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Epistemic and intuitionistic formal systems
Annals of Pure and Applied Logic 32 (1986) 53-60 North-Holland
EPISTEMIC
AND
INTUITIONISTIC
53
FORMAL
SYSTEMS
R.C. F L A G G and H. F R I E D M A N The Ohio State University, Columbus, OH 43210, USA
Communicated by A. Nerode Received 25 February 1985
O. Introduction Recent interest in epistemic formal systems (that is, formal systems based on Lewis' modal logic ($4)) stems from efforts to 'integrate' classical and intuitionistic mathematics - - cf. [1], [4], [9], and [10]. For this purpose the interpretation (.)o of intuitionistic propositional logic into epistemic propositional logic, described by McKinsey and Tarski [5], has been extended to mathematical systems such as first-order arithmetic, type theory and set theory. For each of these it has been shown that (.)n provides a sound and faithful interpretation of the intuitionistic formal system into the corresponding epistemic one. The proofs that (.)n is conservative are quite different for different formal systems. The goal of the present paper is to describe one uniform method which can be used to obtain these conservative extension results for all the formal systems considered thus far. Our arguments are based on a syntactic formulation of Funayama's Theorem which asserts that any complete Heyting algebra can be embedded in a complete Boolean algebra by a map preserving finite meets and arbitrary joins. The second author has observed that another method of embedding a Heyting algebra into a Boolean algebra can be used to obtain our results. In this approach one formally adds Boolean terms in disjunctive normal form (cf. [8, p. 128]). Here the interior operator has a simple syntactic description. The resulting Boolean algebra may not be complete, but, one works with regular open subsets.
1. Propositional Logic For intuitionistic propositional logic (IP) we take as primitives the logical constants _t., v; A and ---~. For the corresponding epistemic system (EP) we add to these the modal operator [3. Formulas are built up in the usual way. We will use p, q, r , . . . as syntactic variables for propositional letters. As syntactic variables ranging over formulas, the letters A, B, C , . . . will be used. We will write -hA for (A--~ _1_). 0168-0072/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
R.C. Flagg, H. Friedman
54
(IP) is formulated as a system of natural deduction as in Prawitz [6] with the following rules: Ao A1 Ao ^ A1 Al ^E A0 ^ A1 Ai
[Ao] [A1] vl
Ai
vE
A0 v A1 '
Ao vA1
B
B
B
[A] --~I
II
B A---> B ' /
--~E
A--->B B
A
A
The natural deduction formulation of (EP) is obtained by deleting the rule .1.~ from the above list and adding the following: Ell
DE
A ElA
provided all open assumptions are of the form DB,
DA , A
J-c
[-aA] ± - - . A
If (S) is one of the systems (IP) or (EP) and C1, • . . , C,, B are formulas of (S), we will write Q , . . . , C,, % B
to indicate that there is a derivation in (S) of B with open assumptions among C1, .. •, Cn. Also we will write AqFsB if A F s B and B~-sA.
1.1. Definition. A formula B of (EP) is called stable if FEeA ~ EtA.
1.2. Lemma. (i) Each formula o f the form E1A is stable. (ii) I f A and B are stable, then so are (A v B) and (A ^ B ). (iii) The following is a permissible inference rule o f (EP): A
DI' DA
provided all open assumptions are stable.
The following interpretation of (IP) into (EP) appears first to be considered by McKinsey and Tarski [5].
Epistemic and intuitionistic formal systems
55
1.3. Definition. For each formula A of (IP) we define, by induction on the complexity of A, a formula A n of (EP) as follows: (i) for each propositional letter p, (ii) _l.n = .1_, (iii) (A v B)n--(A[] v B[]),
el-1 -- [---]p~
(iv) (A ^ B)[]-- (A [] ^ 8 ° ) , (v) (A --, B)[] -- O(A [] ~ B°). From (1.2) it follows at once that each formula of the form A [] is stable. The next lemma follows easily by induction on derivations• 1.4. L e m m a . A1,..
