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Hardegree, Compositional , Chapter 4: Expanded Categorial 1 of 18 Chapter 4 Expanded

Chapter 4 Expanded Categorial Grammar ...... 1 A. Expanded Categorial ...... 2 1. Introduction ...... 2 2. What is Categorial Logic?...... 2 3. Classical Logic ...... 3 4. Intuitionistic Logic...... 3 5. No Monotonic Logic Properly Models Grammatical-Composition...... 4 6. Relevance Logic...... 4 7. Without Identity ...... 5 8. Multi-Linear Logic...... 6 1. Formal Presentation of System ML ...... 6 1. Arrow-Fragment ...... 6 2. Syntax...... 6 3. Derivation System...... 6 4. Definition of Derivation...... 7 5. Assumptions...... 7 6. Inference-Rules ...... 7 7. Cross-Arrow-Fragment of System L ...... 8 8. Syntax...... 8 9. Derivation System [Added Rules] ...... 8 10. Examples of Derivations...... 9 B. Expanded Categorial Semantics...... 11 1. Semantic-Composition...... 11 2. General Composition Rule...... 11 2. Structural Rules...... 11 1. Definition of Type-Logical Derivation...... 11 2. Assumption-Rule ...... 12 3. Lambda-Out ( λO) [-application]...... 12 4. Lambda-In ( λI) [function-generation]...... 12 5. Cross-Out ( ×O) ...... 12 6. Cross-In ( ×I)...... 12 7. Sub-Structural (Index) Rules ...... 12 8. Lambda-Calculus ...... 12 3. Extra Composition Rules ...... 13 4. Examples of Semantic Derivations ...... 13 1. Simple Function-Composition (Transitivity) ...... 13 2. Schönfinkel's Transform...... 14 3. Montague's Transform ...... 15 4. Generalized-Conjunction ...... 15 5. Fail...... 16 6. Appendix – Summary of Systems Considered ...... 18

Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 2 of 18 A. Expanded Categorial Syntax 1. Introduction In the previous chapter we observed that many sentences do not readily submit to the grammatical analysis we developed in the previous two chapters. In response to this, we propose to enlarge the rules of composition. In particular, we propose Categorial Logic , which is intended to be a calculus of grammatical- composition ,1 including syntactic-composition and semantic-composition. 2 We call it a "logic" because, in characterizing it, we utilize logical formalism, and we draw inspiration from existing logical systems – including classical logic, intuitionistic logic, relevance logic, and linear logic. According to standard categorial-grammar, grammatical-composition consists (almost) exclusively in function-application , depicted as follows.

φ a function from A into B AB α an item of type A A φ(α) an item of type B B

If we think of arrow as the logical if-then connective, then this procedure corresponds to the following form, if A then B A B which is the inference-principle known as modus ponens. Whereas standard categorial-grammar admits only one mode of composition, corresponding to modus ponens , expanded categorial-grammar admits infinitely-many modes of composition, each one corresponding to a valid-inference of categorial logic. This principle is officially presented as follows.

Categorial Logic Principle A A Where 0 , …, k are types. A A A 1, …, k combine to form 0 precisely if A A ⊢ A 1 , …, k 0 Here, ‘ ⊢’ is a meta-linguistic corresponding to logical entailment. 3 2. What is Categorial Logic? The obvious question then is – what logical system best models grammatical-composition? In this connection, there are several prominent extant logical systems that serve as candidates, including Classical Logic Intuitionistic Logic Relevance Logic Linear Logic which we examine in the next few sections.

