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Logicality and Semantic Theory Gil Sagi University of Haifa July 26, 2018 Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 1 / 36 Introduction The Concept of Logical Consequence Gil Sagi (Haifa University)

  1. Semantic Constraints The Framework I ( Ann ) = Ann I ( smokes ) = λ x ∈ D . x smokes I ( most ) = {� A , B � ∈ P ( D ) 2 : | A ∩ B | > | A \ B |} I ( even ) ∩ I ( odd ) = ∅ Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

  2. Semantic Constraints The Framework I ( Ann ) = Ann I ( smokes ) = λ x ∈ D . x smokes I ( most ) = {� A , B � ∈ P ( D ) 2 : | A ∩ B | > | A \ B |} I ( even ) ∩ I ( odd ) = ∅ I ( bachelor ) ⊆ I ( unmarried ) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

  3. Semantic Constraints The Framework I ( Ann ) = Ann I ( smokes ) = λ x ∈ D . x smokes I ( most ) = {� A , B � ∈ P ( D ) 2 : | A ∩ B | > | A \ B |} I ( even ) ∩ I ( odd ) = ∅ I ( bachelor ) ⊆ I ( unmarried ) I ( H 2 O ) = I ( water ) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

  4. Semantic Constraints The Framework I ( Ann ) = Ann I ( smokes ) = λ x ∈ D . x smokes I ( most ) = {� A , B � ∈ P ( D ) 2 : | A ∩ B | > | A \ B |} I ( even ) ∩ I ( odd ) = ∅ I ( bachelor ) ⊆ I ( unmarried ) I ( H 2 O ) = I ( water ) I ( wasBought ) = I ( wasSold ) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

  5. Semantic Constraints The Framework I ( Ann ) = Ann I ( smokes ) = λ x ∈ D . x smokes I ( most ) = {� A , B � ∈ P ( D ) 2 : | A ∩ B | > | A \ B |} I ( even ) ∩ I ( odd ) = ∅ I ( bachelor ) ⊆ I ( unmarried ) I ( H 2 O ) = I ( water ) I ( wasBought ) = I ( wasSold ) I ( ∃ ) = { A ⊆ D : A � = ∅} Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

  6. Semantic Constraints The Framework I ( Ann ) = Ann I ( smokes ) = λ x ∈ D . x smokes I ( most ) = {� A , B � ∈ P ( D ) 2 : | A ∩ B | > | A \ B |} I ( even ) ∩ I ( odd ) = ∅ I ( bachelor ) ⊆ I ( unmarried ) I ( H 2 O ) = I ( water ) I ( wasBought ) = I ( wasSold ) I ( ∃ ) = { A ⊆ D : A � = ∅} I ( ∀ ) ∈ {{ B ⊆ D : A ⊆ B } : A ⊆ D } Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

  7. Semantic Constraints The Framework I ( Ann ) = Ann I ( smokes ) = λ x ∈ D . x smokes I ( most ) = {� A , B � ∈ P ( D ) 2 : | A ∩ B | > | A \ B |} I ( even ) ∩ I ( odd ) = ∅ I ( bachelor ) ⊆ I ( unmarried ) I ( H 2 O ) = I ( water ) I ( wasBought ) = I ( wasSold ) I ( ∃ ) = { A ⊆ D : A � = ∅} I ( ∀ ) ∈ {{ B ⊆ D : A ⊆ B } : A ⊆ D } I ( R ) is a symmetric binary relation. Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

  8. Semantic Constraints The Framework I ( Ann ) = Ann I ( smokes ) = λ x ∈ D . x smokes I ( most ) = {� A , B � ∈ P ( D ) 2 : | A ∩ B | > | A \ B |} I ( even ) ∩ I ( odd ) = ∅ I ( bachelor ) ⊆ I ( unmarried ) I ( H 2 O ) = I ( water ) I ( wasBought ) = I ( wasSold ) I ( ∃ ) = { A ⊆ D : A � = ∅} I ( ∀ ) ∈ {{ B ⊆ D : A ⊆ B } : A ⊆ D } I ( R ) is a symmetric binary relation. 0 ∈ I ( naturalNumber ) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

