2. Logic CL4 Episode 16 0 The language of CL4 The rules of CL4 CL4 as a conservative extension of classical logic The soundness and completeness of CL4 The decidability of the blind-quantifier-free fragment of CL4 Other axiomatizations (affine and intuitionistic logics)

3. The language of CL4 16.1 CL4 is by far the most expressive deductive system for computability logic known to be sound and complete at present. Its language is obtained from the full language described in Episode 14 by forbiddingEpisode 14 only the parallel quantifiers and the recurrence operators (together with and ). We refer to the formulas of this language as CL4-formulas. To define CL4, we need to agree on some terminology. (Here) understanding E  F as an abbreviation of  E  F, a negative occurrence of a subformula is one that is in the scope of an odd number of  ’ s. Otherwise, the occurrence is positive. A surface occurrence of a subformula means an occurrence that is not in the scope of a choice operator. A CL4-formula is said to be elementary iff it does not contain general letters and choice operators. Note that elementary formulas = formulas of classical logic. The elementarization of a CL4-formula is the result of replacing, in it, every surface occurrence of the form E ⊓ F or ⊓ xE by ⊤, every surface occurrence of the form E ⊔ F or ⊔ xE by ⊥, every negative surface occurrence of each general atom by ⊤, and every positive surface occurrence of each general atom by ⊥. A CL4-formula is said to be stable iff its elementarization is valid in classical logic, i.e. provable in G3. Otherwise the formula is instable.

4. The rules of CL4 16.2 CL4 has the following four rules, with E,F standing for CL4-formulas and H for a set of CL4-formulas: where F is stable and, whenever F has a positive (resp. negative) surface occurrence of: Rule A:, (i) G 1 ⊓ G 2 (resp. G 1 ⊔ G 2 ), for each i  {1,2}, H contains the result of replacing in F that occurrence by G i (ii) ⊓ xG(x) (resp. ⊔ xG(x)), H contains the result of replacing in F that occurrence by G(y) where y does not occur in F H F where E is the result of replacing in F some negative (resp. Rule B1:, positive) surface occurrence of G 1 ⊓ G 2 (resp. G 1 ⊔ G 2 ) by G i for some i  {1,2} E F where E is the result of replacing in F some negative (resp. Rule B2:, positive) surface occurrence of ⊓ xG(x) (resp. ⊔ xG(x)) by G(t) for some term t that has no bound occurrence in F. E F where E is the result of replacing in F two – one positive and Rule C:, one negative – surface occurrences of some n-ary general letter by an n-ary elementary letter that does not occur in F E F

5. Propositional examples 16.3 CL4 ⊦ F means “F is provable in CL4”, and CL4 ⊬ F means “F is not provable in CL4” CL4 ⊦ P  P  P 1. p  P  p Justification: The elementarization of this formula is p⊤p.p⊤p. And the latter is a classical tautology. So, p  P  p is stable. And it does not contain any choice operators. Hence, it follows from the empty set {} of premises by Rule A (is an “ axiom ” ) 2. P  P  P Justification: From 1 by Rule C CL4 ⊬ P  P  P Reason: The elementarization of the above formula is which is not a tautology. So, P  P  P is instable and hence cannot be derived by Rule A. And it does not contain choice operators, so it cannot be derived by Rules B1 or B2, either. Thus, it could only be derived by Rule C. Then the premise should be p  P  p or p  p  P for some elementary atom p. In either case we deal with an instable formula which does not contain choice operators and contains only one occurrence of a general atom. Such a formula cannot be derived by any of the four rules of CL4. ⊤⊥⊥,⊤⊥⊥, CL4 ⊦ p  p  p This formula, as any other classically valid elementary formula, follows from {} by Rule A (is an “ axiom ” ).