1 The idea of a calculus of (pure) syntactic-composition traces to Joachim Lambek's "The Mathematics of Sentence Structure", American Mathematical Monthly , 65 (1958), 154–170. 2 Here, by syntax, we actually mean just the syntax-semantics interface. In particular, our syntactic-rules mimic our semantic-rules in being completely indifferent to the left-right structure of parse-trees. This is in marked to other approaches to categorial grammar that are more narrowly syntactic in orientation. 3 Logicians use various turnstile-symbols, including ‘ ⊢’ and ‘ ⊨’, to depict various manners of judging entailment; usually ‘ ⊢’ pertains to proof-theoretic entailment, ‘ ⊨’ pertains to model-theoretic entailment. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 3 of 18 3. Classical Logic Since it is the strongest logic of the group, 4 Classical Logic authorizes the maximum number of compositions, but unfortunately it also authorizes compositions that are grammatically implausible. For example, the following is a principle of Classical Logic. 5 (c1) A ; B ⊢ A →B Accordingly, using CL to judge grammatical-composition, the following composition-rule is authorized. S ; S ⊢ SS We read this saying that one may combine two sentences ( S) to form a sentential-adverb ( S- operator). Since this is grammatically highly implausible, categorial logic must reject (c1). In an important sense, to be explained shortly, (c1) follows from the following inference principle, which is also valid in Classical Logic. (c2) B ⊢ A →B This is an inference principle found especially obnoxious by many pioneers of alternative logical systems, including the strict-entailment logics of C.I. Lewis 6 and the relevant-entailment logics of Anderson and Belnap. 7 Grammatically-understood, (c2) is equally obnoxious, since the following instance S ⊢ SS authorizes transforming 8 a sentence into a sentential-adverb ( S-operator), which seems grammatically implausible. In addition to (c1) and (c2), the following CL-principles also yield implausible composition principles. (c3) (A →B) →B ⊢ (B →A) →A (c4) (A →B) →A ⊢ A In light of numerous examples of inadmissible grammatical-compositions based on CL- principles, we conclude that Classical Logic does not properly model grammatical composition. 4. Intuitionistic Logic Intuitionist Logic is weaker than Classical Logic – every argument deemed valid by IL is also deemed valid by CL, but not conversely. For example, (c3) and (c4) above are rejected by IL. On the other hand, both (c1) and (c2) are accepted by IL. Since the latter yield grammatically- implausible compositions, so we conclude that Intuitionistic Logic also does not properly model grammatical composition.

4 If a weaker logic validates an argument form , then a stronger logic also validates , granting the two logics both pass judgment on . 5 Although nothing hinges on this, we use one arrow-symbol ‘→’ for logic, and another arrow-symbol ‘ ’ for . 6 C.I. Lewis, ‘Implication and the Algebra of Logic’, Mind N.S., 21 (1912), 522-31. 7 Alan Anderson and Nuel Belnap, Entailment , Princeton University Press, 1975. 8 A transformation corresponds to a single-premise argument. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 4 of 18 5. No Monotonic Logic Properly Models Grammatical-Composition A principle that is fundamental to nearly every logical system that has been considered over the past few millennia is the principle of monotonicity .9 The basic idea is that adding premises to a valid argument does not result in an invalid argument. The following meta-logical principle is a special case of monotonicity. 10 B ⊢ C ⇒ A ; B ⊢ C In other words, if C follows from B, then C also follows from A and B. Next, suppose we also grant the following identity principle B ⊢ B which is generally regarded as a minimum requirement of any formal system that presumes to model reasoning. Putting monotonicity and identity together, we obtain the following simplification principle .

A ; B ⊢ B Suppose we include this principle in the logic of grammatical-composition. Then the following composition is authorized. D ; S ⊢ S In the proposed composition, we compose a sentence out of a definite-noun-phrase ( D) and a sentence ( S) – presumably by simply discarding the DNP. However, it seems manifestly plausible that a valid grammatical-composition must meaningfully utilize all its inputs in constructing its output. Accordingly, in order for a logic to model grammatical-composition, it cannot contain the simplification principle, and so it cannot be monotonic. In what follows, we examine two prominent non-monotonic logics – Relevance Logic, and Linear Logic. 6. Relevance Logic As its name suggests, Relevance Logic is characterized by sensitivity to matters of relevance – in particular, between premises and conclusions of arguments, and between antecedents and consequents of conditional (if-then) statements. In particular, for Relevance Logic, a conditional statement cannot be true unless there is a connection between the antecedent and the consequent, and an argument cannot be valid unless there is a connection between the premises and the conclusion. Most well-known systems of logic do not satisfy either of these desiderata . At the same time, it seems that relevance desiderata are tailor-made for modeling grammatical-composition. For example, it is presumed that a grammatical functor actually uses its input in generating its output, and it is also presumed that a grammatical-composition uses all its input in producing its output. Relevance Logic does a very decent job of modeling grammatical-composition, but it also authorizes several problematic compositions, which we now review. 1. Contraction A→(A →B) ⊢ A →B If we use this to model grammatical-composition, then we obtain the following grammatical principle D(DS) ⊢ DS according to which a transitive- automatically transforms into an intransitive verb, which seems implausible.