L e t A1, . . . , A n , B be formulas o f
• , An~-IpB,
then
(IP). ff
A ~ , . . . , An[] FEpB []•
1.5. Theorem. For each f o r m u l a A o f (IP), i f }-IpA, then I-EpA [].
To prove the converse of 1.5, McKinsey and Tarski used the algebraic semantics for intuitionistic and epistemic logic and a version of Funayama's Theorem. We will avoid the semantics by refining Funayama's Theorem to obtain an interpretation of (EP) into (IP) which, in a certain sense, is inverse to (.)o. It will be convenient to introduce the notation "3E A -- A--> E
for any two formulas A and E of (IP). The following easily established facts will be used on several occasions below. 1.6. Lemma. Let A , B be formulas o f (IP) and let F be a finite set o f formulas o f (IP). Then (i) A ~-ve-le-aeA, (ii) ~ E A q~'Ip-1E-lr-aEA, (iii) i r A e F, then A q~m /)~ -aE-~EA, EeF
(iv) -~:E(A v B ) ~ - ~ : ~ ( - y ~ A (v) - ~ : ~ ( A ^ B) ~ ( - ~ : ~ A
v-y~Ea),
^-~:~B),
(vi) if (A--> B) ~ r, then (A~B)~Ip/)~ (~E-~A-->~e-~eB), EeF
(vii) the following rule is permissible in (IP): A1, • • • ,An FB
-aE~EA1, . . . , -~E-~EAn ~"aE"lE B"
R.C. Flagg, H. Friedman
56
1.7. Definition. Let F be a finite set of formulas of (IP) and let E e F. For each formula A of (EP), we define, by induction on the complexity of A, a formula A~E) of (IP) as follows:
A~e)=--~e-~eA,
(i)
for A atomic
(ii)
(A v B)~e)=--~E~E(A(rE)V B~-E)),
(iii)
(A ^ B)(re) =-(A~E) ^ B~g)),
(iv)
(A--->B)~E)=--(A~E)---~B~E));
(v)
(DA)
CeF
We will drop the subscript from A~E) if this does not cause confusion. For any formula B of (EP), we evidently have (~B)(E)qFIP~E B(E),
and
~E-1EB(E)qFIpB (E).
From these two facts and Lemma 1.6, the next result follows by induction on derivations. 1.8. Lemma. L e t A 1 , • • • , A n , B b e f o r m u l a s o f (EP). I f A 1 , • • • , A , }-EP B,
then for any set F of formulas of (IP) and any C e F, A (1c), . . . , A(C)FipB(C). 1.9. Theorem. Let B be a formula of (EP). If FEe B,
then for any set F of formulas of (IP) and any C e F,
FIp 1.10. Lemma. Let A be a formula of (IP) and suppose F contains all subformulas
of A. Then A 4 Fir/~ A ~(c). CeF
Proof. The argument is b y induction on the complexity of A. We consider two cases. A - ( A o v A 0 . Then
Am(C~--~c-~c(A~o (C) v Aml (c)).
Epistemic and intuitionistic formal systems
57
Using the fact that Aoa and A~ are stable and Theorem 1.9, we get
An(C3-~FiP~c~c(~c~c~ A°o (E) v ~c7c/I)~ A~(e'). The result follows from the induction hypothesis and Lemma 1.6 (iii), (iv). A -- (A0--->A 1). Then
Am(c) - - ~ c ~ c / ~ (A~ (e) .-~ A~(n)). E
As above, it follows that A D(c) -I ~'Ip"tC'-Ict~ ("l~,~e.Ao-'*-le~e.A1). E
Now we apply Lemma 1.6 (iii), (vi).
[]
1.U. Theorem (McKinsey and Tarksi [5]). For any formula A of (IP),
FreA
if and only if FEpAD.
Proof. Assume I-EpAD. Let F be the set of all subformulas of A. By 1.9, we have Fn,/~e~r A °(c9. The theorem follows by 1.10. []
2. Predicate calculus
The arguments of the last section extend with very little modification to first-order logic with equality. Let (IPC) and (EPC) denote, respectively, the formal systems for intuitionistic and epistemic predicate calculus with equality. These are obtained from the corresponding systems of propositional logic by adding rules for equality and for the quantifiers. It is convenient to formulate (EPC) so that equality is stable. This will follow from Vii if we take the following standard rules: r
t = t,
Sub
s = t A(t) A(t)
2.1. Lemma. (i) For any terms t and s, the formula (t = s) is stable. (ii) f f dp is a stable formula, then so is 3x qb. The interpretation (.)D was extended to first-order logic by Rasiowa and Sikorski [8] and by Prawitz and Malmnas [7].
