Let LedZeppelin Explain… Our shadows taller than our soul. Yes, there are two paths you can go by, but in the long run There’s still time to change the road you’re on. And it makes me wonder. Your head is humming and it won’t go, in case you don’t know, The piper’s calling you to join him, Your stairway lies on the whispering wind? And as we wind on down the road There walks a lady we all know There’s a lady who’s sure all that glitters is gold Who shines white light and wants to show How everything still turns to gold. And if you listen very hard The tune will come to you at last. When all are one and one is all To be a rock and not to roll. And she’s buying a stairway to heaven. It’s just a spring clean for the May Queen. If there’s a bustle in your hedgerow, don’t be alarmed now, And the forests will echo with laughter. And a new day will dawn for those who stand long, And she’s buying a stairway to heaven. When she gets there she knows, if the stores are all closed Ooh, ooh, and she’s buying a stairway to heaven. There’s a sign on the wall but she wants to be sure ’Cause you know sometimes words have two meanings. In a tree by the brook, there’s a songbird who sings, Sometimes all of our thoughts are misgiven. Ooh, it makes me wonder, Ooh, it makes me wonder. There’s a feeling I get when I look to the west, And my spirit is crying for leaving. In my thoughts I have seen rings of smoke through the trees, And the voices of those who stand looking. Ooh, it makes me wonder, Ooh, it really makes me wonder. And it’s whispered that soon, if we all call the tune, Then the piper will lead us to reason. With a word she can get what she came for. Dear lady, can you hear the wind blow, and did you know conjunction 19 negation 3 existential quantifier 15 possibility 2
Let LedZeppelin Explain… Our shadows taller than our soul. There’s still time to change the road you’re on. And it makes me wonder. Your head is humming and it won’t go, in case you don’t know, The piper’s calling you to join him, Dear lady, can you hear the wind blow, and did you know Your stairway lies on the whispering wind? And as we wind on down the road There walks a lady we all know It’s just a spring clean for the May Queen. Who shines white light and wants to show How everything still turns to gold. And if you listen very hard The tune will come to you at last. When all are one and one is all To be a rock and not to roll. And she’s buying a stairway to heaven. Yes, there are two paths you can go by, but in the long run If there’s a bustle in your hedgerow, don’t be alarmed now, There’s a lady who’s sure all that glitters is gold Sometimes all of our thoughts are misgiven. And she’s buying a stairway to heaven. When she gets there she knows, if the stores are all closed With a word she can get what she came for. Ooh, ooh, and she’s buying a stairway to heaven. There’s a sign on the wall but she wants to be sure ’Cause you know sometimes words have two meanings. In a tree by the brook, there’s a songbird who sings, Ooh, it makes me wonder, Ooh, it makes me wonder. And the forests will echo with laughter. There’s a feeling I get when I look to the west, And my spirit is crying for leaving. In my thoughts I have seen rings of smoke through the trees, And the voices of those who stand looking. Ooh, it makes me wonder, Ooh, it really makes me wonder. And it’s whispered that soon, if we all call the tune, Then the piper will lead us to reason. And a new day will dawn for those who stand long, conjunction 19 negation 3 existential quantifier 15 possibility 2
Let LedZeppelin Explain… Then the piper will lead us to reason. And she’s buying a stairway to heaven. The tune will come to you at last. And if you listen very hard How everything still turns to gold. Our shadows taller than our soul. And as we wind on down the road Your stairway lies on the whispering wind? The piper’s calling you to join him, And it makes me wonder. There’s still time to change the road you’re on. There’s a lady who’s sure all that glitters is gold And the forests will echo with laughter. And a new day will dawn for those who stand long, conjunction 19 negation 3 existential quantifier 15 possibility 2 And it’s whispered that soon, if we all call the tune, Sometimes all of our thoughts are misgiven. And she’s buying a stairway to heaven. When she gets there she knows, if the stores are all closed With a word she can get what she came for. There’s a sign on the wall but she wants to be sure ’Cause you know sometimes words have two meanings. Ooh, it makes me wonder, Ooh, it really makes me wonder. In a tree by the brook, there’s a songbird who sings, Ooh, it makes me wonder, Ooh, it makes me wonder. There’s a feeling I get when I look to the west, And my spirit is crying for leaving. In my thoughts I have seen rings of smoke through the trees, And the voices of those who stand looking. If there’s a bustle in your hedgerow, don’t be alarmed now, It’s just a spring clean for the May Queen. Yes, there are two paths you can go by, but in the long run Ooh, ooh, and she’s buying a stairway to heaven. Your head is humming and it won’t go, in case you don’t know, Dear lady, can you hear the wind blow, and did you know There walks a lady we all know Who shines white light and wants to show When all are one and one is all To be a rock and not to roll.
