Semantical Models for Intuitionistic Logics

MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Costa (eds.) © North-Holland Publishing Company, 1980

SEMANTICAL MODELS FOR INTUITIONISTIC LOGICS E.G.K.

L6pez-f¢cob~

ABSTRACT.

It is ironic that intuitionism, whose origins are rooted in the concept of "proofs", shou I d produce so many (apparently) different kinds of models: Kripke models, Beth models, topological models, realizability, Swart models, and so on. Furthermore there appears to be a general view that most of the model! ings are equivalent, al though occasionally it is observed that they are not! In this talk we consider the concept of an abstract semantics for a logic L which we believe satisfies the minimum requirements in order to be called a "truth-value semantics" for L. We then discuss possible notions of equivalence between different semantics for L and in particular we catalogue just about all the truth-value semantics for intuitionistic logic and some of its extensions. We conclude with a Beth-l ike model I ing for the extension CO (constant domains) of intuitionistic logic.

§

O.

INTRODUCTI ON.

One of the distinguishing features between Kripke models and Beth models for the Intuitionistic Predicate Calculus (I PC) is that the latter have constant domains while the former need not. Furthermore it was known, almost from the time of their conception, that the Kripke models with constant domains correspond to the extension of the 1 PC obtained by adding the schema: IJx (P x V Q)

~

IJx P x V Q

where x is not free in Q. Not surprisingly such an extension of the I PC is often called the "logic of constant domains", or simply" CD" (1). Si nce it is often claimed that Kripke models and Beth models are equivalent, it was natural 'to try to determine which Beth models correspond to the loqf e CD. In order to be able to give a meaningful answer to the above problem, it

is

(1) The logic CD has turned up in other contexts; for example in the extension of CD obtained by adding the w- rules is equivalent to the extension obtained by adding the restrictively restricted w- rule.

191

192

E. G. K. LOPEZ-ESCOBAR

first necessary to be more precise about "correspond". In trying to do the latter one is quickly made aware that there are many different types of models for the I pe; even amongst the Kripke models one finds many different definitions, usually accompanied by a remark to the effect that the class of models so defined is equivalent to the one originally defined by Kripke (although no one seems to bother to define what is meant by two classes of models being equivalent). It may be worth observing that a similar situation occurred with classical logic in that a certain moment a large number of extensions of the Classical Predicate Calculus (cr c ) were being investigated. It wasn't until the appearance of lindstrom's 1969 paper, giving a characterization of the epe amongst the co11 ection of ab.6:t'ute:t logic.!>, that some order was imposed on the various extensions of the

cr c .

There was however an essential difference. In the classical case the languages varied but the kinds of models did not, while in the intuitionistic case what changed were the models, the language did not. What we propose to do in this paper is to introduce the concept of an "a.b.6:t'ute:t .6emawc.!>" for (extensions of) the IPe and discuss some of its ramifications. In passing we shall obtain a (non - standard) Beth semantics for the logic cn . §

1.

THE LOG I C I P

e

AND ITS EXTENS IONS.

1.1. The language. We shall assume that we are given a fixed first-order 1anguage La which includes amongst its symbols:

propositional constant: 1 propositional parameters p, q, ••• individual parameters a, b, individual variables x , lj, R, S, relational symbols \f x, 3 x, /\ , V , logical and other symbols

::l , (,) •

If the language L is obtained from the language La by the addition of a set C of individual constants, then we may write "LO(C)" instead of "L". For the most part we shall only consider languages L which have a finite number of relational symbols and propositional parameters. SeVl-t L is the set of sentences in the 1anguage L (i .e. those formulae of L with no individual parameters). 1.2. The c.alc.ullL6. One of the purposes for introducing model theoreti ca 1 methods is to avoid formal proofs. Consequently we shall avoid as much as possible reference to any parti cu1 ar proof-theory of the IP e. We do the 1atter byassuming that we have a consequence relation Cn such that

A E Cn (f) iff the formula A is an intuitionistic consequence of the setf of sentences. The principal requirement we place on Cn is that Cn(0), where 0 is the empty set, be preci se1y the set of theorems of the I p e (say, accordi ng to Kl eene 1952) . 1.3. TheoJUu. Given a set T of sentences in a language L the.theoltem.6 of T, ThmL(T), are those formulae A in the language L such that A E Cn(T). A set T of sentences in the language L is an L-TheOltlj just in case that

193

MODELS FOR INTUITIONISTIC LOGIC

Log~eh. A set H of formulae in a language L is a log~~ iff for all A, if A E Thm (FJ) and B is a formula of L obtained from A by the

1.4.

formulae

L

replacement of relational symbols or propositional parameters by other relational symbols or propositional parameters respectively, then B is also a theorem of H. A theory T(logicH) isB-~on.oLltentiffB'1'ThmL(T)(B'1'Thm L(H». iff it is 1- consistent.

It

is

~on.o..utent

A theory T is L--6a.tuJuLted iff for sentences A,B, hCx of L: (1) if AVBET then either A E T or BET, (2) if h C X E T then for some individual constant ~ of L, C~ E T.