  9. Semantic Constraints The Framework I ( Ann ) = Ann I ( smokes ) = λ x ∈ D . x smokes I ( most ) = {� A , B � ∈ P ( D ) 2 : | A ∩ B | > | A \ B |} I ( even ) ∩ I ( odd ) = ∅ I ( bachelor ) ⊆ I ( unmarried ) I ( H 2 O ) = I ( water ) I ( wasBought ) = I ( wasSold ) I ( ∃ ) = { A ⊆ D : A � = ∅} I ( ∀ ) ∈ {{ B ⊆ D : A ⊆ B } : A ⊆ D } I ( R ) is a symmetric binary relation. 0 ∈ I ( naturalNumber ) I ( prime ) = { 2 , 3 , 5 , ... } Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

  10. Semantic Constraints The Framework I ( Ann ) = Ann I ( smokes ) = λ x ∈ D . x smokes I ( most ) = {� A , B � ∈ P ( D ) 2 : | A ∩ B | > | A \ B |} I ( even ) ∩ I ( odd ) = ∅ I ( bachelor ) ⊆ I ( unmarried ) I ( H 2 O ) = I ( water ) I ( wasBought ) = I ( wasSold ) I ( ∃ ) = { A ⊆ D : A � = ∅} I ( ∀ ) ∈ {{ B ⊆ D : A ⊆ B } : A ⊆ D } I ( R ) is a symmetric binary relation. 0 ∈ I ( naturalNumber ) I ( prime ) = { 2 , 3 , 5 , ... } | I ( Red ) | = 375 (i.e., the size of the extension of Red is 375.) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 19 / 36

  11. Semantic Constraints The Framework I ( P ) ⊆ D Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

  12. Semantic Constraints The Framework I ( P ) ⊆ D I ( John ) ∈ D Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

  13. Semantic Constraints The Framework I ( P ) ⊆ D I ( John ) ∈ D I ( abc ) = T or I ( abc ) = F Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

  14. Semantic Constraints The Framework I ( P ) ⊆ D I ( John ) ∈ D I ( abc ) = T or I ( abc ) = F I ( s ) = T Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

  15. Semantic Constraints The Framework I ( P ) ⊆ D I ( John ) ∈ D I ( abc ) = T or I ( abc ) = F I ( s ) = T I ( d ) � = I ( ∧ ) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

  16. Semantic Constraints The Framework I ( P ) ⊆ D I ( John ) ∈ D I ( abc ) = T or I ( abc ) = F I ( s ) = T I ( d ) � = I ( ∧ ) I ( or ) ∈ { f ∨ , f ⊻ } where f ∨ is the inclusive or function, and f ⊻ is the xor function from pairs of truth values to truth values. Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 20 / 36

  17. Semantic Constraints The Framework The Language and its Models Language Primitive expressions ( terms ) Complex expressions ( phrases ) Models M = � D , I � D (the domain) is a non-empty set. I (the interpretation function) assigns to phrases values from the set-theoretic hierarchy with the members of D ∪ { T , F } as ur-elements. Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 21 / 36

  18. Semantic Constraints The Framework Semantic Constraints A semantic constraint for L is a sentence in the metalanguage that somehow constrains or limits the admissible models for L. Semantic constraints include implicit universal quantification over models (domains and interpretation functions). Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 22 / 36

  19. Semantic Constraints The Framework Semantic Constraints A semantic constraint for L is a sentence in the metalanguage that somehow constrains or limits the admissible models for L. Semantic constraints include implicit universal quantification over models (domains and interpretation functions). Let ∆ be a set of semantic constraints. A ∆ -model is an admissible model by ∆ , i.e. a model abiding by the constraints in ∆ . Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 22 / 36

  20. Semantic Constraints The Framework Logical Consequence Let ∆ be a set of constraints such as those mentioned above. An argument � Γ , ϕ � is ∆ - valid ( Γ | = ∆ ϕ ) if for every ∆ -model M , if all the sentences in Γ are true in M , then ϕ is true in M . So, for instance we have: bachelor ( John ) | = ∆ unmarried ( John ) . Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 23 / 36

  21. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

  22. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism As a criterion for logical terms: Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

  23. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism As a criterion for logical terms: First, we assume that for every (candidate) term t , we have an associated operation O t such that for each set D , O t ( D ) gives the extension of t in models with domain D . Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