a definition
William James, a Tree, a Squirrel and a Man A man walks rapidly around a tree, while a squirrel moves on the tree trunk. Both face the tree at all times, but the tree trunk stays between them. A group of people are arguing over the question: Does the man go round the squirrel or not? : The man goes round the squirrel. : The man doesn’t go round the squirrel. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 9 of 62
William James, a Tree, a Squirrel and a Man A man walks rapidly around a tree, while a squirrel moves on the tree trunk. Both face the tree at all times, but the tree trunk stays between them. A group of people are arguing over the question: Does the man go round the squirrel or not? : The man goes round the squirrel. : The man doesn’t go round the squirrel. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 9 of 62
William James, a Tree, a Squirrel and a Man A man walks rapidly around a tree, while a squirrel moves on the tree trunk. Both face the tree at all times, but the tree trunk stays between them. A group of people are arguing over the question: Does the man go round the squirrel or not? GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 9 of 62 α : The man goes round the squirrel. δ : The man doesn’t go round the squirrel.
William James, a Tree, a Squirrel and a Man Which party is right depends on what you practically mean by ‘going round’ the squirrel. If you mean passing from the north of him to the east, then to the south, then to the west, and then to the north of him again, obviously the man does go round him, for he occupies these successive positions. But if on the contrary you mean being first in front of him, then on the right of him then behind him, then on his left, and finally in front again, it is quite as obvious that the man fails to go round him … Make the distinction, and there is no occasion for any farther dispute. — William James, Pragmatism (1907) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 10 of 62
Resolving a dispute by clarifying meanings Once we disambiguate “going round” no disagreement remains. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 11 of 62 α : The man goes round 1 the squirrel. δ : The man doesn’t go round 2 the squirrel.
Resolving a dispute by clarifying meanings Once we disambiguate “going round” no disagreement remains. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 11 of 62 α : The man goes round 1 the squirrel. δ : The man doesn’t go round 2 the squirrel.
Resolution by translation Perhaps terms and can’t be explicated in terms of prior vocabulary. No matter. could learn while could learn . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 12 of 62 ▶ For James, “going round 1 ” and “going round 2 ” are explicated in other terms of α and δ ’s vocabulary.
Resolution by translation prior vocabulary. No matter. could learn while could learn . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 12 of 62 ▶ For James, “going round 1 ” and “going round 2 ” are explicated in other terms of α and δ ’s vocabulary. ▶ Perhaps terms t 1 and t 2 can’t be explicated in terms of
Resolution by translation prior vocabulary. No matter. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 12 of 62 ▶ For James, “going round 1 ” and “going round 2 ” are explicated in other terms of α and δ ’s vocabulary. ▶ Perhaps terms t 1 and t 2 can’t be explicated in terms of ▶ α could learn t 2 while δ could learn t 1 .
