But each sentence of English is presumably distinct from the potential premise/conclusion indicated with that sentence in a given context. Different speakers can use *I swam today* at different times to indicate various propositions, each of which could be expressed in other languages. Nonetheless, propositions seem to be sentence-like in some respects, especially with regard to being composite.

The conclusion of (1)

(1) Chris swam if Pat swam, and Pat swam; so Chris swam.

is evidently part of the first premise, which has the second premise as another part. But simple propositions, without propositional parts, also seem to have structure. Aristotelian schemata like the following are valid: Every *P* is *D*, and every *S* is a *P* ; so every *S* is *D*. The italicized variables are intended to range over predicates—logical analogs of nouns, adjectives, and other classificatory terms (like *politician, deceitful*, and *senator* ). Simple propositions appear to have subject-predicate structure; where a subject can consist of a predicate and a quantifier (indicated with a word like *every, some*, or *no* ).

Medieval logicians explored the hypothesis that all propositions are composed of simple propositions and a few special elements, indicated with words like *or* and *only*. While they expected some differences between grammatical and propositional structure, the idea was that sentences reflect the important aspects of logical form. The medieval logicians also made great strides in reducing Aristotelian schemata to more basic inferential principles: one concerning replacement of a predicate with a less restrictive predicate, as in *Rex is a brown dog, so Rex is a dog* ; and one concerning converse examples, like *Rex is not a dog, so Rex is not a brown dog*.

Nonetheless, traditional logic/grammar was inadequate. If Juliet kissed Romeo, then Juliet kissed someone. And predicates containing quantifiers were problematic. If *respects some doctor* and *respects some senator* indicate nonrelational proposition-parts, like *is tall* and *is ugly*, then the argument indicated with (2)

(2) Some patient respects some doctor, and every doctor is a senator; so some patient respects some senator.

has the following form, which is not valid: Some *P* is *T*, and every *D* is an *S* ; so some *P* is *U*. One can introduce a variable *R* ranging over relations and offer schemata like the following: Some *P* **R** some *D*, and every *D* is an *S* ; so some *P* **R** some *S*. But this is not a basic inference pattern; and such schemata do not capture the validity of inferences like the following: Every patient who met every doctor is tall, and some patient who met every doctor respects every senator; so some patient who respects every senator is tall. Relative clauses posed difficulties as well. If sentence (3) is true, so is sentence (4):

- (3) Every patient respects some doctor.
- (4) Every old patient respects some doctor.

But in (5) and (6) the direction of valid inference is reversed:

(5) No lawyer who saw every patient respects some doctor.

(6) No lawyer who saw every old patient respects some doctor.

## Functions and Arguments

Gottlob Frege showed how to deal with these examples and more. But on his view, propositions have function-argument structure. Let *S* stand for the successor function. Frege interpreted the arithmetic expression *S* (3) as having a semantic value: the value of the relevant function given the relevant argument; that is, the number four. The division function can be represented as a mapping from ordered pairs of numbers to quotients: Q(x, y) = x/y. Functions can also be specified conditionally; consider the function that maps every even integer onto itself, and every odd integer onto its successor. On Frege's view, *Mary sang* indicates a proposition with the following structure: Sang(Mary). And he took the relevant function to be a conditional mapping from individuals in a given domain to truth values: Sang(x) = **t** if x sang, and **f** otherwise; where for each individual x, Sang(x) = **t** if and only if (iff) x sang, and Sang(x) = **f** iff x did not sing. The proposition that John admired Mary, like the proposition that Mary was admired by John, was said to have the following structure: Admired(John, Mary); where Admired(x, y) = **t** if x admired y, and **f** otherwise.

Frege's treatment of quantification departed more radically from tradition. Let F be the function indicated by *sang*, so that someone sang iff some individual x is such that F(x) = **t** . Using modern notation, someone sang iff ∃x[Sang(x)]; where the quantifier binds the variable. Every individual in the domain sang iff F maps each individual onto **t** ; in modern notation, ∃x[Sang(x)]. With regard to the proposition that some politician is deceitful, subject-predicate grammar suggests the division *Some politician / is deceitful*. But for Frege the logically important division is between the existential quantifier and the rest, with the quantifier binding two occurrences of its variable: ∃x[P(x) & D(x)]; some individual is both a politician and deceitful. Likewise with regard to the proposition that every politician is deceitful: ∀x[P(x)→ D(x)]; everyone is such that if he or she is a politician then he or she is deceitful. In which case, *every politician* does not indicate a constituent of the proposition. Grammar also masks a logical difference between the existential and universal propositions: predicates are related conjunctively in the former, but conditionally in the latter.

