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Restricted -terms and logics
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Definition (1D1) A -term P is called a I-term
iff, for each subterm with the form xM in P, x
occurs free in M at least once.
Example:
I  xx is a I-term;
K  xyx is a non-I-term.
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•
Sometimes unrestricted -terms are called Kterms.
I-terms are terms without vacuous binding.
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• Definition (1D2)
A BCK-term is a -term P such that
(i) for each subterm xM of P, x occurs free in
M at most once,
(ii) each free variable of P has just one
occurrence free in P.
• Examples: the following are BCK-terms:
I  xx, B  xyzx(yz), B'  xyzy(xz),
C  xyzxzy, K  xyx, C0  xyy, C1  xyxy
The following are not:
C2  xyx(xy), S  xyzxz(yz), W  xyxyy
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• Definition (1D3)
A BCI-term or linear -term is a -term P such
that
(i) for each subterm xM of P, x occurs free in
M exactly once,
(ii) each free variable of P has just one
occurrence free in P.
• Every BCI-term is a BCK-term, but the BCKterm K is not a BCI-term.
• A term is a BCI-term iff it is both a I-term and
a BCK-term.
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• Definition (6C2) The implicational fragment of
BCK-logic is defined exactly like intuitionistic
logic in (6A2) on Slide 06.19, except that
multiple discharging is not allowed. That is,
when (I) is used, its discharge-label must
either be vacuous or contain only one
occurrence of .
• BCK-logic is a logic in which an assumption
cannot be used more than once; it is a logic of
non-reusable information.
• Example: the proof of (aac)aac in
(6A2.2) is a BCK-proof.
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• Definition (6C3) The implicational fragment of
BCI-logic is defined exactly like intuitionistic logic
in (6A2) on Slide 06.19, except that both
vacuous and multiple discharging are forbidden.
That is, when (I) is used, its discharge-label
must contain exactly one occurrence of .
• BCI-logic is a relevance logic of non-reusable
information.
• Example: the proof of (aac)aac in
(6A2.2) is also a BCI-proof.
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• Definition (6C1)
The definition of the relevance logic R is
exactly like that of intuitionistic logic in
(6A2) on Slide 06.19, except that vacuous
discharging is forbidden. That is, when
 is the conclusion of rule (I), its
discharge-label must contain at least one
occurrence of .
• Example:
the formula (aac)aac is provable in R (see
6A2.2).
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• Motivation for R:
In one important view of implication, a formula
 should not be provable unless  is in some
way relevant to . In this view the formula
aba is not universally valid, because it says
in essence that if a statement a is true then
every other statement b implies it, even when b
has no connection with the meaning of a.
R is one of the earliest and simplest attempts
to capture the notion of relevant implication. In it,
we can only prove  when  has actually
been used in the deduction of .
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• Refined Curry-Howard Theorem (6C5)
(i) The provable formulas of R, BCK-logic, and BCI-logic
are exactly the types of the following -terms:
R: types of the closed I-terms;
BCK-logic: types of the closed BCK-terms;
BCI-logic: types of the closed BCI-terms.
(ii) The relation 1, ..., n ⊢  holds in R, BCK-logic or
BCI-logic iff there exist M and x1, ..., xn (distinct) such that
x1:1, ..., xn:n ⊢ M:
and M is, respectively, a I-term, BCK-term or BCI-term.
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Axiomatic (Hilbert-style) Systems
• Definition (6D1) Let A be any set of implicational formulas
that are tautologies in the classical truth-table sense. Then
A generates the following Hilbert-style system, which will be
called the corresponding A-logic.
Axioms: the members of A.
Deduction-rules:


(E): 
[often called modus ponens]


(Sub): 
[ if s is a substitution and no variable in Dom(s)
s()
occurs in a non-axiom assumption in the
deduction above the line ]
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Deductions in an A-logic are trees, with axioms and
assumptions at the tops of branches and the conclusion
at the bottom of the tree. The notation
1, ..., n ⊢A 
means that there is a deduction whose non-axiom
assumptions are some or all of 1, ..., n and whose
conclusion is . (1, ..., n need not all be distinct.)
When n = 0, the deduction is called a proof of  and we
call  a provable formula or theorem of the A-logic in
question, and we write
⊢A .
The set of all theorems in an A-logic may be called A⊢.
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• The rule (Sub) is the substitution rule. Its
side-condition says that substitutions may
be made only for variables that occur in
axioms.
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• Example (6D1.2): Let A contain the formulas
C  (abc)bac and K  aba,
and s  [(aba)/b, a/c].
Then the following deduction gives ⊢A aa.
(abc)bac
aba
 (Sub)  (Sub)
(a(aba)a)(aba)aa
a(aba)a
 (E)
(aba)aa
aba
 (E)
aa
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• Definition (6D2) In any A-logic, a substitutionsfirst deduction is a deduction in which the rule
(Sub) is only applied to axioms.
• Lemma (6D2.1) In any A-logic, every deduction
 can be replaced by a substitutions-first
deduction * with the same assumptions,
axioms and conclusion.
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Proof of Lemma (6D2.1):
Suppose the rule (Sub) is applied below an application of
the rule (E), as follows:


(E) 

(Sub)

s()
Then (Sub) can be moved up above (E), thus:


(Sub) 
(Sub) 
s()s()
s()
(E) 
s()
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Two successive (Sub)'s can be combined into one. The
moving-up procedure ends when all (Sub)'s are at the
tops of branches in the deduction tree. By the restriction
on (Sub) in (6D1), the top formula of each of these
branches cannot be a non-axiom assumption, so it must
be an axiom.
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• Definition (6D3) Hilbert-style intuitionistic
logic of implication is the A-logic whose A
has just the following four members:
(B)
(ab)(ca)(cb),
(C)
(abc)bac,
(K)
aba,
(W)
(aab)ab.
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• Definition (6D4) Hilbert-style R is the Alogic whose A has just the following four
members:
(B)
(ab)(ca)(cb),
(C)
(abc)bac,
(I)
aa,
(W)
(aab)ab.
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• Definition (6D5) Hilbert-style BCK-logic of
implication is the A-logic whose A has just
the following three members:
(B)
(ab)(ca)(cb),
(C)
(abc)bac,
(K)
aba.
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• Definition (6D6) Hilbert-style BCI-logic of
implication is the A-logic whose A has just
the following three members:
(B)
(ab)(ca)(cb),
(C)
(abc)bac,
(I)
aa.
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• Example:
By Example (6D1.2) I  aa is provable
in Hilbert-style BCK-logic and in Hilbertstyle intuitionistic logic.
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• (B), (C), (I), (K), and (W) are the principal types of the terms B, C, I, K and W. Each of the formulas also
expresses a property of implication that has its own
interest quite independently of type-theory.
(I)  aa indicates the reflexivity property of implication,
(C)  (abc)bac states that hypotheses can be
commuted,
(K)  aba states that redundant hypotheses can be
added,
(W)  (aab)ab states that duplicates can be
removed,
(B)  (ab)(ca)(cb) indicates a transitivity
property of implication or a "right-handed" replacement
property which says that if ab holds, then a may be
replaced by b in the formula ca.
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• Definition (9F1)
If S is a set of -terms, an S-combination, or applicative
combination of members of S, is a -term built from
some or all of the members of S by application only. An
S-and-variables combination is an applicative
combination of members of S and variables.
For subsets of {B, B', C, I, K, S, W} (see Slide 2) the Scombinations will be called BCK-combinations, BCIWcombinations,etc.
• Examples: If S = {B, C, K} then CKK and B are Scombinations and CKx, xy and CKK are S-and-variables
combinations. But x.BC is neither an S-combination nor
an S-and-variables combination.
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• CKK  ?
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• Curry-Howard Theorem for Hilbert systems (6D7)
Let {C1, C2, ...} be a finite or infinite set of typable
closed -terms and let A = {1, 2, ...} where i 
PT(Ci). Then
(i) the theorems of A-logic are exactly the types of
the typable applicative combinations of C1, C2, ...,
(ii) the relation 1, ..., n ⊢A  holds iff there exist
an applicative combination M of C1, C2, ..., and
some distinct term-variables x1, ..., xn, such that
x1:1, ..., xn:n ⊢ M:.
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• Proof:
Part (i) is a special case of (ii) with n = 0.
We prove (ii). First, the "if"-part.
Let M be an applicative combination of x1, ..., xn, C1, C2, ...,
and let  be a TA-deduction of
x1:1, ..., xn:n ↦ M:.
(1)
Corresponding to each occurrence of a Ci in M there will be
an occurrence of ↦ Ci:s(i) in  for some substitution s.
Remove from  all steps above these occurrences of C1, C2,
..., and replace each formula ↦ Ci:s(i) by the type s(i).
Then replace every other formula in , say  ↦ N:, by the
type . The result is a Hilbert-style deduction giving
1, ..., n ⊢ .
(2)
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Now, we prove the "only if"-part. Let 1, ..., n ⊢  in Alogic. Then by Lemma (6D2.1) there is a deduction  of  in
which (Sub) is only applied to axioms. Change  to a TAdeduction as follows. First choose some distinct termvariables x1, ..., xn and replace each undischarged branchtop occurrence of each i in  by
xi:i ↦ xi:i.
Next, since each application of (Sub) in  will be applied to
an axiom to give, say, ↦ s(k); replace it by a TA-proof of
↦ Ck:s(k). Then replace the logic rule (E) by the TA-rule
(E) throughout. The result is a TA-deduction of (1) for
some term M as required.
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• Theorem (6D8: Hilbert-Gentzen link)
For the intuitionistic logic, R-logic, BCKlogic, and BCI-logic, the relation
1, ..., n ⊢ 
holds in the Natural Deduction version iff it
holds in the Hilbert version.
• Note: This link is usually proved directly using the socalled Deduction Theorem, without going through calculus.
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