Introducing General Scheme GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 13 of 62 L α L δ A A
GregRestall Introducing General Scheme http://consequently.org/presentation/2015/verbal-disputes-oxford/ 13 of 62 L α L δ A A t α t δ t δ ( A ) t α ( A ) L ∗
or coherent ( in bounds ) What is a Language ? + weakening: If http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . then and + cut: If . and then . A syntax + identity: . , which are either incoherent ( out of bounds ) is denied , and each member of is asserted , where each member of positions 14 of 62
or coherent ( in bounds ) What is a Language ? + weakening: If http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . then and + cut: If . and then . + identity: . , which are either incoherent ( out of bounds ) is denied , and each member of is asserted , where each member of positions 14 of 62 ▶ A syntax
or coherent ( in bounds ) What is a Language ? and http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . then and + cut: If . + weakening: If then . + identity: . , which are either incoherent ( out of bounds ) 14 of 62 ▶ A syntax ▶ positions [ X : Y ] , where each member of X is asserted and each member of Y is denied ,
What is a Language ? and http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . then and + cut: If . then + weakening: If . + identity: . or coherent ( in bounds ) 14 of 62 ▶ A syntax ▶ positions [ X : Y ] , where each member of X is asserted and each member of Y is denied , which are either incoherent ( out of bounds ) X ⊢ Y ,
What is a Language ? and http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . then and + cut: If . then + weakening: If . + identity: 14 of 62 ▶ A syntax ▶ positions [ X : Y ] , where each member of X is asserted and each member of Y is denied , which are either incoherent ( out of bounds ) X ⊢ Y , or coherent ( in bounds ) X ̸⊢ Y .
What is a Language ? . http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . then and + cut: If and then + weakening: If 14 of 62 ▶ A syntax ▶ positions [ X : Y ] , where each member of X is asserted and each member of Y is denied , which are either incoherent ( out of bounds ) X ⊢ Y , or coherent ( in bounds ) X ̸⊢ Y . + identity: A ⊢ A .
+ cut: If What is a Language ? and then . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 14 of 62 ▶ A syntax ▶ positions [ X : Y ] , where each member of X is asserted and each member of Y is denied , which are either incoherent ( out of bounds ) X ⊢ Y , or coherent ( in bounds ) X ̸⊢ Y . + identity: A ⊢ A . + weakening: If X ⊢ Y then X, A ⊢ Y and X ⊢ A, Y .
What is a Language ? GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 14 of 62 ▶ A syntax ▶ positions [ X : Y ] , where each member of X is asserted and each member of Y is denied , which are either incoherent ( out of bounds ) X ⊢ Y , or coherent ( in bounds ) X ̸⊢ Y . + identity: A ⊢ A . + weakening: If X ⊢ Y then X, A ⊢ Y and X ⊢ A, Y . + cut: If X ⊢ A, Y and X, A ⊢ Y then X ⊢ Y .
What is a Translation? may be incoherence preserving : . may be coherence preserving : . may be compositional (e.g., , so .) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 15 of 62
What is a Translation? may be incoherence preserving : . may be coherence preserving : . may be compositional (e.g., , so .) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 15 of 62 t : L 1 → L 2
What is a Translation? may be coherence preserving : . may be compositional (e.g., , so .) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 15 of 62 t : L 1 → L 2 ▶ t may be incoherence preserving : X ⊢ L 1 Y ⇒ t ( X ) ⊢ L 2 t ( Y ) .
What is a Translation? may be compositional (e.g., , so .) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 15 of 62 t : L 1 → L 2 ▶ t may be incoherence preserving : X ⊢ L 1 Y ⇒ t ( X ) ⊢ L 2 t ( Y ) . ▶ t may be coherence preserving : X ̸⊢ L 1 Y ⇒ t ( X ) ̸⊢ L 2 t ( Y ) .
GregRestall What is a Translation? http://consequently.org/presentation/2015/verbal-disputes-oxford/ 15 of 62 t : L 1 → L 2 ▶ t may be incoherence preserving : X ⊢ L 1 Y ⇒ t ( X ) ⊢ L 2 t ( Y ) . ▶ t may be coherence preserving : X ̸⊢ L 1 Y ⇒ t ( X ) ̸⊢ L 2 t ( Y ) . ▶ t may be compositional (e.g., t ( A ∧ B ) = ¬ ( ¬ t ( A ) ∨ ¬ t ( A )) , so t ( λp.λq. ( p ∧ q )) = λp.λq. ( ¬ ( ¬ p ∨ ¬ q )) .)
Example Translations , a de Morgan translation. . This is coherence and incoherence preserving , and compositional . , interpreting arithmetic into set theory. This is compositional and coherence preserving , but not incoherence preserving for fol derivability. is true in all models (whether the axioms of pa hold or not). Its translation is a zf theorem but not true in all models. while . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 16 of 62 ▶ t α ( going round ) = going round 1 ; t δ ( going round ) = going round 2 .