9 This usage comes from the definition of monotonic (increasing) functions. Specifically, a function φ is said to be monotonic precisely if φ(x)φ(y) whenever xy. In the of logical systems, the relevant order-relation is set- inclusion, and the relevant function is the consequence function , defined so that (Γ) { α | Γ⊢α}. Then a logical system is monotonic precisely if its associated consequence-function is monotonic. 10 Here, we use the symbol ‘⇒’ as the meta-language's if-then connective. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 5 of 18 2. Duplication A ⊢ A ×A Here, × is the multiplicative-counterpart of →, which we examine in more detail in Section 1.7. If we use this to model grammatical-composition, then we obtain the following grammatical principle. D ⊢ D×D This amounts to saying that a DNP can duplicate itself (repeatedly) and accordingly serve as the input for an unlimited number of functors (e.g., VPs), which seems implausible. 3. Law of Assertion (A →A) →B ⊢ B If we use this to model grammatical-composition, then we obtain the following grammatical principle, where C is the type of common-noun-phrases. (C C) (C C) ⊢ CC From this, we obtain the following composition principle. (C C) (C C) ; C ⊢ C A modifier is a phrase of type AA, where A is any type. For example, a common-noun modifier () is a phrase of type CC, and an adjective-modifier is a phrase of type (C C) (C C). According to the grammatical principle proposed above, an adjective-modifier like ‘very ’ can be combined with a common-noun like ‘dog ’ to produce a common-noun ‘very dog ’, which seems implausible. 4. Tautology A tautology is a formula that is logically-true, alternatively a formula that follows from nothing. 11 Every logic has valid argument forms, but not every logic has tautologies. Relevance logic has tautologies, the simplest of which is depicted in the following principle, which we call ‘Tautology’. ⊢ A→A Note that Assertion follows from Tautology, so the latter must be rejected by categorial logic insofar as the former is rejected. Also, it is implausible to suppose that a phrase can simply enter a semantic-derivation "out of thin air". 7. Linear Logic Without Identity We next consider an even weaker logical system, Linear Logic, which is founded on the idea of resource-usage. 12 In particular, whereas Relevance Logic requires that every supposition be used at least once , Linear Logic requires every supposition to be used exactly once . This adjustment gets rid of Contraction and Duplication, but it does not get rid of Assertion or Tautology. To accomplish this, we also ban arguments with no premises. This restriction makes sense from a grammatical point of view, since we do not want phrases entering a grammatical-construction "out of thin air". The resulting system is sometimes called Linear Logic without Identity. By moving to Linear Logic without Identity, we get rid of many inference principles that make no sense grammatically, but unfortunately we also get rid of inference principles that seem desirable, including the following. (1) A→B ; A →C ⊢ A →(B ×C) [Conditional Multiplication] (2) A→(B →C) ; A →B ⊢ A →C [Conditional Modus Ponens] (3) A→B ; A →C ; (B ×C) →D ⊢ A→D [Generalized-Conjunction] Note that (1) entails the other two in System L, so it is the crux.

11 When no formulas are in front of ⊢, it is understood that the premise set is empty. 12 Linear Logic originates with Jean-Yves Girard, "Linear Logic", Theoretical Computer Science , 50:1, pp. 1-102, 1987. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 6 of 18 8. Multi-Linear Logic As it turns out, we later expand our logical system to include infinitary-categories, and in this expanded system, Conditional-Multiplication is derivable, and so is the Conjunction rule. But, in the meantime, since these rules are very useful, we will go ahead and use them as ad hoc rules. Thus, as it stands, our rules of composition have two modules. (1) Structural Rules Linear Logic Without Identity (2) Ad Hoc Rules Conjunction Conditional Multiplication **Note: one obtains Relevance Logic from Multi-Linear Logic by adding Identity.** 1. Formal Presentation of System ML [Multi-Linear Logic] 1. Arrow-Fragment We begin with the arrow-fragment of ML, which is to say the fragment of ML that pertains exclusively to the conditional operator ‘ →’, which is the starting point in many formal treatments of logic. 2. Syntax Since it involves only one connective, the syntax is very easy to describe. (1) every atomic formula is a formula; (2) if A and B are formulas, then so is ( A→B); (3) nothing else is a formula. 3. Derivation System 1. Preliminaries For us, a derivation is a structure of the following form. line-number formula index annotation

L1 Φ1 I1 A1

L2 Φ2 I2 A2 …

Lk Φk Ik Ak

A line's number indicates its location in the derivation, which is used to the formula in the annotation-column. A line's annotation indicates how that line is justified according to the rules of inference. A line's index basically lists the suppositions (premises, assumptions) on which the line depends. So, for example, the following derivation line

(7) P 1,2 3,5, MP indicates that formula ‘P’, which is line 7, follows from lines 3 and 5 by the inference-rule modus ponens , and depends upon suppositions 1 and 2. 13

13 There are two reasonable ways of indexing suppositions; (1) one can simply use the supposition's line-number as its index. (2) one can number suppositions independently of formulas. We employ the latter approach. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 7 of 18 2. Indices Indices encode sub-structural 14 information, and are officially defined as follows. (1) every non-empty sequence of numerals is an index; (2) nothing else is an index. Indices form an algebra under the operation of sequence-addition, +, which satisfies various algebraic principles depending upon the specific logical system. In System L, the principles are as follows, where i, j, k are indices. (1) i+( j+k) = ( i+j)+ k + is associative (2) i+j = j+i + is commutative 4. Definition of Derivation

Where Φ0, …, Φk are formulas, a derivation of Φ0 from { Φ1, …, Φk} is a sequence of lines as follows. line number formula index annotation

L1 Φ1 1 Pr … … … …

Lk Φk k Pr … … … …

L0 Φ0 1, … , k …

In particular:

(1) the first k-many lines are the premises – Φ1, …, Φk, each one indexed by the line-number k. (2) every remaining line is either (1) an assumption, or (2) follows from previous lines by an inference-rule.