§

2.

2.1.

logic

MODELL! NGS FOR THE LOG I CS IJ. By a semantical interpretation S

Semafttic.a..t ~nteJLpJteta.tion.o.

H we understand a pair:

S

such that AI Msx Sent

s

=

(M S'

for an L-

F S)

is a non-empty set and FS is a subset of the cartesian product

subject to the condition that: L

60Jt a.U

m: EMS'

~6

and

m

(2)

A, B E sentL and a.U B E Cn( IJ U {A}) then

F s B.

m F SA

(we omit the subscripts "SO when there is no chance of confusion; furthermore write "m FA" instead of "(m, A) E F "). An H-semantical interpretation S and a11 sentences A E ThmL( H ):

is

we

-6ound just in casethatforallmEM s

An H- semantical interpretation S is ~omplete iff for all L-sentences A, if for all EMs' F SA, then A is a theorem of H.

m

m

An

H- semantical interpretation

s i stent H - theory if

'ilrrtEM

S

T

S

is -6tJtongly c.omplete iff for every

con-

and all sentences A:

('ilBET(mFSB)

... mFSA)

then

AECn(HUT).

(2) This is what we bel ieve to be the minimal requirements for a "semantical" interpretation: Observe that it excludes mcst real izabil ity interpretations. Another, advantage of the condition is that the set of sentences "true" in a given model Ms will be closed under H-consequence and thus we can talk about the "theory of a model".

mE

E.G.K. LOPEZ-ESCOBAR

194 2.2.

IP C.

Examplu 06 flOund and -6tMngly complete -6emant.<.cal .£nteJr.pJte..t.a.tiOM

60Jt

K = Original Kripke models (Kripke 1965). KO = Rooted Kripke models.

Xl

X2

X 3

= Tree-Kripke models. = Finitely branching tree-Kripke models.

= Countable tree-Kripke models.

B = Original Beth models (Beth 1956). Bo = Strong Beth models (Kripke 1965).

= Existential-realizing Beth models (Gabbay 1977). 1 B = Exploding Beth models (de Swart 1977). 2 8 3 = Intuitionistic Beth models (de Swart 1977). B

2.3.

Model. theMy Jtel.a..ti.ve to a -6emant.<.cal .£nteJr.pJte..t.a.tion.

Suppose that we are given an H-semantical interpretation S = (M S' I=S)' then relative to S we make the following definitions concerning the elements of MS (usua 11 y ca 11 ed modd-6 or -6.tJtuc..tuJtu). Th S

em)

= {A

In == s 1& iff Furthermore given a set T of

E

Th S

sentL :

em}

m

= Th

I=S A}. S

(n ).

L - sentences then we let

Observe that the condition that S be a sound semantics for H may be expressed by:

IJm

E

MS(ThmL(F} n SentL

~ Th S (m)).

If T is an If- theory and m. EMS is such that Th S (m) = T, then m: will be called a T - paJta.d.£gm. The semantics S is a paJta.cUgma..ti.cal -6emant.<.c.a.l .£nteJr.pJte.ta..ti.on 6M H (p. s . t , for short) iff for every consi stent If -theory T there correspond at least onem EMs which is a T-paradigm. The semantics S isana!mo"t paJta.d.£gma..tlcal -6ema.nt.£cal .£nteJtpJte..t.a.tion for H (a. p, s. i .) iff for every consistent H - theory T there corresponds an mE MS such that T ~ Th S (nt). 2•4 •

Re1.a..ti.o M between cU 66eJr.ent "ema.nt.<.cal .£nteJr.pJte..t.a.tio M

•

Given two semantical interpretations for a logic H, S = (M S' 1= S) and , 1= 1 ) ' we now consider some of the ways in which S and I could be con1 sidered to be equivalent.

1= (1.1

2.4.1.

DEFINITION.

S

and

I

are equivalent, in symbols S == I, iff

MODELS FOR INTUITIONISTIC LOGIC 'rim

E

M

31l

s

'ri7t E M I

2.4.2.

(1-1)

0

All (Th S (m) = Th

3mE M

s

DEF IN ITION.

nunc..t£ol!

E

S

MS

oftom

(Th l ('It) " Th

DEFINITION.

S

3n E

'rimE Ms 'rin E MI

3711E

and

and

(1t))

S

(m)).

S =.+ t , '£00 ~uc.h tho.;(: oOft aU EMS:

and I ol'l-to MI

MC e.qu.-[va1.cn,t,

ThS(m) = Th l

2 . 4 . 3.

l

195

theA€..u,

0.

wm)).

aft.c cqu,{vo.te n.t* ,

I

MI (Th[

m

('It) c Th

-

S

(1lL))

S

",* I,

{6n:

and

S MS (Th (m) c- Th[ (lL)) .

Finally we consider two possible orders amongst semantical interpretations. 2.4.4.

DEFINITION. 'rin E M

3'l1lE M (Th S

I

2 . 4 . 5.

DE FIN ITION. 'ri

n

l u weo./Wt thaI!

I

L60

l (n) = Th S ('nf',)).

