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-aap/ 14 of 61 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-aap/ 14 of 61 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-aap/ 14 of 61 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-aap/ 14 of 61 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-aap/ 15 of 61 ▶ 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-aap/ 15 of 61 ▶ 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-aap/ 15 of 61 ▶ 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-aap/ 15 of 61 ▶ 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-aap/ 15 of 61 ▶ 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-aap/ GregRestall . and , , and , For some language iff and ) is A dispute denies and asserts (where , over language of , and of language between a speaker 16 of 61
A General Scheme… and http://consequently.org/presentation/2015/verbal-disputes-aap/ GregRestall . and , , and , For some language iff said to be resolved by translations ) is denies and asserts (where , over language of and 16 of 61 A dispute between a speaker α of language L α ,
A General Scheme… iff http://consequently.org/presentation/2015/verbal-disputes-aap/ GregRestall . and , , and , For some language and said to be resolved by translations ) is denies and asserts (where over 16 of 61 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-aap/ GregRestall . and , , and , iff and said to be resolved by translations ) is denies and asserts (where 16 of 61 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-aap/ 16 of 61 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-aap/ 16 of 61 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-aap/ 16 of 61 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-aap/ 16 of 61 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-aap/ 17 of 61 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-aap/ 17 of 61 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-aap/ 18 of 61
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-aap/ 18 of 61
a method …
… to resolve any dispute by translation. GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 20 of 61
Resolution by DisjointUnion GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 21 of 61
Resolution by DisjointUnion Or, what I like to call “the way of the undergraduate relativist.” GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 21 of 61
Resolution by DisjointUnion GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 21 of 61 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-aap/ 21 of 61 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-aap/ 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. 22 of 61 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-aap/ 22 of 61 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-aap/ 22 of 61 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-aap/ 22 of 61 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-aap/ 23 of 61 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-aap/ 23 of 61 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-aap/ 23 of 61 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-aap/ 23 of 61 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-aap/ 23 of 61 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-aap/ GregRestall 23 of 61 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-aap/ GregRestall 25 of 61 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-aap/ 26 of 61 What is A ∧ B ?
Losing my Conjunction GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 26 of 61 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-aap/ 27 of 61 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-aap/ GregRestall . So, there’s no conjunction in (no). (possible) and then is in (no). (possible) and then is in if then and If 28 of 61 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-aap/ GregRestall . So, there’s no conjunction in (no). (possible) and then if (no). (possible) and then is in if 28 of 61 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-aap/ 28 of 61 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-aap/ 28 of 61 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-aap/ 28 of 61 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
30 of 61 in . http://consequently.org/presentation/2015/verbal-disputes-aap/ 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 ?
30 of 61 in . http://consequently.org/presentation/2015/verbal-disputes-aap/ 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 ?
30 of 61 in . http://consequently.org/presentation/2015/verbal-disputes-aap/ 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:
30 of 61 We can mean many different things by ‘and’. http://consequently.org/presentation/2015/verbal-disputes-aap/ 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 .
30 of 61 We can mean many different things by ‘and’. http://consequently.org/presentation/2015/verbal-disputes-aap/ 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-aap/ 31 of 61 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-aap/ 32 of 61
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-aap/ 34 of 61
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-aap/ 34 of 61 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-aap/ 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 35 of 61
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-aap/ 35 of 61 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-aap/ 35 of 61 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-aap/ 35 of 61 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 ∗ .
No Verbal Disagreement Between Two Conjunctions If the following two conditions hold: Why? GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 35 of 61 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 ∗ . That is, in L ∗ , A ∧ B ⊢ A & B and A & B ⊢ A ∧ B .
No Verbal Disagreement Between Two Conjunctions If the following two conditions hold: Why? GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 35 of 61 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 ∗ . That is, in L ∗ , A ∧ B ⊢ A & B and A & B ⊢ A ∧ B .
Here's why (Since and are both conjunctions in .) GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 36 of 61 Reason as follows inside L ∗ :
Here's why GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 36 of 61 Reason as follows inside L ∗ : A & B ⊢ A & B A ∧ B ⊢ A ∧ B [& ↑ ] [ ∧ ↑ ] A, B ⊢ A & B A, B ⊢ A ∧ B [ ∧ ↓ ] [& ↓ ] A ∧ B ⊢ A & B A & B ⊢ A ∧ B (Since ∧ and & are both conjunctions in L ∗ .)
Equivalence and Verbal Disagreements If http://consequently.org/presentation/2015/verbal-disputes-aap/ GregRestall . and are equivalent, so are / and / Cut Cut . and or and involves distinguishing and deny The only way to coherently assert 37 of 61 If ‘ ∧ ’ and ‘ & ’ are equivalent, then any merely verbal disagreement betwen A ∧ B and A ′ & B ′ cannot be explained by an equivocation between ‘ ∧ ’ and ‘ & ’.
Equivalence and Verbal Disagreements and http://consequently.org/presentation/2015/verbal-disputes-aap/ GregRestall . and are equivalent, so are / / If Cut Cut 37 of 61 If ‘ ∧ ’ and ‘ & ’ are equivalent, then any merely verbal disagreement betwen A ∧ B and A ′ & B ′ cannot be explained by an equivocation between ‘ ∧ ’ and ‘ & ’. The only way to coherently assert A ∧ B and deny A ′ & B ′ involves distinguishing A and A ′ or B and B ′ .
Equivalence and Verbal Disagreements GregRestall http://consequently.org/presentation/2015/verbal-disputes-aap/ 37 of 61 If ‘ ∧ ’ and ‘ & ’ are equivalent, then any merely verbal disagreement betwen A ∧ B and A ′ & B ′ cannot be explained by an equivocation between ‘ ∧ ’ and ‘ & ’. The only way to coherently assert A ∧ B and deny A ′ & B ′ involves distinguishing A and A ′ or B and B ′ . A ′ & B ′ ⊢ A ′ & B ′ [& ↑ ] A ′ , B ′ ⊢ A ′ & B ′ B ⊢ B ′ [ Cut ] A ′ , B ⊢ A ′ & B ′ A ⊢ A ′ [ Cut ] A, B ⊢ A ′ & B ′ [ ∧ ↓ ] A ∧ B ⊢ A ′ & B ′ If A / A ′ and B / B ′ are equivalent, so are A ∧ B and A ′ & B ′ .
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