R.C. Flagg, H. Friedman
58
2.2. Definition. For each formula A of (IPC) we define, by induction on the complexity of A, a formula A D of (EPC) as follows:
(i) (ii) (iii) (iv) (v) (vi) (vii)
if A -
Z
o r (s = t), t h e n A [] = A ,
if A is an atomic formula different from ± a n d (s = t), t h e n A [] - fflA, (A
v
B) m~ (A[]v B•),
(A ^ B) = - (A [] ^ B=),
(A ---)B)D = wl(a [] --) B[]), (:Ix A)D --= 3 x A [], ( ¥ x A ) ° -= rqVxA D.
Again, each formula of the form A [] is stable, so the next theorem can be proved just as before. 2.3. Theorem. For each formula A of (IPC), if Fn,cA, then FEpcAn. The converse of 2.3 was proved by Rasiowa and Sikorski by a straight forward extension of the approach of McKinsey and Tarski, and independently by Prawitz and Malmnas using proof-theoretic arguments. To extend our approach we will need to add the following elementary result to the list 1.6(i)-(vii):
-~E-ae :Ix A q ~'IPC-'IE-'IE3X "aE"aE A . The translation (.)~E), for F a set of formulas of (IPC) and E an element of F, can be extended to the predicate calculus by adding the following two clauses to Defirfition 1.7:
(vi)
(::IxA)~-e) =---~E'-aE::IxA (F'),
(vfi)
(Vx A ) V ) =--Vx A V ).
We can now argue as above. Z.4. Theorem. Let B be a formula of (EPC). ff
~'EPCB,
~,henfor any set r of formulas of (IPC) and any C ~ F FmcB~-c)
Let A be a formula of (I.PC) and suppose F contains all subformulas of A. Then 2.5. ILemma.
A-IFn, Cc/~rA~ (c)
Epistemic and intuitionistic formal systems
59
2.6. Theorem. For any formula A of (IPC),
FxpcA if and only if FEpcA°. 3. Systems of arithmetic A formal system for first-order epistemic arithmetic, denoted by (EA), was introduced by Shapiro [10] and independently by Rienhardt. The descriptive constants are the usual ones: 0, s, +, .. The axioms and rules of (EA) are those of (EPC) together with the usual axioms for zero and successor, the defining equations for + and • and the scheme of induction for all formulas of the language. Shapiro observed that the translation (.)[] provides a sound interpretation of intuitionistic first-order arithmetic, (HA), into (EA). Later Goodman [4] showed, by a proof-theoretic argument using a cut-elimination theorem for an infmitary version of (EA) together with partial reflection principles for (HA), that this interpretation is conservative. We give an alternative proof. 3.1. Theorem (Shapiro [10], Goodman [4]). For any sentence A of (HA),
VnAA
if and only if VEAA m.
Proof. The implication from left to right follows by an easy induction derivations. For the converse, first note, again by an easy induction derivations, that for any formula B of (EA) if FEAB, then for any set F formulas of (HA) and any C e F, ~HAB~c). The result follows by Lemma 2.5.
on on of []
We close with an application of Theorem 3.1 to the metamathematics of (HA). By Markov's Rule we understand the inference rule MR:
Vx -a~3y R(x, y) Vx 3y R(x, y)
where R(x, y) -- (f(x, y) = 0) for some binary primitive recursive function symbol
f. The epistemic version of this rule would be
EMR:
E]Vx @:ly R(x, y) nVx 3y R(x, y)
3.2. Lemma. ( E A ) / s closed under the rule EMR. Proof. Suppose I-EAI-IVX'-I[~-I=:IyR(x,y). By 'erasing' the N's in the proof, we get ~PAVX 3y R(x, y). Hence, by Eli, FEAE]Vx 3y R(X, y). D
3.3. Theorem. (HA)/s closed under the rule MR.
60
R.C. Flagg, H. Friedman
Proof. Suppose knA VX ~-~=ly R(x, y). Then kEA I-']VX-~I-I~3y R ( x , y). By 3.2, I-EA[]VX 3y R(x, y). Hence by 3.1, ki-tAVX 3y R(x, y). [] The second author [3] has given a short proof of Theorem 3.3 by way of a simple interpretation of (HA) into itself. This interpretation can be seen as a special case of the map (-)(rc). From 3.3, it follows that (HA) and (PA) have the same provably recursive functions.