.u., weakeJt * than S,

3rTtE M (Th s

E M[

I:S S,

S,

S (77t)

~

Th

l

I:S

* S,

,£00

('ll)) .

It is clear from the definitions that: (S =.+ [

..,.

(S '" [)

..,.

(S =.'" I)

and it is not difficult to give examples that show that the converse implications do not hold unless some further restrictions are placed on the semantical interpretations. First of all let us agree that by the cardinal ity of s , c.Md(S), we understand the cardinality of the set Ms' Next, given an m E Ms, we let [m]S= {?tEMs ' Th S (m) = Th S ('It)}; [ml is called the S-ei.cmcl'l-tMy UM.6 oom. s 2.4.5.

DEFINITION. A.6emo.l'l-tLc.a1.-£l'l-teJtplte.ta.t{on (.6.L) S = (M FS-U S' homogcncoU.6 -£66 60ft aU m « "s: thc C.Md,(lta1.Uy 06 [ntIs -u cqual. .to c.Md(S). 2.4.6.

LEMMA.

tho.;(: c.Md (S1 ;" c.o.!td [s*J

PROOF.

16 c.Md(S) ~ ~O' thcn theJtc.u, and S

=- S *.

Let /( be c.Md(S). Then to eachm E Ms < /(. Then define M * = MS U um E M S s

objectsm~, ~

0.

homoge.ncoU.6 .6.L

s* .6uc.h

introduce K new distinct < K} and let FS *

{m~; ~

E.G.K. LOPEZ-ESCOBAR

196 be the extension of

1= S such that for allYn EMS' ~

A:

(*)

iff

Finally let S* = (M S*, cMd(S).

< K , and all L - sentences

I=S*

l.

A simple computation shows that

Condition (*) guarantees that S =S*.

2. 4 . 7 . LEMMA. 16 S and I Me homogeneotL6 cMdbUlUxy and S I., then $ +I. .

=

cMd(S*)

•

=

.0. L

06 the .oarne

-in 6.f.n.<.te

PROOF. Assume that S and I. are homogeneous st . of the same cardinality and that S I. Consider the partitions PS and PI of Ms and iiI with respect to elementary equivalence. Using the Cantor-Bernstein Theorem and the condition S = 1 we conclude that there is a (1-1) function 6 from Ps onto PI' Using the homogeneity condition we can extend the function 6 to a (1-1) function from ~'s onto MI' •

=

2.4.8. CUte

COROLLARY. 16 S and I CUte.6.L .ouch tha.t S r" .ouch tha.t S = s* =+ t" = f.

s,«. S* and

=I,

=

then theJte

=

It follows from the corollary that the difference between and + is rather superficial. The situation between - and ='* is not as clear. We content ourselves in remarking that if Sand l are two H - paradigmatic s. i , then S

=*

§

I. .

3.

LOG I CS AND

CAL I NTERPRETAT I ON.

SEM.~NTI

3.1. Given a 1ogi cHand two semanti ca1 i nterpreta ti ons Sand l for H we will now consider how the relations between S and I. introduced in the previous sections affect conditions such as soundness and completeness.

16 3 . 2. LEMMA. a.1..o 0 a .00 und .6ema.ntiC.6

s

.£.0 a .6Ound .6emantiC.6 60!l. H and I. ~ * S then I.

6on:

H.

PROOF. Assume that S is a sound semantics for H and that I ~ * S. be an L-sentence which is a theorem of Hand let7J,E M r Then since I there is an mE MS such that Th

But Sis sound for 3.3. LEMMA.

H, thus

Smn ~

A E Th S ('~).

601t

If.

Let A ~* S,

nf (on). Hence

A E Thl. (7£).

16 S .£.6 a complete .oemantiC.6 601L Hand S ~

a.t.oo a complete .oemantiC.6

.£.0

•

* I.

then I. .£.0

197

MODELS FOR INTUITIONISTIC LOGIC

nE

PROOF. Assume that S is a complete semantics for H and that for all MI, A ETh I (17, ) . Using the condition that S :::;* I we conclude that 'tj

m

~'s (A

E

E

ThS (m)) .

But S was assumed to be a compl ete semantics for H. orem of H. •

Hence A is indeed a the-

nOll

3.4. COROLLARY. In S iJ., a !.Jound and comple:te !.JemafttiC!.J and S == *I, then I M aUo a !.Jound and comple:te !.JemafttiC!.J H.

nOll

3.5. S :::;

*

LEMMA.

InS, I

aJr.e!.J.L

noJr. H and

I

the logic H

Is an a.p.1>.L

then

I.

m

PROOF. Let E ~S be given. Then 1et T = ThS (m). From the assumption that I is an a.p.s.i. we conclude that there is an n E ~II such that T/(n):='T. In other words we have shown that 1J/1f, EM 3?t E M- (TI/(n) c ThS(m)) , S I -

i. e.

S

:::;

3.6. then: (1)

(2) (3)

*

I.

