1. Episode 14 Logic CL4 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)

2. The language of CL4
14.1
Other than CL5, a number of sound and complete systems for various fragments of CoL have been constructed. Of those, we will only take a look at CL4.
CL4-formulas (formulas of the language of CL4) are formulas of the language described in Episode 12 that do not contain any operators other than
   ⊓ ⊔   ⊓ ⊔
Further, we require that  can only be applied to nonlogical atoms. If E is not such an atom, E is to be understood as an abbreviation defined by ⊤=⊥; ⊥=⊤; E=E; (EF)=EF; (EF)=EF; (E⊓F)=E⊓F; (E⊔F)=E⊔F; xE=xE; xE=xE; ⊓xE=⊔xE; ⊔xE=⊓xE.
Similarly, if write EF, it is to be understood as an abbreviation of EF.
We agree that, throughout the present episode, “formula” means “CL4-formula”.

3. Notation and terminology
14.2
A formula is said to be elementary iff it doesn’t contain choice operators and general letters. Formulas of classical logic are nothing but elementary formulas.
A literal is L(t1,…,tn) or L(t1,…,tn), where L(t1,…,tn) is an atom. Such a literal is said to be elementary or general depending on whether the letter L is so.
A surface occurrence of a subformula is an occurrence which is not in the scope of ⊓, ⊔, ⊓, ⊔ or .
The elementarization of a formula F is the result of replacing in F all surface occurrences of ⊓- and ⊓-subformulas by ⊤, all surface occurrences of ⊔- and ⊔-subformulas by ⊥, and all surface occurrences of general literals by ⊥.
A formula is said to be stable iff its elementarization is classically valid (i.e. provable in some standard axiomatization of classical first-order logic). Otherwise it is unstable.
We will write F[E1,…,En] to mean a formula F together with some n fixed surface occurrences of subformulas E1,…,En. Writing F in this form sets a context, in which F[G1,…,Gn] will mean the result of replacing in F[E1,…,En] those occurrences of E1,…,En by G1,…,Gn, respectively.

4. The rules of CL4
14.3
CL4 has the following four rules of inference, where E  F means “from premise(s) E conclude F”. Axioms are not explicitly stated, but the set of premises of the Wait rule can be empty, in which case the conclusion of the rule acts like an axiom.
⊔-Choose: F[Hi]  F[H0⊔H1], where i{0,1}.
⊔-Choose: F[H(t)]  F[⊔xH(x)], where t is either a constant or a variable with no
bound occurrences in the premise.
Wait: E  F, where F is stable, and E is a set of formulas satisfying the following
two conditions:
1. Whenever F has the form F[G⊓H], E contains both F[G] and F[H].
2. Whenever F has the form F[⊓xG(x)], E contains F[G(y)] for some
variable y which does not occur in the conclusion.
Match: F[p(t),p(s)]  F[P(t),P(s)], where P is an n-ary (n0) general game letter,
t and s are n-tuples of terms, and p is an n-ary elementary game letter which
does not occur in the conclusion.

5. Propositional examples
14.4
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, i.e. of (pP)p, is the 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 Wait (is an “axiom”).
p⊤p, i.e. (p⊥)p.
2. PPP
Justification: From {1} by Match.
CL4 ⊬ PPP
which is not a tautology. So, PPP is unstable and hence cannot be derived by Wait. And it does not contain choice operators, so it cannot be derived by ⊔-Choose or ⊔-Choose, either. Thus, it could only be derived by Match. Then the premise should be pPp or ppP for some elementary game letter p. In either case we deal with an unstable formula which does not contain choice operators and contains only one occurrence of a general atom. Such a formula cannot be the conclusion of any of the four rules of CL4.
Reason: The elementarization of the above formula is
⊤⊥⊥,
This formula, as any other classically valid elementary formula,
follows from {} by Wait (is an “axiom”).
CL4 ⊦ ppp

6. More propositional examples
14.5
CL4 ⊦ p⊔(QR)  (p⊔Q)(p⊔R)
1. p  pp
from {} by Wait
2. p  p(p⊔R)
from {1} by ⊔-Choose
3. p  (p⊔Q)(p⊔R)
from {2} by ⊔-Choose
4. qr  qr
from {} by Wait
5. qR  qR
from {4} by Match
6. QR  QR
from {5} by Match
7. QR  Q(p⊔R)
from {6} by ⊔-Choose
8. QR  (p⊔Q)(p⊔R)
from {7} by ⊔-Choose
9. p⊔(QR)  (p⊔Q)(p⊔R)
from {3,8} by Wait
On the other hand, one can show that
CL4 ⊬ P⊔(QR)  (P⊔Q)(P⊔R)

