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In this case we can push the Cut upwards beyond the disjunction rule, as
follows
X(B) C
Y (C) D
Y (X(B) D
[CutC ]
X(A) C
Y (C) D
Y (X(A) D
Y (X(A B) D
[CutC ]
[L]
In this way we have pushed the Cut upwards in

H ILBERT S YSTEMS
CONS
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83
If X(A) = Y (A) then we wish to prove
(Y (A B)
Y (A)
Y (B).
Y (A), and Y (B) "
Y (B), and
If " is present, then we have Y (A) "
Y (A) "
Y (B), and Y (B) "
Y (A) "
Y (B), which
hence Y (A) "
Y (A)"
Y (B),

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11
X A B follows from X; A B. Conversely, if X A B, then if we
apply X to A, then B is a consequence, for A B is a conditional fact given
by X. So, in reasoning about bodies of information in general, the dedu

M ODALITIES
CONS
1999/11/6
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59
of linear logic. We will just work through two representative cases and leave
the rest to the exercises. For conjunction introduction, the intuitionistic rule
XA
XB
X AB
is straightforwardly mapped to the same rule i

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E XAMPLE 8.15 (P OWER S ETS ARE D ISTRIBUTIVE )
It is not too difficult to show that x a (b c) if and only if x (a b) (a c).
So a (b c) = (a b) (a c) for each set a, b and c, and power set

CATEGORIES
CONS
1999/11/6
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223
E XAMPLE 10.27 (F UNCTION O BJECTS IN C ONCRETE CATEGORIES )
[A B] is usually constructed as the set of all homomorphisms from A to B,
equipped with some sort of structure inherited from A and B. For example,
in Lat

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CONS
1999/11/6
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63
We can also relate our negation operators to conjunction and disjunction.
L EMMA 3.21 (O NE DE M ORGAN LAW )
For any split negation we have A B (A B) in any logic in which and
are present (and similarly for in place

I FS , A NDS
O RS
AND
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23
Note that these rules assume that we are in a language with a conditional,
and a system of structures with at least the semicolon. These rules make sense
in any language with at least these resources. We ha

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AND
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1999/11/6
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39
families. The only place where we do not have this substitution is instances of
the axioms. The only leaves which might have changed from the old proof to
the new one are those of the form A A. But these ch

F ORMULAE
AS
T YPES , P ROOFS
AS
T ERMS
CONS
1999/11/6
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143
Similarly, given a Gentzen proof ending in an implication right step, the corresponding natural deduction proof ends in the same step.
N ()
; x:A M :B
N
xM :A B
; x:A M :B
xM :A B
A mo

CONS
1999/11/6
page 71
M ODALITIES
71
cfw_3.13 Is A equivalent to A in R? Are they equivalent when you add the structural rule
M?
cfw_3.14 Show that A !B A, !A A, t !t and !A !A !A in LL.
cfw_3.15 Show that !A is equivalent to t A (!A !A).
cfw_3.16 Show

M ODALITIES
CONS
1999/11/6
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67
D EFINITION 3.35 (S EMICOLON N EGATION R ULES )
The standard rules tying together a split negation with intensional structure are
these:
X; A B
Y B
X; Y A
(I; E)
X; A B
Y B
X; Y A
(I; E)
Note that in these rules, if

CATEGORIES
CONS
1999/11/6
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219
the unique map from A B to A B. It is enough to show that (l , r )pl , pr
!
and pl , pr (l , r ) are identity arrows.
If a category has products, then it is also possible to construct product arrows
in the followin

F ORMULAE
AS
T YPES , P ROOFS
AS
T ERMS
CONS
1999/11/6
page 135
135
So Cut, for proofs, corresponds exactly to substitution of proofs.
The next nice result of term systems is the Decomposition Theorem. It is a
ready consequence of the nature of terms. Gi

D EFINING P ROPOSITIONAL S TRUCTURES
CONS
1999/11/6
page 171
171
E XAMPLE 8.38 (T RUTH VALUES )
As we have already seen, the truth values cfw_t, f form a propositional structure.
We represent the structure graphically by a Hasse diagram. The other operat

F RAMES I: L OGICS
WITH
D ISTRIBUTION
CONS
1999/11/6
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255
x ! A iff A x
The crucial lemma involves showing that this is a sensible relation. That is,
the membership relation in the canonical model satisfies the conditions for an
evaluation.
L EMM