The real power of Frege's logic is most evident in his discussion of how the proposition that every number has a successor is logically related to more basic arithmetic truths. But just consider the following analyses of (3a–6a):

(3a) ∀x{P(x)→ ∃y[D(y) & R(x,y)]}

(4a) ∀x{[O(x) & P(x)]→ ∃y[D(y) & R(x,y)]}

(5a) ¬∃x{Lx & ∀y[P(y)→ S(x,y)] & ∃z[D(z) & R(x,z)]}

(6a) ¬∃x{Lx & ∀y{[O(y) & P(y)]→ S(x,y)} & ∃z[D(z) & R(x,z)]}

Given Frege's rules of inference, (3a) implies (4a), while (5a) follows from (6a). Frege concluded that natural language is not suited to the task of representing propositions perspicuously. On his view, premises/conclusions have function-argument structure, which is often masked in natural language. But one can try to invent languages whose sentences depict true propositional structure.

Frege originally took propositional constituents to be the relevant functions and (ordered n-tuples of) entities that such functions map to truth-values. But he later refined this view, taking the sense of an expression to be a mode of presentation of the corresponding semantic value. Frege identified propositions—or what he called thoughts (*Gedanken* )—with senses of sentences in an ideal language, which allowed him to distinguish the proposition that Hesperus is bright from the proposition that Phosphorus is bright. Thus, Frege could deny that the inference *Hesperus is Hesperus, so Hesperus is Phosphorus* is an instance of the valid form *P, so P*.

## Descriptions and Mismatch

One might think that the logical form of any proposition indicated with *The boy from Canada sang* is *Sang(b)*, where *b* stands for the individual in question. But this makes elements of the description logically irrelevant. And if the boy from Canada sang, then a boy sang. Moreover, *the* implies uniqueness (at least within a context). So Bertrand Russell (1919) held that a proposition expressed with *The boy sang* has the following structure: ∃x{Boy(x) & ∀y[Boy(y)→ y = x] & Sang(x)}; where the middle conjunct is one way, among many, of expressing uniqueness. According to Russell, even if a speaker refers to a certain boy when saying *The boy sang*, that boy is not a constituent of the indicated proposition—which has the form of an existential quantification, as opposed to a function saturated by the boy. In this respect, *the boy* is like *some boy*. Though on Russell's view, not even *the* indicates a propositional constituent. This extended Frege's idea that natural language is misleading, while letting Russell account for the meaningfulness of descriptions that describe nothing.

Let **Frank** be the proposition indicated (now) with *The (present) king of France is bald*. If **Frank** consists of some function saturated by an entity indicated with *The king of France*, there must *be* such an entity. But instead of appealing to nonexistent kings, or ways of presenting them, Russell held that **Frank** is of the form ∃x{K(x) & ∀y[K(y)→ y = x] & B(x)}. In which case, the true negation of **Frank** is not of the form ∃x{K(x) & ∀y[K(y)→ y = x] & ¬B(x)}. This invited the thought, developed by Ludwig Wittgenstein (1922, 1953) and others, that many philosophical puzzles might dissolve if one properly understood the logical forms of one's claims. Russell also held that one bears a special relation to constituents of propositions one can entertain and that one typically does not bear this relation to the individuals one refers to with names. This led Russell to say that names are disguised descriptions. On this view, *Hesperus* is associated with a complex predicate—say, for illustration, of the form E(x) & S(x). Then *Hesperus is bright* indicates a proposition of the form ∃x{[E(x) & S(x)] & ∀y{[E(y) & S(y)]→ y = x]} & B(x)}. It follows that Hesperus exists iff ∃x[E(x) & S(x)]; and this was challenged by Saul Kripke (1980). But Russell could say that "Phosphorus is bright" indicates a proposition of the form ∃x{[M(x) & S(x)] & ∀y{[M(y) & S(y)]→ y = x]} & B(x); where E(x) and M(x) indicate different functions, specified in terms of evenings and mornings, leaving room to discover that E(x) & S(x) and M(x) & S(x) both indicate functions that map Venus alone to the truth-value **t** .