References [1] R. Flagg, Integrating classical and intuitiionistic type theory, Annals Pure Appl. Logic, to appear. [2] R. Flagg, Epistemic set theory is a conservative extension of intuitionistic set theory, J. Symbolic Logic, to appear. [3] H. Friedman, Classically and intuitionistically provably recursive functions, in: G.H. Muller and D.S. Scott, eds., Higher Set Theory, Lecture Notes in Math. 669 (Springer, Berlin, 1978) 21-27. [4] N.D. Goodman, Epistemic arithmetic is a conservative extension of intuitionistic arithmetic, J. Symbolic Logic 49 (1984) 192-203. [5] J. McKinsey and A. Tarski, Some theorems about the sentential calculi of Lewis and Heyting, J.
symbolicLogic13 (1948)pp. 1-15. [6] D. Prawitz, Ideas and results in proof theory, in: J.E. Fenstad, ed., Proceedings of the Second Scandinavian Logic Symposium (North-Holland, Amsterdam, 1971) 235-307. [7] D. Prawitz and P.E. Malmnas, A survey of some connections between classical, intuitionistic and minimal logic, in: H. Arnold Schmidt et al., eds., Contributions to Mathematical Logic (North-Holland, Amsterdam, 1966) 215-229. [8] H. Radiowa and R. Sikorsld, The Mathematics of Metamathematics (Polish Scientific Publishers, Warsaw, 1963). [9] A. g~cdrov, Extending Gfdel's modal interpretation to type theory and set theory, in: S. Shapiro, ed., Intensional Mathematics (North-Holland, Amsterdam, 1985). [10] S. Shapiro, Epistemic and intuitionistic arithmetic, in: S. Shapiro, ed., Intensional Mathematics (North-Holland, Amsterdam, 1985).
EPISTEMIC
AND
INTUITIONISTIC
53
FORMAL
SYSTEMS
R.C. F L A G G and H. F R I E D M A N The Ohio State University, Columbus, OH 43210, USA
Communicated by A. Nerode Received 25 February 1985
O. Introduction Recent interest in epistemic formal systems (that is, formal systems based on Lewis' modal logic ($4)) stems from efforts to 'integrate' classical and intuitionistic mathematics - - cf. [1], [4], [9], and [10]. For this purpose the interpretation (.)o of intuitionistic propositional logic into epistemic propositional logic, described by McKinsey and Tarski [5], has been extended to mathematical systems such as first-order arithmetic, type theory and set theory. For each of these it has been shown that (.)n provides a sound and faithful interpretation of the intuitionistic formal system into the corresponding epistemic one. The proofs that (.)n is conservative are quite different for different formal systems. The goal of the present paper is to describe one uniform method which can be used to obtain these conservative extension results for all the formal systems considered thus far. Our arguments are based on a syntactic formulation of Funayama's Theorem which asserts that any complete Heyting algebra can be embedded in a complete Boolean algebra by a map preserving finite meets and arbitrary joins. The second author has observed that another method of embedding a Heyting algebra into a Boolean algebra can be used to obtain our results. In this approach one formally adds Boolean terms in disjunctive normal form (cf. [8, p. 128]). Here the interior operator has a simple syntactic description. The resulting Boolean algebra may not be complete, but, one works with regular open subsets.