(3) the last line is Φ0, which depends upon all and only the premises. 5. Assumptions There are two kinds of suppositions allowed in derivations – premises, and assumptions. Whereas premises correspond to the input expressions of a grammatical-composition, assumptions arise in sub-derivations in connection with various inference-rules (see below). The following schematically exhibits the assumption-insertion rule.

line-number formula index annotation L Φ m (new) As

Here, Φ is any formula, and m is any numeral that is new, which is to say it does not occur earlier in the derivation. 6. Inference-Rules We follow a derivation scheme according to which every logical-operator is characterized by two rules – a construction-rule and a deconstruction-rule. In the literature, these are generally known as introduction-rules and an elimination-rules; we prefer the more succinct terms ‘in’ and ‘out’.

14 See Greg Restall, An Introduction to Substructural Logics , Routledge, 2000. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 8 of 18

1. Arrow-Out ( →O)

L1 A→B i

L2 A j

L3 B i+j L1, L2, →O 2. Arrow-In ( →I)

L1 A j

L2 B i+j

L3 A→B i L1, L2, →I

Here, i and j are indices (sequences of numerals), and i+j is the sequential-sum of i and j. Whereas arrow-out is just modus ponens , arrow-in corresponds to the following key principle about conditionals. A A A ⊢ B A A ⊢ A B 1 ; … ; k ; ⇒ 1 ; … ; k → Note that, in arrow-in, most derivations introduce A by way of the assumption-rule. 7. Cross-Arrow-Fragment of System L A conditional connective → is said to be residuated 15 precisely if there is an associated multiplication-operator × satisfying the following residual law. 16 (A ×B) →C ⊣⊢ A →(B →C) For example, in Classical and Intuitionistic Logic, the multiplication-operator is simply logical-and . A×B = A&B On the other hand, in Relevance Logic 17 and Quantum Logic 18 , multiplication corresponds to "compossibility", which is defined as follows. 19 A×B ∼(B →∼A) [ ≠ A&B] 8. Syntax (1) every atomic formula is a formula; (2) if A and B are formulas, then so is ( A→B); (3) if A and B are formulas, then so is ( A×B); (4) nothing else is a formula. 9. Derivation System [Added Rules] 1. Cross-In ( ×I)

L1 A i

L2 B j

L3 A × B i+j L1, L2, ×I

15 See T.S. Blyth and M.F. Janowitz, Residuation Theory , Pergamon Press, Oxford, 1972. 16 One can also call this a division principle; to see why, rewrite A →B as B/A. Then the residual law amounts to the following. A/(B ×C) = (A/B)/C A divided by (B times C) equals (A divided by B) divided by C However, the word ‘residual’ pertains to subtraction, in which case the residual law is written thus. A−(B+C) = (A −B) −C 17 Dunn, The Algebra of Intensional Logics , PhD Dissertation, University of Pittsburgh, 1966. 18 Hardegree, "Material Implication in Orthomodular (and Boolean) Lattices", Notre Dame Journal of Formal Logic , 22 (1981), 163-82. 19 The order doesn't matter for relevance logic, in which × is commutative, but it does matter for quantum logic, in which × is not commutative. Also note that, in relevance logic, × is referred to as "fusion", and is usually written using ‘ ◦’. See note 17. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 9 of 18