•

In

COROLLARIES.

in in in

S

and

I

both S

and

I

Me a.p.1>.L

both S

and

Me p.1>.L

both S

and

I I

Me !.J emafttica! intvr.pJte;taUo I1!.J

then

S==* I,

then S == I,

Me a.p.1>.i.

and S

noJr. H,

iJ.,

1>ound and comple:te

noJr. H

1.

then 1>0 -l.!.J

§ 4. SUMMARY. The following are some of the well-known results concerning Kripke and Beth models expressed in terms of the relations between semantical interpreta ti ons.

4.1.

(Kripke 1965)

K == K == K == X 3 O 1

4.2.

(Kripke 1965)

Xl:::; B O

4.3.

4.4.

B O :::; B,

IU

(Gabbay 1977)

i KO B:::; B

1

4.5. B:::; B 2 ' B 2 i B 4.6. (de Swart 1977) B 3 '" B 2 §

5.

NON-STANDARD BETH MODELS.

198

E.G.K. LOPEZ-ESCOBAR

5.1. I NTRODUCT ION. A1 though some authors appear to object to the various attempts to make Kripke (and Beth) models a plausible interpretation of intuitionistic reasoning, it nevertheless is a fact that people are more interested in models which have some semantical interpretation (perhaps that is a reason why Kripke models have been studied much more than algebraic models). The attempts to give a plausible interpretation for the Beth and Kripke models center around the view that intuitionistic logic is a logic of "positivistic research". In such a positivistic research we assume that we have various states of knowledge which form themselves into a partial order. In both Kripke and Beth models if a sentence is asserted to be true on the basis of a given stateofknowledge it will also be asserted to be true on later states of knowledge. The difference appears on how it is asserted to be true: in Krinke models, it depends only on the later states of knowledge, in Beth models it also depends On the paths through the states of knowledge. In an intuitionistically correct Beth model the partial order of the" states of knowledge" forms a OM (classically: a finitely branching tree) and the paths considered are all the paths through the fan. In fact, it is customary to restrict oneself to fans which have no finite (terminating) paths and so the paths in question are the so called .£pfJ ('£'Il6.{VI-Ue£y pft(!.eecUllg fJequelleefJ) admitted by the fan (law) . The Beth models thus described form a complete and sound semantics for the fPC and even an intuitionistically acceptable proof of the completeness is possible (see de Swart 1977). We will call the Beth models in which all the admissible ips are used in the definition of validity the .£./ltuitiOn.{.fJt.<.caity ~tall~d Beth mod~ (or simply: Beth models); the '{nt~On.{.fJt.<.cally 1l01l-fJtall~d Beth mod~ (orsimply: n - s Beth models) will be obtained by restricting the class of ips. We shall see that the n - s Beth models can be used to obtain sound and complete semantics for some extensions of the fPC (the extensions are perhaps non-standard from the intuitionistic viewpoint ~). 5.2. the set of bers, IN, o. If n tion that latter by

SOME CONVENTIONS ABOUT finite sequences of natural we may assume that a fan is is (the code for a) finite II is a node of the fan a

FANS.' By using some standard mapping from numbers (1 - 1) onto the set of natural numdetermined by a number theoretic function sequence of natural numbers then the condiis given by 0(11) ~ 0, we abbreviate the

II Eo.

a

If a is a function then a(k) ~ (0(0) .... , 0(k-1».

is the course of values function

admUted by

'" is an .£p~ (or plLth) for all k, a(k) EO.

0 (or

be£ollg.£.ng

A node 11 of 0 UefJ on the .£.PfJ o (or be£ollgfJ :tJJ and only if there is a k such that 11= a(k) . II,

In ~

:tJJ 0)

0),

just in case

in symbol s:

that

II E '" ,

if

If 11 and m are nodes of a then we say that In is above (or !a.ten thall) in symbols: II::' m, iff there is an ips a such that for some k,j, 11 ~ a(j) a(k) and k;;> j.

An ips a is called a tha t for a11 nodes In of 5.3.

DEFINITION.

fJ~p 0,

If

0

anAoW of

if

II C m

0,

then

just in case there is an n E a such mEa.

is a fan then Path o

is the set of all ips admit-

MODELS FOR INTUITIONISTIC LOGIC

199

ted by a , 5.4. CONVENTION. From now on whenever we consider fans we fans which have no finite paths. 5.5.

DEFINITION.

a camp/tchcru,.ive (ux

at

If

0

is a fan and Fe Path o then

understand

F will be called

path6 on. .ip.6) iff

5.3. Bcth mod~, .6mndMd and non-.6mndMd. Beth structures are usua 11 y defi ned as to cons i st of a fan 0 and a mappi ng V from the nodes of 0 into a set of finite sets of (atomic) formulae. However the satisfaction (forcing) relation also involves the ips of o. Thus we prefer to define the Beth structures so that the required ips form part of the structure.

that

5.3.1. DEFINITION.

,

F:t- ) such

is a fan, is a mapping from the set of nodes of 0;(, into a set of finite sets of atomic formulae such that if n ~ mE o./:.then V~ (nl ~ V"" (m) , II Fg ~ Path Ox- and ~ is a comprehensive set of paths.

(1)

aX.