6. More propositional examples 16.4 CL4 ⊦ p ⊔ (Q  R)  (p ⊔ Q)  (p ⊔ R) 9. p ⊔ (Q  R)  (p ⊔ Q)  (p ⊔ R) 3. p  (p ⊔ Q)  (p ⊔ R) 8. Q  R  (p ⊔ Q)  (p ⊔ R) 2. p  p  (p ⊔ R) 1. p  p  p 6. Q  R  Q  R 7. Q  R  Q  (p ⊔ R) 5. q  R  q  R 4. q  r  q  r from {} by Rule A from 1 by Rule B1 from 2 by Rule B1 from {} by Rule A from 4 by Rule C from 5 by Rule C from 6 by Rule B1 from 7 by Rule B1 from {3,8} by Rule A On the other hand, one can show that CL4 ⊬ P ⊔ (Q  R)  (P ⊔ Q)  (P ⊔ R)

7. Examples with quantifiers 16.5 3. ⊔ y ( P(z)  P(y) ) 2. P(z)  P(z) 1. p(z)  p(z) 4. ⊓ x ⊔ y ( P(x)  P(y) ) from {} by Rule A from 1 by Rule C from 2 by Rule B2 from {3} by Rule A CL4 ⊦ ⊓ x ⊔ y ( P(x)  P(y) ) CL4 ⊬ ⊔ y ⊓ x ( P(x)  P(y) ) Indeed, this instable formula cannot be the conclusion of any rule but B2. If it is derived by this rule, the premise should be for some term t different from x. In turn, ⊓ x ( P(x)  P(t) ) could only be derived by Rule A where, for some variable z different from t, P(z)  P(t) is a (the) premise. The latter is an instable formula and does not contain choice operators, so the only rule by which it can be derived is C, where the premise is p(z)  p(t) for some elementary letter p. Now we deal with a classically non-valid and hence instable elementary formula, and it cannot be derived by any of the four rules of CL4. ⊓ x ( P(x)  P(t) ) CL4 ⊦  y  x ( P(x)  P(y) ) in contrast to the previous example. 1.  y  x ( p(x)  p(y) ) from {} by Rule A (see Slide 5.16) 2.  y  x ( P(x)  P(y) ) from 1 by Rule C

8. Propositional exercises 16.6 Show that: (7) CL4 ⊦ (P ⊔ Q)  (P ⊔ R)  P ⊔ (Q  R) [Hint: see Slide 9.4] (8) CL4 ⊬ P ⊔ (Q  R)  (P ⊔ Q)  (P ⊔ R) (1) CL4 ⊦ P  P (2) CL4 ⊬ P ⊔  P (3) CL4 ⊦ P ⊔ P  P (9) CL4 ⊦ (P  P)  (P  P)  (P  P)  (P  P) [Hint: see Slide 8.6] (10) CL4 ⊦ ( ( (P  (R ⊓ S) ) ⊓ ( (Q  (R ⊓ S) ) ) ⊓ ( ( (P ⊓ Q)  R ) ⊓ ( (P ⊓ Q)  S ) )  (P ⊓ Q)  (R ⊓ S) (4) CL4 ⊬ P  P  P (5) CL4 ⊦ P  Q  P ⊓ Q (6) CL4 ⊬ P ⊓ Q  P  Q

9. Exercises with quantifiers 16.7 Below CL4 ⊦ E  F means “ Both CL4 ⊦ E  F and CL4 ⊦ F  E ”. Show that: (1) CL4 ⊦  xP(x)  ⊓ xP(x) (2) CL4 ⊬ ⊓ xP(x)   xP(x) (7) CL4 ⊦ ⊓ x ⊔ y ( q(x)  p(y) )  ( ⊓ x (  p(x) ⊔ p(x) )  ⊓ x (  q(x) ⊔ q(x) ) ) [See Slide 9.11] (3) CL4 ⊦  x ( P(x)  Q(x) )   xP(x)   xQ(x) (4) CL4 ⊬ ⊓ x ( P(x)  Q(x) )  ⊓ xP(x)  ⊓ xQ(x) (5) CL4 ⊦ ⊓ x  yP(x,y)   y ⊓ xP(x,y) [Similarly for ⊔ instead of ⊓, and/or  instead of  ] (6) CL4 ⊦  xP(x) ⊓  xQ(x)   x ( P(x) ⊓ Q(x) ) [Similarly for ⊔ instead of ⊓, and/or  instead of  ] (8) CL4 ⊬ ( ⊓ x (  p(x) ⊔ p(x) )  ⊓ x (  q(x) ⊔ q(x) ) )  ⊓ x ⊔ y ( q(x)  p(y) ) [See Slide 9.12] (9) CL4 ⊦ ⊓ x ( ( P(x)  ⊓ xQ(x) ) ⊓ ( ⊓ xP(x)  Q(x) ) )  ⊓ xP(x)  ⊓ xQ(x)