1. Propositional Logic For intuitionistic propositional logic (IP) we take as primitives the logical constants _t., v; A and ---~. For the corresponding epistemic system (EP) we add to these the modal operator [3. Formulas are built up in the usual way. We will use p, q, r , . . . as syntactic variables for propositional letters. As syntactic variables ranging over formulas, the letters A, B, C , . . . will be used. We will write -hA for (A--~ _1_). 0168-0072/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
R.C. Flagg, H. Friedman
54
(IP) is formulated as a system of natural deduction as in Prawitz [6] with the following rules: Ao A1 Ao ^ A1 Al ^E A0 ^ A1 Ai
[Ao] [A1] vl
Ai
vE
A0 v A1 '
Ao vA1
B
B
B
[A] --~I
II
B A---> B ' /
--~E
A--->B B
A
A
The natural deduction formulation of (EP) is obtained by deleting the rule .1.~ from the above list and adding the following: Ell
DE
A ElA
provided all open assumptions are of the form DB,
DA , A
J-c
[-aA] ± - - . A
If (S) is one of the systems (IP) or (EP) and C1, • . . , C,, B are formulas of (S), we will write Q , . . . , C,, % B
to indicate that there is a derivation in (S) of B with open assumptions among C1, .. •, Cn. Also we will write AqFsB if A F s B and B~-sA.
1.1. Definition. A formula B of (EP) is called stable if FEeA ~ EtA.
1.2. Lemma. (i) Each formula o f the form E1A is stable. (ii) I f A and B are stable, then so are (A v B) and (A ^ B ). (iii) The following is a permissible inference rule o f (EP): A
DI' DA
provided all open assumptions are stable.
The following interpretation of (IP) into (EP) appears first to be considered by McKinsey and Tarski [5].
Epistemic and intuitionistic formal systems
55
1.3. Definition. For each formula A of (IP) we define, by induction on the complexity of A, a formula A n of (EP) as follows: (i) for each propositional letter p, (ii) _l.n = .1_, (iii) (A v B)n--(A[] v B[]),
el-1 -- [---]p~
(iv) (A ^ B)[]-- (A [] ^ 8 ° ) , (v) (A --, B)[] -- O(A [] ~ B°). From (1.2) it follows at once that each formula of the form A [] is stable. The next lemma follows easily by induction on derivations• 1.4. L e m m a . A1,..
L e t A1, . . . , A n , B be formulas o f
• , An~-IpB,
then
(IP). ff
A ~ , . . . , An[] FEpB []•
1.5. Theorem. For each f o r m u l a A o f (IP), i f }-IpA, then I-EpA [].
To prove the converse of 1.5, McKinsey and Tarski used the algebraic semantics for intuitionistic and epistemic logic and a version of Funayama's Theorem. We will avoid the semantics by refining Funayama's Theorem to obtain an interpretation of (EP) into (IP) which, in a certain sense, is inverse to (.)o. It will be convenient to introduce the notation "3E A -- A--> E
for any two formulas A and E of (IP). The following easily established facts will be used on several occasions below. 1.6. Lemma. Let A , B be formulas o f (IP) and let F be a finite set o f formulas o f (IP). Then (i) A ~-ve-le-aeA, (ii) ~ E A q~'Ip-1E-lr-aEA, (iii) i r A e F, then A q~m /)~ -aE-~EA, EeF
(iv) -~:E(A v B ) ~ - ~ : ~ ( - y ~ A (v) - ~ : ~ ( A ^ B) ~ ( - ~ : ~ A
v-y~Ea),
^-~:~B),
(vi) if (A--> B) ~ r, then (A~B)~Ip/)~ (~E-~A-->~e-~eB), EeF
(vii) the following rule is permissible in (IP): A1, • • • ,An FB
-aE~EA1, . . . , -~E-~EAn ~"aE"lE B"
R.C. Flagg, H. Friedman
56
1.7. Definition. Let F be a finite set of formulas of (IP) and let E e F. For each formula A of (EP), we define, by induction on the complexity of A, a formula A~E) of (IP) as follows:
A~e)=--~e-~eA,
(i)
for A atomic
(ii)
(A v B)~e)=--~E~E(A(rE)V B~-E)),
(iii)
(A ^ B)(re) =-(A~E) ^ B~g)),
(iv)
(A--->B)~E)=--(A~E)---~B~E));
(v)
(DA)
CeF
We will drop the subscript from A~E) if this does not cause confusion. For any formula B of (EP), we evidently have (~B)(E)qFIP~E B(E),
and
~E-1EB(E)qFIpB (E).