(2)

V..l:-

(3)

A Bcthl>t!tuctwte is a triple z,= (oS' Vz

If J;

is a Beth 'structure such that

F:&. = Pat;h~

then:t.-

is

called

a

In terms of the view of positivistic research a comprehensive set of paths one that goes through every possible state of knowledge.

is

.6mndMd Be.th stnuctune , otherwise it is a n-.6 (non-l>mndMdIBe.th .6XILuc:tu/tC.

5.3.2.

DEFINITION.

If,'? = (oz, V~, F~)

is a Beth structure

a

n

node of the fan a; and A a formula of L then A .i.6.tJwe at; n .in Z- (or better still: A.i.6 .6cCUltcd at; n ,[n .i-); in symbols :r;, n 1= A, if and only if : (1)

1 E V~(n)

(2)

A is an atomic formula and II a n E a E

(3) (4)

A = (c 1\ 0) A = (CV 0)

and and

(5)

A = (C ::> 0)

and

(6)

A = 'VxCx

and for all parameters a, ~, n

(7)

A=3xCx

and

or

h,

n 1= C and

'i

:&-,

3 k(A

Va

n£fnE~

nEaE ~

(%-,

((;'" k)),

or C or

n 1= 0 ,

(;'"k ljanEaE~ 3 k(.Jt, Ijm

E V~

1=

g, (;'"

m 1= C .. :t-, m 1= D)

3 k 3 a(z., (;'"(k)

1=

Ca

or

fl 1= 0)

or

or

or

1= C a)

In definition 5.3.1. we allowed the possibil ity that for some nE0J;. ,1 EVz. (n).

E.G. K. LOPEZ- ESCOBAR

200

Of course in terms of positivistic research that is rather doubtful since it would correspond that at some state of knowledge we have concluded the absurd (on second thought it is perhaps not too far fetched). However the possibility of 1 E V~ (n) permits us to obtain an intuitionistically acceptable completeness theorem(3). We could of course simply avoid that by requiring in the definition of a Beth structure £.- that for all n E Ox. l'f V (n}, in which case (1) of defi nition z 5.3.2. becomes moot. 5.3.2. DEF I NITl ON. If;g, is a Beth structure, it- = (0:e. ' V»> F ~) and A and L - f'orrsul a, then A.u., tnue. -<-n :t (or: L>eCU/led -<-n:&-) iff for all n E ~, t- , n F A. We express that A is secured in ~ by ":t. FA" • 5.4.

Some bCl..6.{c plWpe.!l.-tieL>

on

Beth L>:tJw.c..:tuJ1.eL>.

that we are dealing with a single Beth structure. 5.4.1.

LEMMA.

Tn

s.

n FA

In this subsection we assume

n~mEo~

and

then~,

m FA.

PROOF. By induction on the logical complexity of the formula A. If A is either a conjuction, conditional or universal formula then the result follows immediately from the induction hypothesis. For atomic, disjunctions andexistentia11~ quantified formulae one needs to use the fact that one is dealing witha cOll'prehensive set of paths. More specifically, one should first verify that no matter what property P( ) : 'V

t< nEt< L

Fz. 3 x p( a (Q)) f\ 'Vex mEt< E F;t;.

5.4.2.

COROLLARY.

5.4.3.

THEOREM.

empty L>equence. qu..{vaLent•

:t

F A -<-on

FOI1. each n

E

n ~ m E 0!Lr

3k.f(a(Q)).

=>

•

~,O F A, wheJLe 0 Ls .the code 0011. the

"z:

(1)

.t-,I'lFA,

(2)

'VexI'lEt
the nou.ow-<-ng .two condU1onL>

aJz.e

e-

PROOF. The condition that Fx. is a comprehensive set of paths gives us immediately that (1) => (2). That (2) ~ (1) is proven by induction on the complexity of the formula A. If A is an.a!omic formula then it follows from the definition of ~, n FA. Of the rema'mt nq cases the only one which might not be immediate is the one corresponding to the existential quantifier. Thus assume that

(3) See, for exanp l e de Swart 1977. For another context in whi,ch plays an important rBle, see Gabbay 1976.

1 E V

(n)

201

MODELS FOR INTUITIONISTIC LOGIC

3k(~,a(k)1=3XCx). Then, from the definition of

lfanEaEF;&

secu-

rability we obtain that:

Using the properties of a comprehensive set of paths we then obtain: If a

nEa E Fg.

3:t 3 a( ~ , a:t 1= C a ) .

From which it follows that:

t-, 5.4.4

DEFINITION.

5.4.5. THEOREM.

Beth

n

1=

3xCx.

•

= {~: ~ is a Beth structure}. Ls a .6ormd and eomptete .6emanlie.6 60IL

Beth,

IPC.

PROOF.

Since Beth includes the original Beth structures it follows that B, the or iq ina l Beth semantics. Since the original Beth semantics is known to be a complete semantics for IrC, it follows from Lemma 3.3. that Beth is also a complete semantics for the IPC. It only remains to show that Beth is a sound semantics for the IPC. That is, we have to show that for all 1tYE Beth and all

Beth

~

theorems A of IPC, :c.- 1= A. One way to do the latter is to choose some formalization for the IPC and then (1) verify it for the axioms and (2) check that the rules of inference preserve that property. If one chooses the formalization given in Kleene 1952 then (1) and (2) are straightfoward enough. •

§

6.