  24. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism As a criterion for logical terms: First, we assume that for every (candidate) term t , we have an associated operation O t such that for each set D , O t ( D ) gives the extension of t in models with domain D . Examples: O ∃ ( D ) = { A ⊆ D : A � = ∅} Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

  25. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism As a criterion for logical terms: First, we assume that for every (candidate) term t , we have an associated operation O t such that for each set D , O t ( D ) gives the extension of t in models with domain D . Examples: O ∃ ( D ) = { A ⊆ D : A � = ∅} O ∀ ( D ) = { D } Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

  26. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism As a criterion for logical terms: First, we assume that for every (candidate) term t , we have an associated operation O t such that for each set D , O t ( D ) gives the extension of t in models with domain D . Examples: O ∃ ( D ) = { A ⊆ D : A � = ∅} O ∀ ( D ) = { D } O ∃ ℵ 0 ( D ) = { A ⊆ D : | A | ≥ ℵ 0 } Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 24 / 36

  27. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism As a criterion for logical terms: Let M = � D , I � and M ′ = � D ′ , I ′ � be models, and let f : D → D ′ be a bijection. Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 25 / 36

  28. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism As a criterion for logical terms: Let M = � D , I � and M ′ = � D ′ , I ′ � be models, and let f : D → D ′ be a bijection. f can be naturally extended to f + over elements in the set-theoretic hierarchy over D ∪ { T , F } and D ′ ∪ { T , F } . Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 25 / 36

  29. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism As a criterion for logical terms: Let M = � D , I � and M ′ = � D ′ , I ′ � be models, and let f : D → D ′ be a bijection. f can be naturally extended to f + over elements in the set-theoretic hierarchy over D ∪ { T , F } and D ′ ∪ { T , F } . Definition (invariance under isomorphisms: terms) A term t is invariant under isomorphisms if for any sets D and D ′ and a bijection f : D → D ′ , f + ( O t ( D )) = O t ( D ′ ) . Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 25 / 36

  30. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism As a criterion for semantic constraints: Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 26 / 36

  31. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism As a criterion for semantic constraints: Definition (isomorphic models) We say that M = � D , I � is isomorphic to M ′ = � D ′ , I ′ � ( M ∼ = M ′ ) if there is a bijection f : D → D ′ such that f + ( I ( p )) = I ′ ( p ) for every phrase p in L. Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 26 / 36

  32. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism As a criterion for semantic constraints: Definition (isomorphic models) We say that M = � D , I � is isomorphic to M ′ = � D ′ , I ′ � ( M ∼ = M ′ ) if there is a bijection f : D → D ′ such that f + ( I ( p )) = I ′ ( p ) for every phrase p in L. Definition (invariance under isomorphism: semantic constraints) A semantic constraint C is invariant under isomorphisms if for any models M and M ′ such that M ∼ = M ′ , then if M is a { C } -model, then M ′ is a { C } -model. ( cf. [Zimmermann, 2011]) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 26 / 36

  33. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints The following constraints are invariant under isomorphisms: Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 27 / 36

  34. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints The following constraints are invariant under isomorphisms: I ( allRed ) ∩ I ( allGreen ) = ∅ I ( Red ) ∩ I ( Big ) = ∅ | I ( Red ) | = 375 (i.e., the size of the extension of Red is 375.) I ( John ) ∈ I ( Bachelor ) I ( s ) = T I ( ∃ ) = { A ⊆ D : A � = ∅} I ( ∀ ) ∈ {{ B ⊆ D : A ⊆ B } : A ⊆ D } Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 27 / 36

  35. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints The following constraints are invariant under isomorphisms: I ( allRed ) ∩ I ( allGreen ) = ∅ I ( Red ) ∩ I ( Big ) = ∅ | I ( Red ) | = 375 (i.e., the size of the extension of Red is 375.) I ( John ) ∈ I ( Bachelor ) I ( s ) = T I ( ∃ ) = { A ⊆ D : A � = ∅} I ( ∀ ) ∈ {{ B ⊆ D : A ⊆ B } : A ⊆ D } The following constraints are not invariant under isomorphisms: 0 ∈ I ( naturalNumber ) I ( prime ) = { 2 , 3 , 5 , ... } I ( Even ) ∩ I ( Prime ) = { 2 } I ( Ann ) = Ann I ( smokes ) = λ x ∈ D . x smokes Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 27 / 36

