Chapter 18: Supplementary Formal Material

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Hardegree, Compositional Semantics, Chapter 18: Supplementary Formal Material 1 of 10 Chapter 18: Supplementary Formal Material Chapter 18: Supplementary Formal Material...1 A. Formal Languages...2 B. Set Theory...2 C. Mereology...2 1. Minimal Principles...2 2. Elementary Definitions and Notation...2 3. Additional (Optional) Principles...2 4. Semantic Principles...3 5. Basic Theorems...3 6. Distributive Predicates...6 7. Appendix Notation for Logical Derivations...10 8. Conventions About Using Mereological Symbols...10 D. Direct Semantics [for Loglish]...10

Hardegree, Compositional Semantics, Chapter 18: Supplementary Formal Material 2 of 10 A. Formal Languages +++ forthcoming +++ B. Set Theory +++ forthcoming +++ C. Mereology 1. Minimal Principles Mereology 1 concerns the logic of part-whole relations. At a minimum, a part-whole relation is a partial-order relation, which is to say a relation satisfying the following conditions. (p1) a a reflexivity (p2) a b & b c. a c transitivity (p3) a b & b a. a = b anti-symmetry Most part-whole relations satisfy additional conditions. The ones relevant to our enterprise are listed after we introduce various definitions and notation. 2. Elementary Definitions and Notation (d1) α < β α β & β α proper part 2 (d2) α β ~ x{ x α & x β } disjoint (d3) {ν Φ} the set of all ν such that Φ set-abstract (d4) {α, β} {x x=α x=β} doubleton (d5) Σ{ν Φ} the sum of all ν such that Φ infinitary-sum (d6) ΣνΦ Σ{ν Φ} alternate notation (d7) α + β Σ{α, β} binary-sum (d8) α β Σ{x xα & xβ} binary-intersection (d9) α β Σ{x xα & x β} difference (d10) α 3 ~ x[x < α] atom (d11) F!α Fα & ~ x{x α & Fx} "only" 3. Additional (Optional) Principles (p4) a<b!x{x a & x+a=b} unique complements 4 (p5) y[y = ΣxFx] xfx existence of sums (p6) xfx. x{fx x ΣxFx} UB property 5 1 The morpheme mer comes from the Greek meros, which means part. Perhaps the best known word containing this morpheme is polymer, which means many parts. 2 This is the general definition for ordering relations, allowing for the possibility that is not anti-symmetric. 3 This definition is peculiar to mereology. In lattice theory, an atom is defined to be an element that has no proper part other than the zero-element, which bears to every element, but which may not exist. Mereology has no zeroelement, except for trivial (one-element) models. 4 More precisely, relative complements. When α<β, the posited item is the complement of α relative to β.

Hardegree, Compositional Semantics, Chapter 18: Supplementary Formal Material 3 of 10 (p7) xfx. y{ x(fx xy) ΣxFx y } LUB property 6 (p8) a = Σ{x xa & x} (p9) x y[y < x] is atomic is non-atomic 4. Semantic Principles (A0) Mass-Entities form a structure satisfying conditions (p1)-(p7)+(p9); no mass-entity is a singular-entity, although it may constitute a singular-entity. (A1) Singular-Entities [individuals] form a structure satisfying conditions (p1)-(p3), where is the natural part-whole relation. (A2) Count-Entities form a structure satisfying conditions (p1)-(p8); every atomic count-entity is a singular-entity, and conversely; every non-atomic count-entity is a plural-entity. 5. Basic Theorems See Appendix for glossary for logical proof notation. 1. T1 (1) a < b. a b & a b D (2) CD (3) a < b As. (4) a b & a b 5a, 6, SL (5) a b & b a 3, Def < (6) a b ID (7) a = b As. (8) 9, 10, SL (9) b b 5b, 7, IL (10) b b reflexive (11) CD. (12) a b & a b As. (13) a < b 12a, 14, Def < (14) b a ID (15) b a As. (16) 12b, 17, SL (17) a = b 12a, 15, anti-sym 2. T2 (1) ~[ab & a b] ID (2) ab & a b As. (3) 4, 6, SL (4) ~ x{xa & xb} 2b, Def (5) aa reflexive (6) x{xa & xb} 2a, 5, QL 3. T3 (1) a b & b c. a c CD 5 UB is short for upper bound. This is a second-order formula, being universally-quantified over F. 6 LUB is short for least upper bound. This is a second-order formula, universally-quantified over F.