BETH STRUCTURES AND CLASSICAL LOGIC.

6.1. In:t!Loduc.:t.i.on. We now turn to consider Beth structures in relation to extensions of the IPC. In this section we consider the Classical Predicate Calculus, cr C. The crc can be obtained from the rrc by simply adding all the Lformulae of the form A V"lA (where"l A is an abbreviation for A:J 1).

Suppose that }& = (0 ~, VI'-' FI'-) is a Beth structure such that all the ips in F~ are sharp arrows (see Definition 5.2.), then such a Beth structure will be called a Sha!Lp Beth stnuc.:twr.e. Then we let SB be the semantics of a.e..e. (standard and non-standard) sharp Beth structures and SB be s the semantics of tre .6tandaILd Beth structures. The following lemma is immediate. 6.2. ShMp Beth .6:tJwc.:twr.e.

6.2.1. LEMMA.

SB

s

s SB.

Since the CPC is obtained from the Irc by the addition of the schema AV,A, the following lemma is preparatory to showing that SB is a sound and compl ete semantics for CPC : 6.2.2. LEMMA.

16 Z=

(ax.' V,j:.' Fx,.)

E

SB, a

E

F~ and A .u,

an L-6O!L-

E.G.K. LOPEZ-ESCOBAR

202 thVte. .w a k a (k ) <.=.. II, :t.,

mula that

tlle.11

6uch that UthVt .ftI-, a (1<) 1= A n II A •

nOJt

au.

11 E

(J~

6uch

n must be a sharp arrow. Let k be such that for then IlEn). • In 1i- E SBthe.n nOJt aU. noliJnU.f.a.e A, J= A V I A •

Since n E (if If" (Ie) <.=.. 6. 2 • 3 • COROLLARY.

11

6.2.4. THEOREM.

SO J.1> a 60ulld alld eomplete. 6e.mantiN nOlL the

PROOF.

all

OIL

F~,

IlE(J~

:v

CP C.

PROOF. From the fact that Beth is a sound semantics for the IPC and Corollary 6.2.3. we obtain that SB is a sound semantics for the CPC. To prove that SB is a complete semantics we shall make use of Lemma 3.3. For that purpose let ( =MrestrictM be the usual Tarskian semantics. Now it is well known that M w ed to countable structures) is a complete semantics for the CPC. But any (J[EMw can be made into a ~ E SB by simply taking the positive atomic diagram of (J[ and placing it on a fan consisting of a single sharp arrow. •

In a similar fashion we can prove that SOs is also a sound and complete semantics for the IPC.

§

7.

BETH STRUCTURES AND THE LOr, I C OF CONSTANT DOMA I NS.

7.1. TnbtOductioll. The sharp Beth structures, SB, might, at first sight, appear to be a sound and compl ete semantics for the logic CD. However the results of the previous section show it to be a sound but not complete semantics (it suffices to observe that A V I A is in general not a theorem of CD).

One of the nice things about the class SO is that it was defined without reference to the assignment mapping, that is the condition for a Beth structure )6 = (o~, l!~, F:c-) to be a member of SO does not depend on the mapping V~. In fact, let us agree to call a class C of Beth structures a 6.tJtue.tMilly den.£lled e£.a66 iff the condition ~ E C can be expressed without reference to the assignment mapping lI.1" Using the fact that SO is a sound semantics for CP C, it can be shown that there cannot be a structurally defined Beth semantics which is both sound and complete for the logic CD (4) . VeMn.£te Beth 6.tJtuetUJte6.

7.2.

7.2.1.

DEFINITION.

A Beth structure ~ = (0;e., V~, FJr,) is a den.£nUe Cx , all ips n E F~ we have that:

Beth 6.tJtuc.:tuJte iff for all L - formulae

(*)

.£n: nOlL

au.

pMamete.Jt6

J&.-,

a thVte J.1> a a(k)

1=

k 6ueh that

Ca

(4) The argument is simply as fol lows: Since the semantics cannot be SB there must be a structure in the semantics with at least one non-sharp Ips. Choosing then an appropriate V one can then fail to satisfy a sentence of the form 'if x (Px V Q.) ::> 'if x P x V Q.

MODELS FOR INTUITIONISTIC LOGIC then:

thVte.u, a

Iz !>uch that 60lt aU piVl.ameteM ~.u(r.:}

F

203

a

Ca

If we use Beth's idea of using the natural numbers as the parameters of language then the condition (*) can be expressed as follows: 'tj P E :IN 3

Iz

E :IN

(it, u ( II)

7.2.2. DEFINITION. a sharp arrow of a.

A fan

7 .2.3. DEF I NI TI ON.

