1. Slides 07 1 Restricted -terms and logics 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. Sometimes unrestricted -terms are called K- terms. I-terms are terms without vacuous binding.

2. Slides 07 2 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, C 0  xy  y, C 1  xy  xy The following are not: C 2  xy  x(xy), S  xyz  xz(yz), W  xy  xyy

3. Slides 07 3 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.

4. Slides 07 4 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.

5. Slides 07 5 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.

6. Slides 07 6 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).

7. Slides 07 7 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 .

8. Slides 07 8 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 x 1,..., x n (distinct) such that x 1 :  1,..., x n :  n ⊢ M:  and M is, respectively, a I-term, BCK -term or BCI -term.

9. Slides 07 9 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 ]

10. Slides 07 10 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 ⊢.

11. Slides 07 11 The rule (Sub) is the substitution rule. Its side-condition says that substitutions may be made only for variables that occur in axioms.

12. Slides 07 12 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

13. Slides 07 13 Definition (6D2) In any A-logic, a substitutions- first 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.

14. Slides 07 14 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(  )

15. Slides 07 15 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.

16. Slides 07 16 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.

17. Slides 07 17 Definition (6D4) Hilbert-style R  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, (I) a  a, (W)(a  a  b)  a  b.

18. Slides 07 18 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.

19. Slides 07 19 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.

20. Slides 07 20 Example: By Example (6D1.2) I  a  a is provable in Hilbert-style BCK-logic and in Hilbert- style intuitionistic logic.

21. Slides 07 21 (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.

22. Slides 07 22 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 S- combinations will be called BCK-combinations, BCIW- combinations,etc. Examples: If S = {B, C, K} then CKK and B are S- combinations and CKx, xy and CKK are S-and-variables combinations. But x.BC is neither an S-combination nor an S-and-variables combination.

23. Slides 07 23 CKK  ?

24. Slides 07 24 Curry-Howard Theorem for Hilbert systems (6D7) Let {C 1, C 2,...} be a finite or infinite set of typable closed -terms and let A = {  1,  2,...} where  i  PT(C i ). Then (i)the theorems of A-logic are exactly the types of the typable applicative combinations of C 1, C 2,..., (ii)the relation  1,...,  n ⊢ A  holds iff there exist an applicative combination M of C 1, C 2,..., and some distinct term-variables x 1,..., x n, such that x 1 :  1,..., x n :  n ⊢ M: .

25. Slides 07 25 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 x 1,..., x n, C 1, C 2,..., and let  be a TA -deduction of x 1 :  1,..., x n :  n ↦ M: . (1) Corresponding to each occurrence of a C i in M there will be an occurrence of ↦ C i :s(  i ) in  for some substitution s. Remove from  all steps above these occurrences of C 1, C 2,..., and replace each formula ↦ C i :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)

26. Slides 07 26 Now, we prove the "only if"-part. Let  1,...,  n ⊢  in A- logic. 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 term- variables x 1,..., x n and replace each undischarged branch- top occurrence of each  i in  by x i :  i ↦ x i :  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 ↦ C k :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.

27. Slides 07 27 Theorem (6D8: Hilbert-Gentzen link) For the intuitionistic logic, R  -logic, BCK- logic, 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 so- called Deduction Theorem, without going through - calculus.