2. Cross-Out ( ×O)

L1 A×B h

L2 A i

L3 B j

L4 C i+j+k

L5 C h+k L1, L2-L4, ×O This rules states that, if one has a cross-formula A×B, and one has a sub-derivation of C from { A,B}, then one can infer C. See example below. 10. Examples of Derivations In what follows, we write sequences without commas. 20 We also take associativity and commutativity for granted, which allows us to write all indices in simple numerical order. 1. Transitivity B→C ; A →B ⊢ A →C (1) B→C 1 Pr (2) A→B 2 Pr (3) A 3 As (4) B 23 2,3, →O (5) C 123 1,4, →O (6) A→C 12 3,5, →I 2. Permutation A→(B →C) ⊢ B →(A →C) (1) A→(B →C) 1 Pr (2) B 2 As (3) A 3 As (4) B→C 13 1,3, →O (5) C 123 2,4, →O (6) A→C 12 3,5, →I (7) B→(A →C) 1 2,6, →I 3. Secondary Modus Ponens A→(B →C) ; B ⊢ A →C (1) A→(B →C) 1 Pr (2) B 2 Pr (3) A 3 As (4) B→C 13 1,3, →O (5) C 123 2,4, →O (6) A→C 12 3,5, →I 4. Lifting [Montague's Law 21 ] A ⊢ (A →B) →B (1) A 1 Pr (2) A→B 2 As (3) B 12 1,3, →O (4) (A →B) →B 1 2,4, →I

20 Except when this practice conflicts with the usual numeral morphology; for example, we may need to distinguish the numeral ‘23’ (twenty-three) from the sequence ‘2,3’ (two, three). 21 This particular transformational principle is so called because one instance of it [ D ⊢ (D →S) →S] corresponds to Montague's proposal to treat DNPs, including proper-nouns, as second-order predicates. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 10 of 18 5. Inflection 22 (A →C) →D ; A →B ⊢ (B →C) →D (1) (A →C) →D 1 Pr (2) A→B 2 Pr (3) B→C 3 As (4) A 4 As (5) B 24 2,4, →O (6) C 234 3,5, →O (7) A→C 23 4,6, →I (8) D 123 1,7, →O (9) (B →C) →D 12 3,8, →I 6. Permutivity (A →C) →D ; A →(B →C) ⊢ B →D (1) (A →C) →D 1 Pr (2) A→(B →C) 2 Pr (3) B 3 As (4) A 4 As (5) B→C 24 2,4, →O (6) C 234 3,5, →O (7) A→C 23 4,6, →I (8) D 123 1,7, →O (9) B→D 12 3,8, →I 7. Addition B →C ; A ×B ⊢ A ×C (1) B→C 1 Pr (2) A×B 2 Pr (3) A 3 As (4) B 4 As (5) C 14 1,4, →O (6) A×C 134 3,5, ×I (7) A×C 12 2,3-6, ×O 8. Residual Law [Schönfinkel's Law 23 ] (A ×B) →C ⊣⊢ A →(B →C) ⊢ (1) (A ×B) →C 1 Pr (2) A 2 As (3) B 3 As (4) A×B 23 2,3, ×I (5) C 123 1,4, →O (6) B→C 12 3,5, →I (7) A→(B →C) 1 2,6, →I ⊣ (1) A→(B →C) 1 Pr (2) A×B 2 As (3) A 3 As (4) B 4 As (5) B→C 13 1,3, →O (6) C 134 4,5, →O (7) C 12 2,3-6, ×O (8) (A ×B) →C 1 2,4, →I

22 We call this principle ‘inflection’ because it authorizes numerous compositions involving case-inflection, which we introduce in a later chapter. 23 Named after Moses Schönfinkel ["Uber die Bausteine der mathematischen Logik", Math. Ann. 92 (1924), 305-316], who first proposed that one can treat a two-place function as a family of one-place functions. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 11 of 18 B. Expanded Categorial Semantics 1. Semantic-Composition Having dealt with type-composition, we now turn to semantic-composition. 24 2. General Composition Rule

ℇ1, …, ℇk compose to form ℇ0 precisely if there is a

derivation of ℇ0 from { ℇ1, …, ℇk} in Categorial-Logic .25

Here, ℇ0, …, ℇk are Loglish expressions, which provide semantic-values (translations) for various phrases in natural language. Categorial-Logic consists of four main components. (1) Structural Rules – Linear Logic without Identity (2) Lambda-Calculus Rules (3) Extra Rules 2. Structural Rules 1. Definition of Type-Logical Derivation

A derivation of ℇ0 from { ℇ1, …, ℇk } is a sequence of lines as follows. line expression type index annotation number

(1) ℇ1 ℑ(ℇ1) 1 Pr … … … … …

(k) ℇk ℑ(ℇk) k Pr … … … … …

? ℇ0 ℑ(ℇ0) 1+…+ k ? In particular:

(1) the first k lines are the premises – ℇ1, …, ℇk.

(2) the last line is ℇ0. (3) every remaining line is either (1) an assumption, or (2) follows from previous lines by an inference-rule. (4) ℑ(α) is the type of α (5) indices are sequences of numerals; + is sequential-sum; for example 〈3,2,1 〉 + 〈5,4,3,2 〉 = 〈3,2,1,5,4,3,2 〉.