~ 3 II. E IN 'tj P E :IN (~,

F Cp}

u (1<)

is wllu!>t-!>hMY.' iff every node of

o

the

F Cp} .

a

lies

on

A Beth structure ~ = (al'-' v~, FJI-) is almo!>:t-.6hMp

if and only if the underlying fan

a

J¢-

is almost sharp.

The use of almost-sharp Beth structure is that they can be used to obtain definite structures. For given an almost sharp Beth structure ~= (ax..' V~, F~) , then the Beth structure .to = (ax,.' Vx,.' Slf,) where S~ is the collection of sharp arrows of "s » will be a sharp Beth structure and hence a defi ni te Beth structure (5). Note however that not all definite structures need be sharp, e.g.

-,

THEOREM.

7.2.4. :tllel1.

J?

PROOF. then

16.t-

'tjx (Ax V B) ~ if x A x V B

'tjQ

11.

E

=

(ax.'

(wheJte

v~, F)C.) .u, x

dOe!>

It suffices to prove that for all EF H(J&.,u(lz} F 'tjx A x or Q ' it-

that ~,11. F ~x(AxVB) have that

11. E

and that

1?,

11.

a de6-{.nUe Beth

11.0t OCCUlt

6ltee -i11.

AX. • if

~,u (Iz)

!>:tltuctUite

B) ~ va..Ud

1?,11.

F if x(Ax VB}

F B}.

Thus assume

Then for all parameters F

.£11.

a

Aa V B.

(5) The almost-sharp Beth structures are sound and complete for the extension of the IPe obtained by adding the schema ifx'l 'lAx ~'l'l'tj xAx.

we

204

E.G.K. LOPEZ-ESCOBAR

In particular we obtain that for all parameters a there corresponds a ~ such that (either Z , a (~) 1= Aa or Z-, a (~) 1= B). If there is a I< such that it- , a (I<) 1= B then we are through. Otherwise we obta in tha t for all parameters a there is a I< such that z., a( I<} 1= s:«; Making use of the assumption that ~ is a definite Beth structure we then obtain a I< such that for all parameters a,1&-, a (I<) 1= Aa, i.e. ~,a(l<) 1= IJxAx . • 7.2.5.

DEFINITION.

7.2.6.

THEOREM.

§

8.

DB

DB is the class of all definite Beth structures.

DB.iJ., a .6QLLYld -6emant.<:C6 non CD.

AS A COMPLETE SEMANTI CS FOR CD.

8.1. In..t!todu.cUoYl. In this section we shall prove that DB is a complete semantics for the logic CD of constant domains. In the proof we shall use of the fact that the semantics K 5 of Kripke structures of constant domains is complete for CD (see, for example, Gornemann 1971) and the transformation of Kripke structures into equivalent (standard) Beth structures given in Kripke 1965.

8.2. (1)

DEFINITIONS. K = {(!,¢): (!,¢) 5 }.; a fan}

and

rela-

In Kripke 1965 a mapping r * is defined which transforms an arbitrary structure a. into a standard Beth structure T * (zz ) such that

Kripke

K

5

11-)

JN

where 11- is the Kripke satisfaction (forcing)

(2)

(K 5' tion.

is a Kripke structure with constant domain

Th(Ol) = Th(T * (Ol

If we restrict ourselves to Kripke models in the class plified as follows: 8.3.

DEFINITION. if (l:, ¢ ) E K5' then as follows: First we let

t

C

0 = i

K5 then

~

can be sim-

is the mapping from K5 to Beth structures such that = (u, V, Path u) where o and V are obtained

is an ips of !

O .;: i 1 .;: ••• .;: it - 1} ,

is the relation of initial segment restricted to elements of

Thus (M,

r*

T(~,¢}

M = {(a(.£o)' a(i 1 ) , .•• , a(i ) } : " t_ 1 &

».

) can be viewed as a fan o .

M.

Observe that each node of

m of

o is

205

MODELS FOR INTUITIONISTIC LOGIC

a finite sequence of nodes of ~; we let t(m) be the last element of m. Thus for each mEa, t(m) E L. Finally V is defined on a so that for each mE a, V(m)

= ep(t(m)).

The following theorem can be obtained from Kripke 1965: 8.3.1.

THEOREM.

01- = (r , ep) E K 5

16

then 6o~ eV~lf 6o~u1a A and QLUva.tent : (1)

eV~lf

it?

and

= (a, V, Path a) = T (~ ,ep ) a~e e-

mEa, the 6oUow-ing two c.ond-itiolt6

Jif-,m FA, 01-, t (m) [f- A.

(2)

Unfortunately }? need not be a definite Beth structure. We shall :& so as to obtain a (non-standard) definite Beth structure. 8.3.2. DEFINITION. Given (r, ep) E K 5 Fr={J}:J}EPath a & 3kVt

let

of

and T(r, ep)

now modify

(a, V, Path

a

)

t>k(fl(t)=fl(k»).

It is trivial to check that the Fr so defined is a comprehensive set of paths a. Thus (a, V, Fr) is a non-standard Beth structure. We define the mapping so that (L, ep) = (a, V, Fr ) . 8.3.3.