Example Translations preserving , and compositional . , interpreting arithmetic into set theory. This is compositional and coherence preserving , but not incoherence preserving for fol derivability. is true in all models (whether the axioms of pa hold or not). Its translation is a zf theorem but not true in all models. while . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 16 of 62 ▶ t α ( going round ) = going round 1 ; t δ ( going round ) = going round 2 . ▶ dm : L [ ∧ , ∨ , ¬ ] → L [ ∨ , ¬ ] , a de Morgan translation. dm ( A ∧ B ) = ¬ ( ¬ dm ( A ) ∨ ¬ dm ( B )) . This is coherence and incoherence
Example Translations preserving , and compositional . This is compositional and coherence preserving , but not incoherence preserving for fol derivability. is true in all models (whether the axioms of pa hold or not). Its translation is a zf theorem but not true in all models. while . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 16 of 62 ▶ t α ( going round ) = going round 1 ; t δ ( going round ) = going round 2 . ▶ dm : L [ ∧ , ∨ , ¬ ] → L [ ∨ , ¬ ] , a de Morgan translation. dm ( A ∧ B ) = ¬ ( ¬ dm ( A ) ∨ ¬ dm ( B )) . This is coherence and incoherence ▶ s : L [ 0, ′ , + , × ] → L [ ∈ ] , interpreting arithmetic into set theory.
Example Translations preserving , and compositional . This is compositional and coherence preserving , but not incoherence preserving for fol is a zf theorem but not true in all models. while . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 16 of 62 ▶ t α ( going round ) = going round 1 ; t δ ( going round ) = going round 2 . ▶ dm : L [ ∧ , ∨ , ¬ ] → L [ ∨ , ¬ ] , a de Morgan translation. dm ( A ∧ B ) = ¬ ( ¬ dm ( A ) ∨ ¬ dm ( B )) . This is coherence and incoherence ▶ s : L [ 0, ′ , + , × ] → L [ ∈ ] , interpreting arithmetic into set theory. derivability. ( ∀ x )( ∃ y )( y = x + 1 ) is true in all models (whether the axioms of pa hold or not). Its translation ( ∀ x ∈ ω )( ∃ y ∈ ω )( ∀ z )( z ∈ y ≡ ( z ∈ x ∨ z = x ))
Example Translations preserving , and compositional . This is compositional and coherence preserving , but not incoherence preserving for fol is a zf theorem but not true in all models. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 16 of 62 ▶ t α ( going round ) = going round 1 ; t δ ( going round ) = going round 2 . ▶ dm : L [ ∧ , ∨ , ¬ ] → L [ ∨ , ¬ ] , a de Morgan translation. dm ( A ∧ B ) = ¬ ( ¬ dm ( A ) ∨ ¬ dm ( B )) . This is coherence and incoherence ▶ s : L [ 0, ′ , + , × ] → L [ ∈ ] , interpreting arithmetic into set theory. derivability. ( ∀ x )( ∃ y )( y = x + 1 ) is true in all models (whether the axioms of pa hold or not). Its translation ( ∀ x ∈ ω )( ∃ y ∈ ω )( ∀ z )( z ∈ y ≡ ( z ∈ x ∨ z = x )) ⊢ ( ∀ x )( ∃ y )( y = x + 1 ) while ̸⊢ t [( ∀ x )( ∃ y )( y = x + 1 )] .