From these two facts and Lemma 1.6, the next result follows by induction on derivations. 1.8. Lemma. L e t A 1 , • • • , A n , B b e f o r m u l a s o f (EP). I f A 1 , • • • , A , }-EP B,
then for any set F of formulas of (IP) and any C e F, A (1c), . . . , A(C)FipB(C). 1.9. Theorem. Let B be a formula of (EP). If FEe B,
then for any set F of formulas of (IP) and any C e F,
FIp 1.10. Lemma. Let A be a formula of (IP) and suppose F contains all subformulas
of A. Then A 4 Fir/~ A ~(c). CeF
Proof. The argument is b y induction on the complexity of A. We consider two cases. A - ( A o v A 0 . Then
Am(C~--~c-~c(A~o (C) v Aml (c)).
Epistemic and intuitionistic formal systems
57
Using the fact that Aoa and A~ are stable and Theorem 1.9, we get
An(C3-~FiP~c~c(~c~c~ A°o (E) v ~c7c/I)~ A~(e'). The result follows from the induction hypothesis and Lemma 1.6 (iii), (iv). A -- (A0--->A 1). Then
Am(c) - - ~ c ~ c / ~ (A~ (e) .-~ A~(n)). E
As above, it follows that A D(c) -I ~'Ip"tC'-Ict~ ("l~,~e.Ao-'*-le~e.A1). E
Now we apply Lemma 1.6 (iii), (vi).
[]
1.U. Theorem (McKinsey and Tarksi [5]). For any formula A of (IP),
FreA
if and only if FEpAD.
Proof. Assume I-EpAD. Let F be the set of all subformulas of A. By 1.9, we have Fn,/~e~r A °(c9. The theorem follows by 1.10. []
2. Predicate calculus
The arguments of the last section extend with very little modification to first-order logic with equality. Let (IPC) and (EPC) denote, respectively, the formal systems for intuitionistic and epistemic predicate calculus with equality. These are obtained from the corresponding systems of propositional logic by adding rules for equality and for the quantifiers. It is convenient to formulate (EPC) so that equality is stable. This will follow from Vii if we take the following standard rules: r
t = t,
Sub
s = t A(t) A(t)
2.1. Lemma. (i) For any terms t and s, the formula (t = s) is stable. (ii) f f dp is a stable formula, then so is 3x qb. The interpretation (.)D was extended to first-order logic by Rasiowa and Sikorski [8] and by Prawitz and Malmnas [7].
R.C. Flagg, H. Friedman
58
2.2. Definition. For each formula A of (IPC) we define, by induction on the complexity of A, a formula A D of (EPC) as follows:
(i) (ii) (iii) (iv) (v) (vi) (vii)
if A -
Z
o r (s = t), t h e n A [] = A ,
if A is an atomic formula different from ± a n d (s = t), t h e n A [] - fflA, (A
v
B) m~ (A[]v B•),
(A ^ B) = - (A [] ^ B=),
(A ---)B)D = wl(a [] --) B[]), (:Ix A)D --= 3 x A [], ( ¥ x A ) ° -= rqVxA D.
Again, each formula of the form A [] is stable, so the next theorem can be proved just as before. 2.3. Theorem. For each formula A of (IPC), if Fn,cA, then FEpcAn. The converse of 2.3 was proved by Rasiowa and Sikorski by a straight forward extension of the approach of McKinsey and Tarski, and independently by Prawitz and Malmnas using proof-theoretic arguments. To extend our approach we will need to add the following elementary result to the list 1.6(i)-(vii):
-~E-ae :Ix A q ~'IPC-'IE-'IE3X "aE"aE A . The translation (.)~E), for F a set of formulas of (IPC) and E an element of F, can be extended to the predicate calculus by adding the following two clauses to Defirfition 1.7:
(vi)
(::IxA)~-e) =---~E'-aE::IxA (F'),
(vfi)
(Vx A ) V ) =--Vx A V ).
We can now argue as above. Z.4. Theorem. Let B be a formula of (EPC). ff
~'EPCB,
~,henfor any set r of formulas of (IPC) and any C ~ F FmcB~-c)
Let A be a formula of (I.PC) and suppose F contains all subformulas of A. Then 2.5. ILemma.