  36. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism: terms and constraints Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 28 / 36

  37. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism: terms and constraints Consider the constraint C t that “fixes” t : C t : I ( t ) = O t ( D ) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 28 / 36

  38. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism: terms and constraints Consider the constraint C t that “fixes” t : C t : I ( t ) = O t ( D ) Example: C = : I ( = ) = {� a , a � : a ∈ D } Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 28 / 36

  39. Semantic Constraints Invariance Criteria Criteria for Semantic Constraints Invariance under isomorphism: terms and constraints Consider the constraint C t that “fixes” t : C t : I ( t ) = O t ( D ) Example: C = : I ( = ) = {� a , a � : a ∈ D } Proposition. Let t be a term, O t an associated operation and C t an associated constraint. Then t is invariant under isomorphisms iff C t is invariant under isomorphisms. Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 28 / 36

  40. Semantic Constraints Invariance Criteria Can we apply the generalized criterion of invariance under isomorphisms to natural language semantics, and in this way demarcate the relation of logical consequence in natural language? Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 29 / 36

  41. Glanzberg B Partiality in Semantic Theory [S]emantics, narrowly construed as part of our linguistic competence, is only a partial determinant of content. Likewise, semantic theories in linguistics function as partial theories of content. I shall go on to offer an account of where and how this partiality arises, which focuses on how lexical meaning combines elements of distinctively linguistic competence with elements from our broader cognitive resources. This account shows how we can accommodate some partiality in semantic theories without falling into skepticism about semantics or its place in linguistic theory. (Glanzberg, 2014) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 30 / 36

  42. Glanzberg B Uninformative Semantic Clauses a. � Ann � = Ann Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 31 / 36

  43. Glanzberg B Uninformative Semantic Clauses a. � Ann � = Ann b. � smokes � = λ x ∈ D e : x smokes Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 31 / 36

  44. Glanzberg B Uninformative Semantic Clauses a. � Ann � = Ann b. � smokes � = λ x ∈ D e : x smokes c. � most � ( A , B ) ⇔ | A \ B | < | A ∩ B | Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 31 / 36

  45. Glanzberg B Uninformative Semantic Clauses a. � Ann � = Ann b. � smokes � = λ x ∈ D e : x smokes c. � most � ( A , B ) ⇔ | A \ B | < | A ∩ B | d. � tall � ( x ) = d ∈ S ( S a scale with the dimension of height ) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 31 / 36

  46. Glanzberg B Uninformative Semantic Clauses a. � Ann � = Ann b. � smokes � = λ x ∈ D e : x smokes c. � most � ( A , B ) ⇔ | A \ B | < | A ∩ B | d. � tall � ( x ) = d ∈ S ( S a scale with the dimension of height ) e. � short � ( x ) = d ∈ S ′ ( S ′ the inverse of S ) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 31 / 36

  47. Glanzberg B Uninformative Semantic Clauses a. � Ann � = Ann b. � smokes � = λ x ∈ D e : x smokes c. � most � ( A , B ) ⇔ | A \ B | < | A ∩ B | d. � tall � ( x ) = d ∈ S ( S a scale with the dimension of height ) e. � short � ( x ) = d ∈ S ′ ( S ′ the inverse of S ) [T]he use of disquotation in semantic theories precisely marks the places where they lose their explanatory force. Insofar as disquotation plays an ineliminable role in building theories of content, semantic theories can be at best partial theories of content... [D]isquotation is a guide to where linguistic meaning contains pointers to extra-linguistic elements of content. (Glanzberg, 2014) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 31 / 36

  48. Glanzberg B Separating the Explanatory from the Non-Explanatory I ( Ann ) = Ann Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 32 / 36

  49. Glanzberg B Separating the Explanatory from the Non-Explanatory I ( Ann ) = Ann I ( Ann ) ∈ D Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 32 / 36

  50. Glanzberg B Separating the Explanatory from the Non-Explanatory I ( Ann ) = Ann I ( Ann ) ∈ D I ( smokes ) = λ x ∈ D . x smokes Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 32 / 36