A General Scheme… said to be resolved by translations http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . and , , and , For some language iff and ) is A dispute denies and asserts (where , over language of , and of language between a speaker 17 of 62
A General Scheme… and http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . and , , and , For some language iff said to be resolved by translations ) is denies and asserts (where , over language of and 17 of 62 A dispute between a speaker α of language L α ,
A General Scheme… iff http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . and , , and , For some language and said to be resolved by translations ) is denies and asserts (where over 17 of 62 A dispute between a speaker α of language L α , and δ of language L δ ,
A General Scheme… For some language http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . and , , and , iff and said to be resolved by translations ) is denies and asserts (where 17 of 62 A dispute between a speaker α of language L α , and δ of language L δ , over C
A General Scheme… is said to be resolved by translations and iff For some language , , and , and . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 17 of 62 A dispute between a speaker α of language L α , and δ of language L δ , over C (where α asserts C and δ denies C )
A General Scheme… For some language , , and , and . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 17 of 62 A dispute between a speaker α of language L α , and δ of language L δ , over C (where α asserts C and δ denies C ) is said to be resolved by translations t α and t δ iff
A General Scheme… and . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 17 of 62 A dispute between a speaker α of language L α , and δ of language L δ , over C (where α asserts C and δ denies C ) is said to be resolved by translations t α and t δ iff ▶ For some language L ∗ , t α : L α → L ∗ , and t δ : L δ → L ∗ ,
GregRestall A General Scheme… http://consequently.org/presentation/2015/verbal-disputes-oxford/ 17 of 62 A dispute between a speaker α of language L α , and δ of language L δ , over C (where α asserts C and δ denies C ) is said to be resolved by translations t α and t δ iff ▶ For some language L ∗ , t α : L α → L ∗ , and t δ : L δ → L ∗ , ▶ and t α ( C ) ̸⊢ L ∗ t δ ( C ) .
…and its Upshot Given a resolution by translation, The position (in ) is coherent. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 18 of 62 there is no disagreement over C in the shared language L ∗ .
…and its Upshot Given a resolution by translation, GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 18 of 62 there is no disagreement over C in the shared language L ∗ . The position [ t α ( C ) : t δ ( C )] (in L ∗ ) is coherent.
Taking Disputes to be Resolved by Translation To take a dispute to be resolved by translation is to take there to be a pair of translations that resolves the dispute. (You may not even have the translations in hand.) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 19 of 62
Taking Disputes to be Resolved by Translation To take a dispute to be resolved by translation is to take there to be a pair of translations that resolves the dispute. (You may not even have the translations in hand.) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 19 of 62
a method …
… to resolve any dispute by translation. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 21 of 62
Resolution by DisjointUnion GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 22 of 62
Resolution by DisjointUnion Or, what I like to call “the way of the undergraduate relativist.” GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 22 of 62
Resolution by DisjointUnion GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 22 of 62 L α L δ C C t α t δ t α ( C ) t δ ( C ) L α | δ = L α ⊔ L δ L α | δ = L α ⊔ L δ
Resolution by DisjointUnion GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 22 of 62 L α L δ C C t α t δ C C L α | δ = L α ⊔ L δ L α | δ = L α ⊔ L δ
Resolution by DisjointUnion This is a coherence relation. http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall and coherence and incoherence preserving too. This ‘translation’ is structure preserving, with no interaction. The vocabularies slide past one another . or iff , For coherence on are the obvious injections. 23 of 62 L α | δ is the disjoint union L α ⊔ L δ , and t α : L α → L α | δ , t δ : L δ → L α | δ
Resolution by DisjointUnion are the obvious injections. This is a coherence relation. The vocabularies slide past one another with no interaction. This ‘translation’ is structure preserving, and coherence and incoherence preserving too. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 23 of 62 L α | δ is the disjoint union L α ⊔ L δ , and t α : L α → L α | δ , t δ : L δ → L α | δ For coherence on L α | δ , ( X α , X δ ⊢ Y α , Y δ ) iff ( X α ⊢ Y α ) or ( X δ ⊢ Y δ ) .
Resolution by DisjointUnion are the obvious injections. This is a coherence relation. The vocabularies slide past one another with no interaction. This ‘translation’ is structure preserving, and coherence and incoherence preserving too. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 23 of 62 L α | δ is the disjoint union L α ⊔ L δ , and t α : L α → L α | δ , t δ : L δ → L α | δ For coherence on L α | δ , ( X α , X δ ⊢ Y α , Y δ ) iff ( X α ⊢ Y α ) or ( X δ ⊢ Y δ ) .
Resolution by DisjointUnion are the obvious injections. This is a coherence relation. The vocabularies slide past one another with no interaction. This ‘translation’ is structure preserving, and coherence and incoherence preserving too. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 23 of 62 L α | δ is the disjoint union L α ⊔ L δ , and t α : L α → L α | δ , t δ : L δ → L α | δ For coherence on L α | δ , ( X α , X δ ⊢ Y α , Y δ ) iff ( X α ⊢ Y α ) or ( X δ ⊢ Y δ ) .