F ORMULAE
AS
T YPES , P ROOFS
AS
T ERMS
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becomes the Gentzen proof
G(2 )
(0) M :C
G(1 )
(z:t) let z be in M :C
N :t
() let N be in M :C
[Cut]
Conversely, the Gentzen proof
(0) M :C
(z:t) let z be in M :C
becomes the natural de

CONS
1999/11/6
page 127
Chapter 7
Formulae as Types, Proofs as Terms
The reduction of Egypt
was immediately followed by the Persian War.
Edward Gibbon, The History of the Decline and Fall of the Roman Empire, 1776
A closer analysis of proofs than what w

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CONS
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Closed under !. That is, if a ! b and a F then b F .
Closed under , if is present in the structure. That is, if a, b F then
a b F.
Contains , if is in the structure.
An ideal is a fi

CONS
1999/11/6
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D EFINING P ROPOSITIONAL S TRUCTURES
187
cfw_8.8 Consider the divisibility structure for R, Example 8.43, introduced on page 174.
Show that this structure has is no residual for extensional conjunction, despite the fact
that the

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CONS
1999/11/6
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167
D EFINITION 8.28 (N EGATIVE M ODAL O PERATORS )
A pair of unary operations and is an n-type negative pair on an ordered set
if and only if a ! b if and only if b ! a. A pair of unary operat

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CONS
1999/11/6
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91
The t condition is t is true. The implication condition is if A B is true,
then if A is true so is B. The non-triviality conditions mean simply that is
true and is not.
In traditional, classical logics, many of these c

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CONS
1999/11/6
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107
At this point, we should note that since we are allowing empty consequents
in our consecutions, our rules should be understood in such a way as to allow
this. In some of the rules of the system, the consequen

I NTRODUCTION
CONS
1999/11/6
page 3
3
E XAMPLE 1.2 (R ESOURCE C ONSCIOUSNESS )
This is not the only way to restrict premise combination. Girard [96] introduced
linear logic as a model for processes and resource use. The idea in this account
of deduction

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CONS
1999/11/6
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115
which satisfies the non-proliferation condition. Both X and Y are parameters
in the conclusion, but they are not congruent with one another, not even in
the case in which X = Y . Remember: Congruence entails

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CONS
1999/11/6
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183
For the structural rules, if we have Y X then if Z(X) A, we have
[Z(X)] ! [A], and since [Y ] ! [X], we have [Z(Y )] ! [Z(X)][A], as desired.
"
The converse is only marginally more difficul

T HEORIES
CONS
1999/11/6
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99
D EFINITION 5.27 (M ODAL C ONFUSION )
A modal confusion of the propositions in the set is defined inductively as
follows.
t, and any element of are confusions of .
C1 .
If C1 and C2 are confusions of , so are C1 C2

F ORMULAE
AS
T YPES , P ROOFS
AS
T ERMS
CONS
1999/11/6
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131
If M is a term of type B then xA M is a term of type A B. The free
variables of xA M are those in M , except for xA , which is now bound.
; x:A M :B
xM :A B
( abst)
In this rule we req

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CONS
1999/11/6
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159
partial order on the structure by lines. The ordering relation ! is read upwards:
a point a is below b in the order if and only if there is an upward path from a to
b in the diagram.
4
2
3

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CONS
1999/11/6
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191
E XAMPLE 9.4 (BN4 AND RM3 )
There are no homomorphisms from BN4 to RM3 , as there are no maps which
will preserve both negation and conjunction and . If there were such a map
f , consid

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O RS
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1999/11/6
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right identity (with respect to the binary punctuation mark ;) if it satisfies the
following two structural rules:
X
X; 0
X; 0
X
Right Push
Right Pop
If 0 is a left identity, then applying 0 to X (on the

I FS , A NDS
AND
O RS
CONS
1999/11/6
page 15
15
We have defined string algebras to be any sort of thing which satisfies these
criteria. However, merely listing a set of criteria does not ensure that there is
something which satisfies them. After all, our

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WITH
D ISTRIBUTION
CONS
1999/11/6
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Similarly, SR(Y ; Z) if and only if for some prime theories y, z, where Ryzx,
SR(Y )[x := y] and SR(Z)[x := z]. That is equivalent to there being prime
theories y and z, where Ryzx, Y (x1

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9.5 The Kleene Star in Structures
The intended interpretation of the Kleene star in a logic is infinitary iteration. In
2
3
a propositional structure,
! nwe would like a to be the propo