10. The soundness and completeness of CL4 16.8 Theorem 16.1. For any CL4-formula F, we have CL4 ⊦ F iff F is valid iff F is uniformly valid. Furthermore: Uniform-constructive soundness: There is an effective procedure that takes any CL4-proof of any formula F and constructs a uniform solution for F. Strong completeness: If CL4 ⊬ F, then F* is not computable for some interpretation * which interprets every elementary letter as a ( , ,  )- combination of  1 -predicates, and interprets every general letter as a ( ⊓,⊔ )-combination of such combinations. Here “  1 -predicate ” means a predicate of the form  yp(y,x 1,...,x n ), where p(y,x 1,...,x n ) is a decidable predicate.

11. The decidability of the ,  -free fragment of CL4 16.9 Fact 16.2. CL4 is a conservative extension of classical first-order logic. That is in the sense that an elementary formula (formula of classical logic) is provable in CL4 if and only if it is valid in the classical sense (i.e. provable in G3). The above fact is established by the simple observation that, when F is elementary, it can only be derived from {} by Rule A, which, in turn, means that F is stable; but stability for an elementary formula simply means its validity in the classical sense. Thus, G ö del ’ s completeness theorem for classical logic is a relatively simple special case of our Theorem 16.1. Specifically, it is Theorem 16.1 restricted to only the elementary fragment of CL4. Classical first-order logic, as we know, is undecidable, which implies that CL4 is not decidable, either. Yet, as it turns out, it is only the blind (rather than the choice) quantifiers that make trouble: Theorem 16.3. For the formulas that do not contain blind quantifiers (but may contain choice quantifiers), the question on provability in CL4 and hence the question on validity or uniform validity is decidable. Thus, not only does CL4 provide a systematic, never-failing effective tool for telling what is valid, but --- in the case of ,  -free formulas --- also for telling what is not valid.

12. Other axiomatizations 16.10 Both CL5 and CL4 are new and proof-theoretically rather unusual logics, created within the program of finding sound and complete axiomatizations for various fragments of computability logic. Here we very briefly survey two other, well known logics that existed long before computability logic was introduced. One is affine logic, a variation of Girard ’ s linear logic. Affine logic, in its full first-order language, turns out to be sound but incomplete with respect to the semantics of computability logic (validity or uniform validity), with the additives understood as choice operators, the multiplicatives as parallel connectives, and exponentials as either sort of our recurrence operators. The other one is intuitionistic logic. It, in the full first-order language, has been shown to be sound with respect to the semantics of computability logic when the intuitionistic “ absurd ” is understood as ⊥, implication as, and the other intuitionistic operators (including quantifiers) as choice operators. At the same time, just as in the case of affine logic, there is no completeness. Yet, the positive ( ⊥ -free) propositional fragment of intuitionistic logic turns out to be complete. Furthermore, completeness extends to the full propositional fragment if we understand “ absurd ” not as ⊥ but rather as $, a “ problem of universal strength ”. The implicative fragment of intuitionistic logic also turns out to be sound and complete with implication seen as. The soundness theorems for both affine and intuitionistic logics hold in the strong, uniform-constructive sense.