Hardegree, Compositional Semantics, Chapter 18: Supplementary Formal Material 4 of 10 (2) a b & b c As. (3) a c 4, Def (4) ~ x{x a & x c} ID (5) x{x a & x c} As, (6) 7, 10, SL (7) ~ x{xb & xc} 2b, Def (8) d a & d c 5, O (9) d b 2a, 8a, Transitive (10) x{xb & xc} 8b, 9, QL 4. T4 (1) x{fx Gx} ΣxFx ΣxGx CD (2) x{fx Gx} As. (3) ΣxFx ΣxGx 5, LUB (4) x{gx x ΣxGx} UB (5) x{fx x ΣxGx} 2, 4, QL 5. T5 (1) a b a 2, def (2) Σ{x xa & xb} a 4, 5, IL (3) x{xa & xb. xa} QL (4) Σ{x xa & xb} Σ{x xa} 3, T4 (5) Σ{x xa} = a T4 6. T6 (1) a b b 2, def (2) Σ{x xa & xb} b 4, 5, IL (3) x{xa & xb. xb} QL (4) Σ{x xa & xb} Σ{x xb} 3, T4 (5) Σ{x xb} = b T4 7. T7 (1) a b & a c. a b c CD (2) a b & a c As. (3) a b c 5, Def (4) a Σ{x xb & xc} 2, 6, QL (5) x{x b & x c} 2, QL (6) x{x b & x c. x Σ{x xb & xc}} 5, UB 8. T8 (1) a b=a ab D (2) CD (3) a b = a As. (4) ab 3, 5, IL (5) a b b T6 (6) CD (7) ab As. (8) a b = a 13, 14, anti-sym (9) aa reflexive (10) aa & ab 7, 9, SL (11) x{xa & xb} 10, QL (12) a Σ{x xa & xb} 11, UB (13) a a b 12, Def (14) a b a T5

Hardegree, Compositional Semantics, Chapter 18: Supplementary Formal Material 5 of 10 9. T9 (1) Σ{a} = a 2, def {a} (2) Σ{x x=a} = a 4, 6, anti-sym (3) x{x=a x Σx[x=a]} UB (4) a Σx[x=a] 3, IL (5) x[xa]. y{ x{xa xa} Σx[xa]a} LUB [F=λx[xa]] (6) a a reflexive (7) Σx[xa] a 5, 6, QL 10. T10 (1) a = Σ{x xa} 5,6, anti-sym (2) F x { Fx x ΣxFx } UB (3) x { xa x Σ{x xa} } 2, SOL (4) aa reflexivity (5) a Σ{x xa} 3, 4, QL (6) Σ{x xa} a LUB 11. T11 [ is subtractive 7 ] (1) a b x{xa & x b} CD (2) a b As. (3) x{xa & x b} SC (4) c1: a b As. (5) x{xa & x b} 4,6,QL (6) aa reflexive (7) c2: ~[a b] As. (8) x{xa & x b} 14-16, QL (9) x{xa & xb} 7, Def (10) x[x = a b] 9, def, Exist-Sum (11) a b a 2, T8 (12) a b a T5 (13) a b < a 11, 12, T1 (14)!x{xa & x a b & x+(a b)=a} 13, complement (15) ca & c a b & c+(a b)=a 14, [!]O (16) c b Def (17) ~ x{xc & xb} ID (18) x{xc & xb} As. (19) (20) d c & d b 18, O (21) d a 15a, 20a, transitive (22) d a b 20b, 21, T7 (23) d a b 15b, 20a, T3 COROLLARY (1) a b x[x = a b] CD (2) a b As. (3) x[x = a b] 5, Def (4) x{xa & x b} 2, T11 (5) x[x = Σx{xa & x b}] 4, exist-sum 12. T12 (1) xfx. a ΣxFx x{fx & ax} CD 2 7 So called because it ensures that the subtraction operator is well-defined; see corollary.