24 The connection between proof-theory and the lambda-calculus traces to Haskel Curry and William Alvin Howard , who proposed what is known as the Curry-Howard Isomorphism. See, e.g., W. Howard. “The formulae-as-types notion of construction”, in Seldin and Hindley (eds), To H.B. Curry: Essays on , and Formalism . Academic Press Limited, 1980, 479-490.. Our own calculus is closely allied with the compositional- calculus implicit in the CH-isomorphism, although our lambda-calculus is bigger and our categorial logic is smaller. 25 As it turns out, the number of premises cannot be zero, since the logic we employ is Linear Logic Without Identity , which has no tautologies. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 12 of 18 2. Assumption-Rule

L α ℑ(α) m ( new ) As Here, α any open expression every variable of which is new, which is to say it does not occur unbound earlier in the derivation, and m is any numeral that is new, which is to say it does not occur earlier in the derivation.

3. Lambda-Out ( λO) [function-application]

L1 φ AC i …

L2 α A j …

L3 [φ]〈α〉 C i+j L1, L2, λO Here, φ is any function. 4. Lambda-In ( λI) [function-generation]

L1 α A i …

L2 Ω B i+j …

L3 λ:α:Ω AB i L1, L2, λI 5. Cross-Out ( ×O)

L0 ℇ1 × ℇ2 A×B i … 26 L1-L2 any sub-derivation of ℇ3 from { ℇ1,ℇ2}

L4 ℇ3 C i+j L0 L1-L2, ×O

6. Cross-In ( ×I)

L1 ℇ1 A i …

L2 ℇ2 B j …

L3 ℇ1 × ℇ2 A×B i+j L1, L2, ×I

7. Sub-Structural (Index) Rules (1) (i + j) + k = i + ( j + k) + is associative (2) i + j = j + i + is commutative These conditions correspond to the sub-structural logic known as Linear Logic without Identity .27 8. Lambda-Calculus 1. Lambda-Conversion ( λC)

[λνℇ]〈σ〉 // ℇ[σ/ν] ν is any variable. ℇ is any expression. σ is any expression of the same type as v; ℇ[σ/ν] results from substituting σ for every occurrence of ν that is free in ℇ for σ. This schema is understood as a bi-directional rule, licensing inter-substitution of the flanking expressions in all contexts.

26 A sub-derivation of γ from { α,β} is like a conditional derivation in sentential logic; in particular, α and β are posited as assumptions [requiring new indices]. 27 One obtains "identity" [tautologies] by permitting an empty index ∅ [ ∅+i=i]. One obtains Relevance Logic, by adding contraction: i + i = i Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 13 of 18 2. Alphabetic Variance (AV)

ℇ1 // ℇ2

Here, ℇ1 , ℇ2 are expressions that result from each other by bound-variable permutation. In other words, there is a permutation (1–1 function) π on the class V of variables, whose inverse is π–1, such that:

ℇ1 results by substituting π(u) for every bound occurrence of u in ℇ2, and –1 ℇ2 results by substituting π (v) for every bound occurrence of v in ℇ1. An occurrence of a variable ν is bound precisely if that occurrence lies within the of an operator ν – i.e., ∀ν, ∃ν, ν, λν. Otherwise, that occurrence is free. A variable ν is free in an expression ℇ precisely at least one occurrence of ν in ℇ is free in ℇ.

3. Extra Composition Rules 28 1. Conjunction (VVV) 29

1 k 1 1 k L1 λv …λv Φ A …A S i … 1 k 2 1 k L2 λv …λv Φ A …A S j … 1 k 1 λv …λv { Φ & 1 L , L , L A …AkS i+j 1 2 3 Φ2} Conj. v1,… vk is any sequence (possibly null) of variables of any types, and Φ1 and Φ2 are formulas. 2. Conditional-Multiplication

1 k 1 1 k L1 λv …λv ℇ A …A B i … 1 k 2 1 k L2 λv …λv ℇ A …A C j … 1 k λv …λv 1 L , L , C- L A …Ak.B×C i+j 1 2 3 {ℇ1×ℇ2} M v1,… vk is any sequence of variables, of any types, and ℇ1 and ℇ2 are expressions of any types.

4. Examples of Semantic Derivations In the next few sections, we provide examples of derivations illustrating how semantic- composition operates within Categorial Logic. 1. Simple Function-Composition (Transitivity) In set theory, if one has two functions F : BC i.e., F is a function from B into C G : AB i.e., G is a function from A into C one can compose F and G into a composite-function, F◦G, from A into C, defined (implicitly) as follows. [F◦G]( x) = F(G(x)) From the viewpoint of categorial logic, this is simply an instance of the logical principle of transitivity. In particular, the following re-constructs function-composition using categorial logic.