Positing unexpected logical forms thus had payoffs. But if mismatches between sentential and propositional structure are severe, one wonders how one manages to indicate propositions. This worry was exacerbated by increasing suspicion that talk of propositions is (at best) a way of talking about how one should regiment one's verbal behavior for purposes of scientific inquiry and that one should regiment natural language in first-order predicate calculus. From this perspective, associated with Willard Van Orman Quine (1950), mismatches between logical and grammatical form are to be expected. Another strand of thought, inspired by Wittgenstein's later work, also suggested that a single sentence could be used (on different occasions) to express different kinds of propositions. Peter Strawson (1950) argued, contra Russell, that a speaker could use an instance of *The F is G* to express a singular proposition about the F in the context at hand. Keith Donnellan (1966) contended that a speaker could even use an instance of *The F is G* to express a singular proposition about an individual that is not an F. Various considerations suggested that relations between spoken sentences and propositions are at best very complex and mediated by speakers' intentions.

With hindsight, though, one can see that the divergence between logical and grammatical form was exaggerated. Consider again the proposed regimentation of the proposition indicated with *Some boy sang* : ∃x[Boy(x) & Sang(x)]. With restricted quantifiers, one can offer another logical paraphrase that parallels the grammatical division between *some boy* and *sang*. Let ∃x:Boy(x) be an existential quantifier that binds a variable ranging over boys in the domain. Then ∃x:Boy(x)[Sang(x)] means that for some individual x such that x is a boy, x sang. Likewise, ∀x:[Tall(x) & Boy(x)]{Sang(x)} is logically equivalent to ∀x{[Tall(x) & Boy(x)]→ Sang(x)}. And ∃x:[Boy(x) & ∀y:Boy(y)[x = y]{Sang(x)} means that for some boy x such that x is identical with every boy, x sang. Richard Montague (1974) offered a similar rewrite of Russell's hypothesis about the logical form of *The boy sang*. On this view, *The boy* corresponds to a propositional constituent, even though the boy referred to (if such there be) does not.

Still, the subject-predicate structure of *Mary trusts every doctor* diverges from the function-argument structure of ∀y:Doctor(y)[Trusts(Mary, y)]. Grammatically, *trusts* and *every doctor* form a phrase; though logically, *trusts* combines with *Mary* and a variable to form a complex predicate that in turn combines with a restricted quantifier. Given Montague's (1974) techniques, one can provide algorithms that systematically associate quantificational sentences of natural language (described in subject-predicate terms) with Fregean propositional structures. But it seemed that mismatches between grammatical and logical form remained, at least in cases of complex predicates with quantificational constituents.

## Transformational Grammar and lf

One must not, however, assume a naive conception of grammar when thinking about its relation to logic. For example, the grammatical form of a sentence need not be determined by the order of its words. Using brackets to indicate phrasal structure, one can distinguish sentence (7) from the homophonous sentence (8).

- (7) {Mary [saw [the [boy [with binoculars]}
- (8) {Mary [[saw [the boy] [with binoculars]}

The direct object of (7) is *the boy with binoculars*, while in (8), *saw the boy* is modified by an adverbial phrase. And a leading idea of modern linguistics is that many grammatical structures are transformations of others.

Expressions often appear to be displaced from positions canonically associated with certain grammatical relations. In (9), *who* seems to be associated with the direct-object position of *saw*.

- (9) Mary wondered who John saw

And (9) can be glossed as *Mary wondered which person is such that John saw him*. This invites the hypothesis that the structure of (9) is as shown in (9-SS), reflecting a transformation of the simpler expression shown in (9-DS):

- (9-SS) {Mary [wondered [who
_{i}{John [saw ( _ )_{i}]}]} - (9-DS) {Mary [wondered {John [saw who]}]}

where coindexing indicates a grammatical relation between the coindexed positions. The idea was that each sentence has a surface structure and a deep structure and that the former will differ from the latter when expressions like *who* are displaced as in (9). As an illustration of the kind of data relevant to such hypotheses about grammar, note that (10–12) are perfectly fine sentences, while (13) is not:

- (10) The boy who sang was happy
- (11) Was the boy who sang happy
- (12) The boy who was happy sang
- (13) Was the boy who happy sang