  51. Glanzberg B Separating the Explanatory from the Non-Explanatory I ( Ann ) = Ann I ( Ann ) ∈ D I ( smokes ) = λ x ∈ D . x smokes I ( smokes ) ∈ { f : f : D → { T , F }} Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 32 / 36

  52. Glanzberg B Separating the Explanatory from the Non-Explanatory I ( Ann ) = Ann I ( Ann ) ∈ D I ( smokes ) = λ x ∈ D . x smokes I ( smokes ) ∈ { f : f : D → { T , F }} I ( tall )( x ) ∈ S ( S a scale with the dimension of height ) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 32 / 36

  53. Glanzberg B Separating the Explanatory from the Non-Explanatory I ( Ann ) = Ann I ( Ann ) ∈ D I ( smokes ) = λ x ∈ D . x smokes I ( smokes ) ∈ { f : f : D → { T , F }} I ( tall )( x ) ∈ S ( S a scale with the dimension of height ) I ( tall )( x ) ∈ S ( S a scale on the vertical axis....) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 32 / 36

  54. Glanzberg B Separating the Explanatory from the Non-Explanatory I ( Ann ) = Ann I ( Ann ) ∈ D I ( smokes ) = λ x ∈ D . x smokes I ( smokes ) ∈ { f : f : D → { T , F }} I ( tall )( x ) ∈ S ( S a scale with the dimension of height ) I ( tall )( x ) ∈ S ( S a scale on the vertical axis....) I ( short )( x ) ∈ S ( S ′ the inverse of S ) Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 32 / 36

  55. Glanzberg B Separating the Explanatory from the Non-Explanatory I ( Ann ) = Ann I ( Ann ) ∈ D I ( smokes ) = λ x ∈ D . x smokes I ( smokes ) ∈ { f : f : D → { T , F }} I ( tall )( x ) ∈ S ( S a scale with the dimension of height ) I ( tall )( x ) ∈ S ( S a scale on the vertical axis....) I ( short )( x ) ∈ S ( S ′ the inverse of S ) I ( most ) = {� A , B � ∈ P ( D ) 2 : | A ∩ B | > | A \ B |} Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 32 / 36

  56. Glanzberg B Logical Consequence in Natural Language The generalized criterion of invariance under isomorphisms: A semantic constraint is logical if it is invariant under isomorphisms. Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 33 / 36

  57. Glanzberg B Logical Consequence in Natural Language The generalized criterion of invariance under isomorphisms: A semantic constraint is logical if it is invariant under isomorphisms. Conjecture: The logic of natural language is precisely the explanatory part of semantic theory for natural language. Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 33 / 36

  58. Conclusion Refined Criterion for Logicality Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 34 / 36

  59. Conclusion Refined Criterion for Logicality A connective is a logical connective if and only if it follows from the meaning of the connective that it is invariant under arbitrary bijections. [McGee, 1996, p. 578] Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 34 / 36

  60. Conclusion Refined Criterion for Logicality A semantic constraint is a logical semantic constraint if and only if it follows from the meaning of the semantic constraint that it is invariant under arbitrary bijections. Cf. [McGee 1996, p. 578] Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 35 / 36

  61. Conclusion Refined Criterion for Logicality A semantic constraint is a logical semantic constraint if and only if it follows from the semantic theory for the language and it is invariant under arbitrary bijections. Cf. [McGee 1996, p. 578] Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 36 / 36

  62. Conclusion Refined Criterion for Logicality A semantic constraint is a logical semantic constraint if and only if it follows from the semantic theory for the language and it is invariant under arbitrary bijections. Cf. [McGee 1996, p. 578] Thank you! Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 36 / 36

  63. Conclusion Fox, D. (2000). Economy and Semantic Interpretation, Linguistic Inquiry Monographs 35 . MITWPL and MIT Press, Cambridge, MA. Fox, D. and Hackl, M. (2006). The universal density of measurment. Linguistics and Philosophy , 29:537–586. Gajewski, J. (2002). On analyticity in natural language. Manuscript. Harman, G. (1984). Logic and reasoning. Synthese , 60:107–127. Lycan, W. (1984). Logical Form in Natural Language . The MIT Press, Cambridge, MA. Gil Sagi (Haifa University) Logicality and Semantic Theory July 26, 2018 36 / 36

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