28 These rules are ad hoc at present, but later they turn out to be derivable in our expanded categorial scheme. 29 ‘VVV’ is short for ‘veni vidi vici’ [I came, I saw, I conquered] – said by Julius Caesar reporting his campaign in Gaul. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 14 of 18

(1) F BC 1 Pr

(2) G AB 2 Pr (3) x A 3 As

(4) G(x) [G]〈x〉 B 23 2,3,λO

(5) F(G(x)) [F]〈[G]〈x〉〉 C 123 1,4,λO

(6) λx F(G(x)) λx [F]〈[G]〈x〉〉 AC 12 3,5, λI Notice that the resulting function is exactly what the received set-theoretic definition entails. 2. Schönfinkel's Transform In 1924 [ 30 ], Moses Schönfinkel proposed that a two-place function F : ( A×B)C which is a function from the Cartesian-product A×B into C, can be transformed into a "family" of one-place functions, F* : A(BC) the latter being a function from A into the set BC of functions from B into C. The converse transformation is also admissible. From the viewpoint of categorial logic, these transformations correspond to the following categorial equivalence, which we duly call Schönfinkel's Law. (A×B)C ⊣⊢ A(BC) The following derivations reconstruct these two transformations.

(1) F (A×B)C 1 Pr (2) x A 2 As (3) y B 3 As (4) x × y A×B 23 2,3, ×I (5) F(x × y) C 123 1,4, λO (6) λy F(x × y) BC 12 3,5, λI (7) λx λy F(x × y) A(BC) 1 2,6, λI

Note that if α and β are type-theoretic items, then the product α×β is just their un -ordered pair

(1) F A(BC) 1 Pr (2) x×y A×B 2 As (3) x A 3 As (2a) (4) y B 4 As (2b)

(5) F(x) BC 13 1,3,λO

(6) [F(x)]( y) C 134 4,5, λO

(7) [F(x)]( y) C 12 2,2-6, ×O

(8) λ(x×y) [F(x)]( y) (A×B)C 12 2,7, λI

Note that the resulting item λ(x×y) φ(x)( y) involves an expanded lambda-abstract; in this particular case, it is a function that takes an un-ordered pair x×y as input.

30 Moses Schönfinkel, "Uber die Bausteine der mathematischen Logik" [Math. Ann. 92 (1924), 305-316]. Schönfinkel is known in some circles as the "father of the combinator" (http://www.cis.upenn.edu/~steedman/moses.html). Given its historical priority, Heim and Kratzer ( Semantics in ) propose the term ‘Schönfinkelization’ for this technique, in place of the "Brontosaurian" term ‘currying’ (after Haskel Curry). Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 15 of 18 3. Montague's Transform In 1973 [ 31 ], proposed that we treat both proper-names and - phrases as second-order predicates of the following type. (D S) S This involves a novel approach to QPs, but also a novel approach to proper-names, according to which a name such as ‘ Kay ’ does not denote an individual (entity), but rather a set of properties, in this case the set of all properties instantiated by the individual ostensibly denoted by ‘ Kay ’. From the viewpoint of categorial logic, this maneuver transforms an item of type D (entity) into an item of type (D S) S, in accordance with the following logical principle, which we call Montague's Law. A ⊢ ( A→B)→B The following derivation shows how categorial logic reconstructs Montague's transform.

(1) K D 1 Pr (2) P DS 2 As (3) PK S 12 1,2, λO (4) λP:PK (D S) S 1 2,3, λI

Notice that this transforms the individual K into the function λP:P( K), which takes a predicate P and yields the result of applying P to K. 4. Generalized-Conjunction In 1983 [ 32 ], Partee and Rooth proposed that an operator of type [S×S]S can be promoted to an operator of type, [( AS) × ( AS)] ( AS) where A is any type. Furthermore, by recursion, one can promote the latter to a function of type [( B(AS)) × ( B(AS))] ( B(AS)) where B is any type, and so forth. This procedure corresponds to what mathematicians call the formation of function-spaces ; for example, if one can add numbers, then one can, by , "add" number-valued functions in accordance with the following (implicit) definition. (F+G)( x) F(x) + G(x) Each such transform corresponds to a valid inference of categorial logic. For example, the following derivation illustrates how categorial logic enables one to combine two transitive with ‘and ’. The following derivation, which involves two predicates, shows how GC follows from Categorial Logic. For the sake of simplifying what rules we need, we convert ‘ and ’ into its Schönfinkel form. and = λΦλΨ{Φ & Ψ}