( ’s assertion of is coherent) and ( ’s denial of is coherent) then (Asserting -from- and denying -from- is coherent.) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 24 of 62 This ‘resolves’ the dispute over C If C ̸⊢ L α
and ( ’s denial of is coherent) then (Asserting -from- and denying -from- is coherent.) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 24 of 62 This ‘resolves’ the dispute over C If C ̸⊢ L α ( α ’s assertion of C is coherent)
( ’s denial of is coherent) then (Asserting -from- and denying -from- is coherent.) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 24 of 62 This ‘resolves’ the dispute over C If C ̸⊢ L α ( α ’s assertion of C is coherent) and ̸⊢ L δ C
(Asserting then -from- and denying -from- is coherent.) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 24 of 62 This ‘resolves’ the dispute over C If C ̸⊢ L α ( α ’s assertion of C is coherent) and ̸⊢ L δ C ( δ ’s denial of C is coherent)
(Asserting -from- and denying -from- is coherent.) GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 24 of 62 This ‘resolves’ the dispute over C If C ̸⊢ L α ( α ’s assertion of C is coherent) and ̸⊢ L δ C ( δ ’s denial of C is coherent) then C ̸⊢ L α | δ C
http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall 24 of 62 This ‘resolves’ the dispute over C If C ̸⊢ L α ( α ’s assertion of C is coherent) and ̸⊢ L δ C ( δ ’s denial of C is coherent) then C ̸⊢ L α | δ C (Asserting C -from- L α and denying C -from- L δ is coherent.)
… and its cost
http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall 26 of 62 Nothing α says has any bearing on δ , or vice versa .
Losing my Conjunction There’s no such sentence in ! GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 27 of 62 What is A ∧ B ?
Losing my Conjunction GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 27 of 62 What is A ∧ B ? There’s no such sentence in L α | δ !
The Case of the Venusians Suppose aliens land on earth speaking our languages and familiar with our cultures and tell us that for more complete communication it will be necessary that we increase our vocabulary by the addition of a 1-ary that certain restrictions to our familiar inferential practices will need to be — Lloyd Humberstone, The Connectives §4.34 GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 28 of 62 sentence connective V … concerning which we should note immediately imposed. As these Venusian logicians explain, ( ∧ E) will have to be curtailed. Although for purely terrestrial sentences A and B , each of A and B follows from their conjunction A ∧ B , it will not in general be the case that V A follows from V A ∧ B , or that V B follows from A ∧ V B …
Losing our Conjunction if http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . So, there’s no conjunction in (no). (possible) and then is in (no). (possible) and then is in if then and If 29 of 62 If some statements A (from L α ) and B (from L δ ) are both deniable (so ̸⊢ A , and ̸⊢ B ) then no sentence in L α | δ entails both A and B .
Losing our Conjunction is in http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall . So, there’s no conjunction in (no). (possible) and then if (no). (possible) and then is in if 29 of 62 If some statements A (from L α ) and B (from L δ ) are both deniable (so ̸⊢ A , and ̸⊢ B ) then no sentence in L α | δ entails both A and B . If C ⊢ A and C ⊢ B then
Losing our Conjunction if is in then (possible) and (no). So, there’s no conjunction in . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 29 of 62 If some statements A (from L α ) and B (from L δ ) are both deniable (so ̸⊢ A , and ̸⊢ B ) then no sentence in L α | δ entails both A and B . If C ⊢ A and C ⊢ B then ▶ if C is in L α then C ⊢ A (possible) and ⊢ B (no).
So, there’s no conjunction in Losing our Conjunction . GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 29 of 62 If some statements A (from L α ) and B (from L δ ) are both deniable (so ̸⊢ A , and ̸⊢ B ) then no sentence in L α | δ entails both A and B . If C ⊢ A and C ⊢ B then ▶ if C is in L α then C ⊢ A (possible) and ⊢ B (no). ▶ if C is in L δ then C ⊢ B (possible) and ⊢ C (no).