Hardegree, Compositional Semantics, Chapter 18: Supplementary Formal Material 6 of 10 13. T13 (1) xfx. x{fx a x} a ΣxFx CD 2 6. Distributive Predicates 1. Introduction Most predicates are distributive, but a few are not, being rather essentially plural. The most prominent examples of the latter are the numerical adjectives. That Jay and Kay are two does entail that Jay is two and Kay is two. The initial idea is that a distributive predicate F satisfies the following condition. (d) x y{ F[x+y]. F[x] & F[y] } This says that F distributes over binary-sum, and hence finitary-sum. More generally, a distributive predicate F distributes over infinitary-sum, which is to say it satisfies the following second-order condition. (D) P: F[ΣxPx] x{px Fx} Thus, our official definition goes as follows. 2. Theorems DIST[F] P: F[ΣxPx] x{px Fx} Our first theorem show how finitary-distribution is a special case of infinitary-distribution. 1. D1 (1) DIST[F] [ P: F[ΣxPx] x{px Fx} ] As. (2) F[Σ{x x=a x=b}] x{x=a x=b. Fx} 1, SOL (3) x y{ F[x+y]. F[x] & F[y] } U2BD (4) CD (5) F[a+b] As (6) F[a] & F[b] 8, IL (7) F[Σ{x x=a x=b}] 4, Def + (8) x{x=a x=b. Fx} 2, 7, SL (9) CD (10) F[a] & F[b] As. (11) F[a+b] 12, Def + (12) F[Σ{x x=a x=b}] 2, 13, SL (13) x{x=a x=b. Fx} 10, IL 2. D2 (1) DIST[F] [ P: x{px Fx} F[ΣxPx] ] As. (2) y{fy x{xy Fx}} UCD (3) Fa As. (4) x{xy Fa} 6, 7, SL (5) a = Σ{x xa} T10 (6) F[Σ{x xa}] 3, 5, IL (7) x{xa Fx} F[Σx[xa]] 1, SOL 3. D3 (1) DIST[F] [ P: x{px Fx} F[ΣxPx] ] As. (2) F[ΣxFx] 1, 3, SOL (3) x{fx Fx} QL

Hardegree, Compositional Semantics, Chapter 18: Supplementary Formal Material 7 of 10 4. D4 (1) DIST[F] As. (2) F!a {x F!x} = {a} CD (3) F!a As. (4) Fa & ~ x{x a & Fx} 3, def! (5) {x F!x} = {a} 6, ST (6) x{f!x x=a} 7, 19, QL (7) x{f!x x=a} UCD (8) F!b As. (9) Fb & ~ x{x b & Fx} 8, def! (10) b=a ID (11) b a As (12) 9b, 18, QL (13) a b b a 11, anti-sym (14) a b WOLOG 8 (15) x{xa & x b} 14, subtractive (16) ca & c b 15, O (17) Fc 1, 4a, 16a, D2 (18) x{x b & Fx} 16b, 17, QL (19) x{x=a F!x} 4a, IL 5. D5 (1) DIST[F] As. (2) F!a x{fx xa} CD (3) F!a As. (4) Fa & ~ x{x a & Fx} 3, Def! (5) x{fx xa} UCD (6) Fb As. (7) ba ID (8) b a As (9) 4b, 13, SL (10) x{xb & x a} 8, subtractive (11) cb & c a 10, O (12) Fc 1, 6, 11a, D2 (13) x{x a & Fx} 11b, 12, QL 6. D6 (1) DIST[F] As. (2) F!a a = ΣxFx D (3) CD (4) F!a As (5) a = ΣxFx 7, 9, anti-sym (6) x{fx xa} 4, D5 (7) ΣxFx a 6, LUB (8) Fa 4, Def! (9) a ΣxFx 8, LUB (10) CD (11) a = ΣxFx As. (12) F!a 11, Def! (13) Fa & ~ x{x a & Fx} &D 8 Short for without loss of generality. Often used when the two cases in a separation-of-case argument are symmetrical.