7. Examples with quantifiers
14.6
1. p(z)  p(z)
from {} by Wait
CL4 ⊦ ⊓x⊔y(P(x)P(y))
2. P(z)  P(z)
from {1} by Match
3. ⊔y(P(z)P(y))
from {2} by ⊔-Choose
4. ⊓x⊔y(P(x)P(y))
from {3} by Wait
Indeed, this unstable formula cannot be the
conclusion of any rule but ⊔-Choose. If it is derived
by this rule, the premise should be
CL4 ⊬ ⊔y⊓x(P(x)P(y))
⊓x(P(x)P(t))
for some constant or variable t different from x. In turn, ⊓x(P(x)P(t)) could only be
derived by Wait where, for some variable z different from t, P(z)P(t) is a (the)
premise. The latter is an unstable formula and does not contain choice operators, so the
only rule by which it can be derived is Match, where the premise is p(z)p(t) for some
elementary game letter p. Now we deal with a classically non-valid and hence unstable
elementary formula, and it cannot be derived by any of the four rules of CL4.
1. yx(p(x)p(y)) from {} by Wait
CL4 ⊦ yx(P(x)P(y))
2. yx(P(x)P(y)) from {1} by Match
in contrast to the previous example.

8. Propositional exercises
14.7
Show that:
(1) CL4 ⊦ PP
(2) CL4 ⊬ P⊔P
(3) CL4 ⊦ P⊔PP
(4) CL4 ⊬ PPP
(5) CL4 ⊦ PQP⊓Q
(6) CL4 ⊬ P⊓QPQ
(7) CL4 ⊦ (P⊔Q)(P⊔R)  P⊔(QR) [Hint: see Slide 7.4]
(8) CL4 ⊬ P⊔(QR)  (P⊔Q)(P⊔R)
(9) CL4 ⊦ (PP)(PP)  (PP)(PP) [Hint: see Slide 6.6]
(10) CL4 ⊦ (((P(R⊓S))⊓((Q(R⊓S)))⊓(((P⊓Q)R)⊓((P⊓Q)S))  (P⊓Q)(R⊓S)

9. Exercises with quantifiers
14.8
Below CL4⊦EF means “Both CL4⊦EF and CL4⊦FE”. Show that:
(1) CL4 ⊦ xP(x)⊓xP(x)
but CL4 ⊬ ⊓xP(x)xP(x)
(2) CL4 ⊦ x(P(x)Q(x))  xP(x)xQ(x) but
CL4 ⊬ ⊓x(P(x)Q(x))  ⊓xP(x)⊓xQ(x)
(3) CL4 ⊦ xy(y=f(x)) but CL4 ⊬ ⊓x⊔y(y=f(x))
(4) CL4 ⊦ ⊓xyP(x,y)  y⊓xP(x,y) (Similarly for ⊔ and/or  instead of ⊓,)
(5) CL4 ⊦ xP(x)⊓xQ(x)  x(P(x)⊓Q(x)) (Similarly for ⊔ and/or  instead
of ⊓,)
(6) CL4 ⊦ ⊓x⊔y(q(x)p(y))  (⊓x(p(x)⊔p(x))  ⊓x(q(x)⊔q(x)))
[See Slide 7.11]
(7) CL4 ⊬ (⊓x(p(x)⊔p(x))⊓x(q(x)⊔q(x)))  ⊓x⊔y(q(x)p(y))
[See Slide 7.12]
(8) CL4 ⊦ ⊓x((P(x)⊓xQ(x)) ⊓ (⊓xP(x)Q(x)))  ⊓xP(x)  ⊓xQ(x)

10. The soundness and completeness of CL4
14.9
Theorem 14.9.a For any CL4-formula F, we have
CL4 ⊦ F iff F is logically valid
Furthermore:
Uniform-constructive soundness: There is an effective procedure that takes any CL4-proof of any formula F and constructs a logical solution for F.
Conjecture 14.9.b For any CL4-formula F such that F does not contain nonlogical elementary game letters, we have
CL4 ⊦ F iff F is nonlogically valid

11. The decidability of the ,-free fragment of CL4
14.10
Fact a 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.
The above fact is established by the simple observation that, when F is elementary, it can only be derived from {} by Wait, which, in turn, means that F is stable; but stability for an elementary formula simply means its validity in the classical sense.
Thus, Gödel’s completeness theorem for classical logic is a relatively simple special case of our Theorem 14.9.a. Specifically, it is Theorem 14.9.a 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 choice) quantifiers that make trouble:
Theorem b 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 logical validity is decidable.
Thus, not only does CL4 provide a systematic tool for telling what is valid, but --- in the case of ,-free formulas --- also for telling what is not valid.

12. Other axiomatizations
14.11
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.
On this slide 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, with the additives understood as choice operators, the multiplicatives as parallel connectives, and the 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 like in the case of affine logic, there is no completeness. Yet, the positive (⊥-free) propositional fragment of intuitionistic logic turns out to be complete. 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.