A-IFn, Cc/~rA~ (c)
Epistemic and intuitionistic formal systems
59
2.6. Theorem. For any formula A of (IPC),
FxpcA if and only if FEpcA°. 3. Systems of arithmetic A formal system for first-order epistemic arithmetic, denoted by (EA), was introduced by Shapiro [10] and independently by Rienhardt. The descriptive constants are the usual ones: 0, s, +, .. The axioms and rules of (EA) are those of (EPC) together with the usual axioms for zero and successor, the defining equations for + and • and the scheme of induction for all formulas of the language. Shapiro observed that the translation (.)[] provides a sound interpretation of intuitionistic first-order arithmetic, (HA), into (EA). Later Goodman [4] showed, by a proof-theoretic argument using a cut-elimination theorem for an infmitary version of (EA) together with partial reflection principles for (HA), that this interpretation is conservative. We give an alternative proof. 3.1. Theorem (Shapiro [10], Goodman [4]). For any sentence A of (HA),
VnAA
if and only if VEAA m.
Proof. The implication from left to right follows by an easy induction derivations. For the converse, first note, again by an easy induction derivations, that for any formula B of (EA) if FEAB, then for any set F formulas of (HA) and any C e F, ~HAB~c). The result follows by Lemma 2.5.
on on of []
We close with an application of Theorem 3.1 to the metamathematics of (HA). By Markov's Rule we understand the inference rule MR:
Vx -a~3y R(x, y) Vx 3y R(x, y)
where R(x, y) -- (f(x, y) = 0) for some binary primitive recursive function symbol
f. The epistemic version of this rule would be
EMR:
E]Vx @:ly R(x, y) nVx 3y R(x, y)
3.2. Lemma. ( E A ) / s closed under the rule EMR. Proof. Suppose I-EAI-IVX'-I[~-I=:IyR(x,y). By 'erasing' the N's in the proof, we get ~PAVX 3y R(x, y). Hence, by Eli, FEAE]Vx 3y R(X, y). D
3.3. Theorem. (HA)/s closed under the rule MR.
60
R.C. Flagg, H. Friedman
Proof. Suppose knA VX ~-~=ly R(x, y). Then kEA I-']VX-~I-I~3y R ( x , y). By 3.2, I-EA[]VX 3y R(x, y). Hence by 3.1, ki-tAVX 3y R(x, y). [] The second author [3] has given a short proof of Theorem 3.3 by way of a simple interpretation of (HA) into itself. This interpretation can be seen as a special case of the map (-)(rc). From 3.3, it follows that (HA) and (PA) have the same provably recursive functions.
References [1] R. Flagg, Integrating classical and intuitiionistic type theory, Annals Pure Appl. Logic, to appear. [2] R. Flagg, Epistemic set theory is a conservative extension of intuitionistic set theory, J. Symbolic Logic, to appear. [3] H. Friedman, Classically and intuitionistically provably recursive functions, in: G.H. Muller and D.S. Scott, eds., Higher Set Theory, Lecture Notes in Math. 669 (Springer, Berlin, 1978) 21-27. [4] N.D. Goodman, Epistemic arithmetic is a conservative extension of intuitionistic arithmetic, J. Symbolic Logic 49 (1984) 192-203. [5] J. McKinsey and A. Tarski, Some theorems about the sentential calculi of Lewis and Heyting, J.
symbolicLogic13 (1948)pp. 1-15. [6] D. Prawitz, Ideas and results in proof theory, in: J.E. Fenstad, ed., Proceedings of the Second Scandinavian Logic Symposium (North-Holland, Amsterdam, 1971) 235-307. [7] D. Prawitz and P.E. Malmnas, A survey of some connections between classical, intuitionistic and minimal logic, in: H. Arnold Schmidt et al., eds., Contributions to Mathematical Logic (North-Holland, Amsterdam, 1966) 215-229. [8] H. Radiowa and R. Sikorsld, The Mathematics of Metamathematics (Polish Scientific Publishers, Warsaw, 1963). [9] A. g~cdrov, Extending Gfdel's modal interpretation to type theory and set theory, in: S. Shapiro, ed., Intensional Mathematics (North-Holland, Amsterdam, 1985). [10] S. Shapiro, Epistemic and intuitionistic arithmetic, in: S. Shapiro, ed., Intensional Mathematics (North-Holland, Amsterdam, 1985).