Hardegree, Compositional Semantics, Chapter 18: Supplementary Formal Material 8 of 10 (14) Fa 11, 15, IL (15) F[ΣxFx] 1, D3 (16) ~ x{x a & Fx} ID (17) x{x a & Fx} As. (18) 19a, 21, T2 (19) b a & Fb 17, O (20) b ΣxFx 19b, UB (21) b a 11, 20, IL 7. D7 (1) DIST[F] As. (2) xf!x ΣxF!x = ΣxFx CD (3) xf!x As. (4) ΣxF!x = ΣxFx 9, 10, IL (5) F!a 3, O (6) {x F!x} = {a} 1, 5, D4 (7) Σ{x F!x} = Σ{a} 6, IL (8) Σ{a} = a T9 (9) Σ{x F!x} = a 7, 8, IL (10) a = ΣxFx 1, 5, D6 8. D8 (1) DIST[F] & DIST[G] Premise (2) x{f!x & Gx}. xfx & x{fx Gx} D (3) CD (4) x{f!x & Gx} As. (5) F!a & Ga 4, O (6) Fa & ~ x{x a & Fx} 5a, Def! (7) xfx & x{fx Gx} &D (8) xfx 6a, QL (9) x{fx Gx} UCD (10) Fb As. (11) Gb 5a, 11, 1b, D2 (12) ba ID (13) b a As (14) 6b, 12, SL (15) x{xb & x a} 13, subtractive (16) cb & c a 15, O (17) Fc 10, 16a, 1a, D2 (18) x{x a & Fx} 16a, 17, QL

Hardegree, Compositional Semantics, Chapter 18: Supplementary Formal Material 9 of 10 (19) CD (20) xfx & x{fx Gx} As. (21) x{f!x & Gx} 27, 27, QL (22) y[y = ΣxFx] 20a, Exist-Sum (23) a = ΣxFx 22, O (24) F[ΣxFx] 1a, D3 (25) Fa 23, 24, IL (26) Ga 20b, 25,QL (27) F!a 28, Def! (28) Fa & ~ x{x a & Fx} 25, 29, SL (29) ~ x{x a & Fx} ID (30) x{x a & Fx} As (31) 33, 34, T2 (32) b a & Fb 30, O (33) b ΣxFx 23, 32, IL (34) b ΣxFx 32b, UB 9. D9 (1) DIST[F] Premise (2) y{f!y & x{xy Gx}}. xfx & x{fx Gx}} D (3) CD (4) y{f!y & x{xy Gx}} As. (5) xfx & x{fx Gx}} 7a, 8, SL (6) F!a & x{xa Gx} 4, O (7) Fa & ~ x{x a & Fx} 6a, Def! (8) x{fx Gx}} UCD (9) Fb As. (10) Gb 6b, 11, QL (11) b a 6a, 9, 1, D5 (12) CD (13) xfx & x{fx Gx}} As. (14) y{f!y & x{xy Fx}} (15) y[y = ΣxFx] 13a, Exist-Sum (16) F![ΣxFx] & x{xσxfx Fx} &D (17) F![ΣxFx] 18, Def! (18) F[ΣxFx] & ~ y{y ΣxFx & Fy} &D (19) F[ΣxFx] 1, D3 (20) ~ y{y ΣxFx & Fy} ID (21) y{y ΣxFx & Fy} As. (22) 23a, 24, T2 (23) a ΣxFx & Fa 21, O (24) a ΣxFx 23b, UB (25) x{xσxfx Fx} UCD (26) b ΣxFx As. (27) Fb 26, 28, 1, D2 (28) F[ΣxFx] 1, D3

Hardegree, Compositional Semantics, Chapter 18: Supplementary Formal Material 10 of 10 7. Appendix Notation for Logical Derivations CD conditional derivation ID indirect derivation (reductio) UCD universal conditional derivation UBD universal biconditional derivation SC separation of cases O existential "out"(instantiation) &D ampersand derivation D biconditional derivation SL sentential logic QL quantifier logic IL identity logic SOL second-order logic, including [covert] lambda-abstraction ST set theory 8. Conventions About Using Mereological Symbols We have many part-whole relations, but only one symbol, so we need to adopt conventions about which part-whole relation is being alluded to. Mass[α] & Mass[β] αβ : mass part-whole Single[α] & Single[β] αβ : natural part-whole Plural[α] Plural[β] αβ : count part-whole Mass[α] & Single[β] αβ : constitution D. Direct Semantics [for Loglish] ++ forthcoming ++

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