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When we are talking about a language Lang(AForm; Conn) and we do not
particularly care about AForm or Conn, we will simply refer to it as Lang. We
use A, B and other capitals from the beginning of the alpha
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Chapter 5
Theories
Dear friend,
theory is all grey,
And the golden tree of life is green.
Goethe, Faust
Logic is not merely about individual propositions and consequence relations between them. Theories are interesting too. Theori
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If we add these rules we have a converse contraposition result.
L EMMA 3.27 (C ONVERSE C ONTRAPOSITION )
In any logic containing both double negation elimination rules, the two-way contraposition rule
A B
=
B A
holds
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by Mints and Dunn, though their work was on Gentzen-style consecution systems (with introduction rules for connectives in the antecedent and consequent,
instead of introduction and elimination rules in the
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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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which are the same backwards as forwards). The string abab is a concatenation
of two two-letter strings, ab and ab, so it has type A . It is also a concatenation
of two palindromes, aba and b, so it has type B . Is t
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Chapter 4
Hilbert Systems
What is now proved
was once only imagind.
William Blake, The Marriage of Heaven and Hell
We have examined natural deduction systems for the last two chapters. These
are not the only way to present a logic
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By the Step Lemma, each xn+1 , yn+1 is a -pair if its predecessor xn , yn
is, for there is always a choice (left or right) for placing Cn while keeping the
result!"
a -pair. So,
on n, each xn , yn is a -pair. It follo
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Note the following result.
L EMMA 4.27 (C ONFUSION C LOSURE )
In any Hilbert system defined above, H B whenever B is a confusion of .
P ROOF By induction on the complexity of B. For the base case, H t H ,
and
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D EFINITION 4.1 (E XPRESSIVITY OF A L ANGUAGE )
A language Lang is expressive of a set of structures Struct if and only if
For every identity punctuation mark 0 in Struct, Lang has a corresponding
truth consta
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For example, the collection of all inferences of the from
X; (Y ; Z) A
(X; (Y ; W ); Z A
is a structural rule. For the A in the consequent position is arbitrary, and so
are the formulae which appear in the
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any action of type A, composed with it, results in an action of type f . The order
of actions is important.
So, if f = (so there are no actions of type f ), an action is of type A if
it cannot be followed by an actio
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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
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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
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These conditions are characteristic of possibility-style operators. If A B is
possible (true in some related state) then one of A and B must be possible too.
Similarly, there is no way that can be possible, as it is
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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
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Practice
cfw_2.1 In the table below, match each consecution with the structural rule needed to
prove it.
AAA
B A (B C) (A C)
A (A B) B
ABB
A B (B C) (A C)
K
M
Bc
WI
B
cfw_2.2 Show that in any system contai
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as we eat up l 1 of the l in the front of the sum to beef up the l-numbered
value of the sum in the interior to nm . This means that the hypothesis holds
of A too, given that it holds of the subformulae of
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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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Here is another proof, using C.
A (B C) A (B C) A A
(E)
A (B C); A B C
!
"
A (B C); A ; B C
!
"
A (B C); B ; A C
A (B C); B A C
BB
(E)
[C]
(I)
A (B C) B (A C)
(I)
The structural rule C gives us strong commu
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P ROOF The positive modality and properties are simple to prove:
A
A
AB
A
B
A
B
(I)
(E)
(E)
(E)
For the disjunction property, we have
AA
AB AB
A
A
BB
(I)
A
A
B
B
B
(I)
(A B)
A
B
(I)
B
A
B
(A B)
A
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a string of type B. I leave it to you to consider how you should read fusion in
other applications.
Whatever the interpretation of fusion, we have a number of important behaviours. For example, we can prove
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Chapter 2
Ifs, Ands and Ors
A syllogism is a form of words
in which when certain assumptions are made,
something other than what has been assumed
necessarily follows
from the fact that the assumptions are such
Aristotle, Prior Anal
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It will be shown in Chapters 9 and 1110 that adding the comma with these
structural rules does not alter the properties of any connectives other than
and . This is not obvious, because there just might be
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L EMMA 4.24 (F USION A XIOMS )
In the presence of B and Bc you can replace the fusion rules with (A B C)
(A (B C).
P ROOF It is sufficient to show that these are provable in the natural deduction
system, give
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D EFINITION 2.17 (I NFERENCES AND R ULES )
An inference is a pair consisting of a set of consecutions (the premises of the
inference) and a single consecution (the conclusion of the inference). The
premises
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numbers to themselves. So any f S is a function which, when given a number
x, returns a number f (x). The function f : x " 3x + 1, when given 5, returns
16, for example.
We will assume that the set S is cou
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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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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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theory containing A (B C) but not (A B) (A C). If there were such a
theory, it would contain either A B or A C. But since the theory contains
A (B C), it must contain A and B C, and by primeness, either B or C. This
me
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C OROLLARY 4.18 (P ROVING C ONSECUTIONS IN HS)
If (X A) is provable in the Hilbert system HS, then X A is provable in S.
P ROOF If (X A) is provable in the Hilbert system, then our previous theorem shows that t
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P ROOF In these proofs we use Lemma 3.38 and double negation elimination
liberally, sliding between A and A t, and between B and B at will.
A B A B
(A B); A B t
(A B); A); B t
(A; B); (A B) t
[B, C]
A; B (A B) t
A B
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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
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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
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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