31 Richard Montague, “The proper treatment of quantification in ordinary English”, In J. Hintikka et al., editors, Approaches to Natural Language , pages 221-242. Reidel, 1973. Reprinted in R. Thomason, editor, Formal Philosophy . Yale University Press, Yale, 1974. 32 Barbara Partee and Mats Rooth. 1983, “Generalized Conjunction and Type ”. In R. Bauerle, C. Schwarze, and A. von Stechow (eds.), Meaning, Use, and Interpretation of Language , 361-366. Berlin: Walter de Gruyter. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 16 of 18 λΨλΦ{Φ & Ψ} ⊢ λPλQλx{P x & Q x} (1) λΨλΦ{Φ & Ψ} S(S S) 1 Pr (2) P [= λxPx] DS 2 As (3) Q [= λxQx] DS 3 As (4) λx{P x × Q x} D(S ×S) 23 2, 3, C-M (5) x D 4 As (6) Px × Q x S×S 234 4, 5, λO (7) Px S 5 As (8) Qx S 6 As (9) λΦ{Φ & Q x} SS 16 1, 8, λO (10) Px & Q x S 156 7, 9, λO (11) Px & Q x S 1234 6, 7-10, ×O (12) λx{P x & Q x} DS 123 5, 11, λI (13) λQλx{P x & Q x} (D S) (D S) 12 3, 12, λI (14) λPλQλx{P x & Q x} (D S) [(D S) (D S)] 1 2, 13, λI

5. Fail In the previous example, we treat transitive verbs, in the customary fashion, as having type [D(D S)]. Unfortunately, when combined with the full power of categorial logic, this has a disastrous consequence. To see this, we observe that both of the following derivations are admissible. λyλxRxy ; K ⊢ λxRxK (1) λyλxRxy D(D S) 1 Pr

(2) K D(D S) 2 Pr

(3) λxRxK DS 12 1,2, λO

λyλxRxy ; K ⊢ λyRKy

(1) λyλxRxy D(D S) 1 Pr

(2) K D(D S) 2 Pr (3) y D 3 As (4) λxRxy DS 13 1,3, λO

(5) RKy S 123 3,4, λO

(6) λyRKy DS 12, 3,5, λI

This means, in turn, that the following constructions are equally admissible

RJK RKJ [i.e., Jay respects Kay] [i.e., Kay respects Jay]

J λxRxK J λxRKx Jay Jay

λyλxRxy K λyλxRxy K respects Kay respects Kay

In other words, according to this account, one reading of ‘ Jay respects Kay ’ is that Kay respects Jay! What is worse, one can generate indefinitely-many examples just like this; for example – one reading of ‘ every dog is a mammal ’ is that every mammal is a dog! Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 17 of 18 By way of avoiding this pitfall, we enlarge categorial grammar by including case-markers. So, when we combine ‘Kay ’ with ‘respects ’, we must indicate whether ‘Kay ’ is the subject or the object of the verb, which is exactly what case-markers are designed to do. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 18 of 18 6. Appendix – Summary of Systems Considered 1. Arguments Valid in all Systems [L, R, I, C] (1) A→B ; A ⊢ B [Modus Ponens] (2) B→C ; A →B ⊢ A →C [Transitivity] (3) A→(B →C) ⊢ B →(A →C) [Permutation] (4) A→(B →C) ; B ⊢ A →C [Secondary Modus Ponens] (5) A ⊢ (A →B) →B [Montague's Law] (6) (A →C) →D ; A →B ⊢ (B →C) →D [Inflection] (7) (A →C) →D ; A →(B →C) ⊢ B →D [Permutivity] (8) B→C ; A ×B ⊢ A ×C [Addition] (9) (A ×B) →C ⊣⊢ A →(B →C) [Schönfinkel's Law] 2. C-Valid Arguments Rejected by I (1) (A →B) →B ⊢ (B →A) →A [Łukasiewicz's Law 33 ] (2) (A →B) →A ⊢ A [Peirce's Law 34 ] 3. I-Valid Arguments Rejected by R (1) A ⊢ B →A [Positive Paradox] (2) A→B ⊢ A →(A →B) [Expansion] (3) A ⊢ B →B [Irrelevance] (4) A×B ⊢ A, B [Simplification] 4. R-Valid Arguments Rejected by L (1) A→(A →B) ⊢ A →B [Contraction] (2) A ⊢ A ×A [Duplication] (3) ⊢ A →A [Tautology] (4) (A →A) →B ⊢ B [Assertion]

33 Named after Jan Łukasiewicz (1878-1956), who invented multi-valued logic. In his system, one can define disjunction in terms of conditional via: A ∨B (A →B) →B. 34 Named after Charles Sanders Peirce (1839-1914); first presented in "On the Algebra of Logic: A Contribution to the Philosophy of Notation", American Journal of Mathematics 7, 180–202 (1885).