Losing our Conjunction GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 29 of 62 If some statements A (from L α ) and B (from L δ ) are both deniable (so ̸⊢ A , and ̸⊢ B ) then no sentence in L α | δ entails both A and B . If C ⊢ A and C ⊢ B then ▶ if C is in L α then C ⊢ A (possible) and ⊢ B (no). ▶ if C is in L δ then C ⊢ B (possible) and ⊢ C (no). So, there’s no conjunction in L α | δ .
preservation
31 of 62 in . http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall ”.) and . There is no sentence “ (There is no conjunction in and We can mean many different things by ‘and’. , , for all and and iff: Let’s say that ‘and’ is a conjunction in Have we got conjunction in L ?
31 of 62 in . http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall ”.) and . There is no sentence “ (There is no conjunction in and We can mean many different things by ‘and’. , , for all and and iff: Let’s say that ‘and’ is a conjunction in Have we got conjunction in L ?
31 of 62 in . http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall ”.) and . There is no sentence “ (There is no conjunction in and We can mean many different things by ‘and’. , , for all and and Have we got conjunction in L ? Let’s say that ‘and’ is a conjunction in L iff:
31 of 62 We can mean many different things by ‘and’. http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall ”.) and . There is no sentence “ (There is no conjunction in Have we got conjunction in L ? Let’s say that ‘and’ is a conjunction in L iff: X, A, B ⊢ Y = [ and ↕ ] = = = = = = = = = = = X, A and B ⊢ Y for all X , Y , A and B in L .
31 of 62 We can mean many different things by ‘and’. http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall Have we got conjunction in L ? Let’s say that ‘and’ is a conjunction in L iff: X, A, B ⊢ Y = [ and ↕ ] = = = = = = = = = = = X, A and B ⊢ Y for all X , Y , A and B in L . (There is no conjunction in L α | δ . There is no sentence “ A and B ”.)
Preservation GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 32 of 62 A translation t : L 1 → L 2 is conjunction preserving if a conjunction in L 1 is translated by a conjunction in L 2 .
Preservation seems like a good idea Translations should keep some things preserved. Let’s see what we can do with this. GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 33 of 62
examples
Conjunction Obviously, there some disagreements can resolved by a disambiguation of different senses of the word ‘and.’ ‘and ’ ‘ ’ ‘and ’ ‘and then ’ GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 35 of 62
Conjunction Obviously, there some disagreements can resolved by a disambiguation of different senses of the word ‘and.’ GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 35 of 62 t α t δ → ‘ ∧ ’ → ‘and then ’ − − ‘and α ’ ‘and δ ’
No Verbal Disagreement Between Two Conjunctions That is, in http://consequently.org/presentation/2015/verbal-disputes-oxford/ GregRestall Why? . and , . If the following two conditions hold: then ‘ ’ and ‘ ’ are equivalent in are both conjunction preserving . , and 2. , and and ‘ ’ is a conjunction in 1. ‘ ’ is a conjunction in 36 of 62
No Verbal Disagreement Between Two Conjunctions If the following two conditions hold: 2. , and are both conjunction preserving . then ‘ ’ and ‘ ’ are equivalent in . That is, in , and . Why? GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 36 of 62 1. ‘ ∧ ’ is a conjunction in L 1 and ‘ & ’ is a conjunction in L 2 , and
No Verbal Disagreement Between Two Conjunctions If the following two conditions hold: then ‘ ’ and ‘ ’ are equivalent in . That is, in , and . Why? GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 36 of 62 1. ‘ ∧ ’ is a conjunction in L 1 and ‘ & ’ is a conjunction in L 2 , and 2. t 1 : L 1 → L ∗ , and t 2 : L 2 → L ∗ are both conjunction preserving .
No Verbal Disagreement Between Two Conjunctions If the following two conditions hold: That is, in , and . Why? GregRestall http://consequently.org/presentation/2015/verbal-disputes-oxford/ 36 of 62 1. ‘ ∧ ’ is a conjunction in L 1 and ‘ & ’ is a conjunction in L 2 , and 2. t 1 : L 1 → L ∗ , and t 2 : L 2 → L ∗ are both conjunction preserving . then ‘ ∧ ’ and ‘ & ’ are equivalent in L ∗ .
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