£=

LEMMA. '16

(r,ep)EK S ' Z-= (a, V, Patha)=T(r,ep) and r ) then 6M aU 6o~u1ae A and aU nod~ mEa:

(L, ep ) = (a , V, F

4C-,m

1=

A

£,

-i66

m

1=

A.

PROOF. By induction on the length of A. We shall only consider the case when A is an atomic formula. If i6-, m F A then A E ep(.t(m»). Thus VflmEflEFL 3 teA E VW(t). Hence 1;, m FA. Conversely, if m 1= A then

.c,

VflmEJ}E F 3t(A E v(if(t)). Let -y E Path a be such that m =;Y (Uh(m)) and for L all t;;. Uh(m), -Yet) = -y(Uh(m»). Then -y E FL and thus for some k, A E V(;Y(k)). But then A E (ep t(;Y(k). So A E ep(.t(m), from which it follows that AE V(m) and hence that J:-, m FA. • 8.3.4. THEOREM.

.c -u a deMnae

16

or

= p.:, ep) E K ' S

); = (r, ep)

Beth ,sbw.cXu!te.

(a, V, Fr) then

PROOF. In view of the previous lemma it only remains to prove that ,£ is a definite Beth structure. So assume that i3 E a and that P(x) is a formula such that: V n 3t(.l,

Let

k

r

(t)

be the natural number such that

F \j

P(n».

t;/;> k (fl (t) =

i3 (Iz)). According to Lemma

206

E.G.K. LOPEZ-ESCOBAR

8.3.3.

we then have that Ii n 3.t(~.i3 (t)

where:? is the standard Beth structure Aplying Lemma 8.3.2.

t= P(n)) ,

(0, V, Path o ) '

we then obtain that Ii n 3

But then using the fact that

aa , L( 73 (:t)) 11E F~

~

P(n)).

we obtain

Ii n(OL , ~ (1<) 11- P(n)).

However crt

E

K 5 ' and thus we concl ude

or ,

~

(I<) 11-

Ii x P (x) ,

whichin turn 1eads to ;;, if(l<+l)

8.3.5.

COROLLARY.

t= lixp(x) . •

DB J.1, a c.ompLe:tc .6eman.tiC1>

60ft

CD.

PROOF. The above lemmata show that K ':S DB and since it is known that 5 is a complete semantics for CD applying Lemma 3.3 we conclude that DB is a complete semantics for CD. •

K5

REFERENCES. E. W. Beth 1956

Seman.tic. COn.61nue:t£on 06 In:tuit{onJ.1,:t£c Akad. van Wet. Afd. Lett. N°ll.

Log~c,

Mededelingen der Kon.

Ned.

D. M. Gabbay 1972 1976 1977

Apptlea:t£on.6 06 :tnCC.6 :to ~n:tenmc~c Log~C.6, J.S.L. vol. 37, pp. 135-138. CompLe:tcnC.6.6 pMpC/l.-tlC.6 06 Hey:t£ng 1.6 plted-lca:te eaLc.u1.u6 wah ltC.6pec:t :to RE mode,U, J.S.L. vol. 41, pp. 81- 94. A new VeM~on 06 Be:th .6eman.tiC1> 60ft ~n:tuit{onJ.1,:t£e loMc, J.S.L. vol. 42 , pp. 306 - 308.

S. Gornemann 1971

A LoMe .6:tMngeJt. :than

~n:tLU.tionJ.1,m,

J.S.L. vol. 36, pp, 249 - 261.

MODELS FOR INTUITIONISTIC LOGIC

207

S. C. Kleene 1952

Introduction

to Metamathematics, Van Nostrand Publ ishing Co.

S. A. Kripke 1965

Sema.ntLcat ana1.y.s.u,

06 .£n:tJ.U.tion.i..1.:Ue .tog.£e, in

Formal systems and re-

J.N. Crossley, M.A.E. Dummett, editors. North-Holland Publishing Co. Amsterdam. pp. 92 - 130. cursive functions.

P. LindstrBm 1969

Ort

ex.:teYl
06

E.temen.t:aJty Log,[c.,

Theoria vol , 35, pp. 1 - 11.

C. A. Smorynski 1973

06 KJUpke mode£.6, Chapter V in: Metamathernatical investigations of intuitionistic arithmetic and analysis. A. S.

AppUc.a:UoYl
Troelstra editor.

Springer Verlag Publishing Co.

H. C. M. de Swart 1977

Art .£n:tJ.U.tion.i..1.:Uc.aUy p.taUA.£b.te '[Yl:teJtpJte:ta:UOI1 06 .£n:tJ.U.tion.i..1.:Uc. .tog,[c., J. S. L. vol. 42, pp. 564 - 578.

A. S. Troelstra 1977

A-6pew 06 c.On6bw.c.:Uve ma:thema:Uc.-6 in: Handbook of Mathematical Logic. J. Barwise, editor. North-Holland Publishing Co.

1978

Some JtemMM on :the c.omptexuy 06 Henk.£n - KJUpke mode£.6, Indagationes Mathematicae, vol. 40, pp, 296 - 302.

Department of Mathematics University of Maryland College Park, Maryland 20742 U. S. A.