The ill-formedness of (13) is striking, since one can ask whether or not the boy who was happy sang. This suggests that (11-SS) is the result of a permissible transformation, but (13-SS) is not:

- (11-SS) Was
_{i}{[the [boy [who sang] [ ( _ )_{i}happy]} - (13-DS) Was
_{i}{[the [boy [who [ ( _ )_{i}happy] sang}

As transformational grammars were elaborated, many linguists posited another level of grammatical structure—LF, intimating logical form—obtained by displacing quantificational expressions. In particular, it was proposed that structures like (14-SS) were transformed, as in (14-LF):

- (14-SS) {Pat [trusts [every doctor]}
- (14-LF) {[every doctor]
_{i}{Pat [ trusts ( _ )_{i}]}}

Clearly, (15-LF) does not reflect the pronounced word order in English. But there is independent evidence for covert (inaudible) quantifier-raising in natural language. The suggestion was that each sentence has a PF (intimating phonological form) that determines pronunciation, and an LF that determines interpretation. On this view, the scope of a quantifier must be determined at LF, as in (14-LF). And one can say this, while also saying that the pronunciation of *Pat trusts every doctor* reflects the untransformed surface structure (14-SS). Many apparent examples of grammar-logic mismatches were thus rediagnosed as mismatches between different aspects of grammatical structure. This preserves the idea that surface appearances are often misleading with regard to propositional structure. But it also suggests that grammatical form and logical form converge, once one moves beyond traditional subject-predicate conceptions of structure with regard to both logic and grammar. And further simplification may be possible.

Given a conception of grammar according to which each sentence has a PF and an LF, perhaps involving different transformations, it is not obvious that one needs to posit other levels of grammatical analysis. Each expression of a natural language may just be a PF-LF pair that can be generated in accordance with certain constraints on how expressions can be combined and transformed. One can hypothesize that a sentence like (9) is formed in stages, including stages like those depicted in (9-DS) and (9-SS), without saying that any one stage is special in ways that deep structure and surface structure were said to be. On this view, (10–12) correspond to natural ways of associating a PF with an LF, but the string of words in (13) does not. From this perspective, urged by Noam Chomsky and others, talk of PFs and LFs need not be understood in terms of interlevel transformations (Chomsky 1995, Hornstein 1995). Rather, PFs and LFs can be viewed simply as generable linguistic structures that reflect pronunciation and meaning. In which case questions about grammatical form and linguistic meaning are largely questions about LFs.

Nonetheless, there is still an important conceptual distinction between the linguist's notion of LF and the logician's notion of logical form. The LF of a sentence may, in various ways, underdetermine the structure of the proposition a speaker expresses with that sentence (in a given context). The LF may, however, provide a scaffolding that can be elaborated in particular contexts, with little or no mismatch between basic sentential and propositional structure. These issues remain unsettled. But discoveries of rich grammatical structure reinvigorated the idea that natural languages are semantically compositional.

Prima facie, *Every tall sailor respects some doctor* and *Some short boy likes every politician* exhibit common modes of linguistic combination. So a natural hypothesis is that the meaning of each sentence is somehow fixed by these modes of combination, given the word meanings. Inspired by Alfred Tarski's development of Frege in 1956, Donald Davidson (1967) conjectured that there are recursively specifiable theories of truth for natural languages. And while there are many apparent objections, the conjecture has been fruitful. This raises the possibility that talk of logical forms should be construed in terms of the structure(s) that speakers impose on words to understand natural language systematically. From this tendentious perspective, the phenomenon of valid inference would be largely a reflection of semantic compositionality.

At this point, many issues become germane. Given any sentence of natural language, one can ask interesting questions about its grammatical structure and what it can be used to say. (Modal claims and propositional attitude reports have been studied intensively.) It is not obvious how one should characterize meanings or logical relations. (Are theories of meaning theories of truth? Which valid inferences, if any, cannot be captured in first-order terms?) The role of context is large and ill understood. But it seems clear that the traditional questions—what kinds of structures do propositions and sentences exhibit, and how do thinkers who also speak relate these structures—must be addressed in terms of increasingly sophisticated conceptions of logic and grammar.

** See also ** Events in Semantic Theory; Modality and Quantification; Semantics; Syntax.

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*Paul M. Pietroski (2005)*

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