embeddability, syntax, and semantics in accounts of scientific theories

PETER TURNEY

EMBEDDABILITY, SYNTAX, AND SEMANTICS IN

ACCOUNTS OF SCIENTIFIC THEORIES

ABSTRACT. Recently several philosophers of science have proposed what has come to be known as the semantic account of scientific theories. It is presented as an improvement on the positivist account, which is now called the syntactic account of scientific theories. Bas van Fraassen claims that the syntactic account does not give a satisfactory definition of “empirical adequacy” and “empirical equivalence”. He contends that his own semantic account does define these notations acceptably, through the concept of “embeddability”, a concept which he claims cannot be defined syntactically. Here, I define a syntactic relation which corresponds to the semantic relation of “embeddability”. I suggest that the critical differences between the positivist account and van Fraassen’s account have nothing to do with the distinction between semantics and syntax.

INTRODUCTION

Recently several philosophers of science, including Bas van Fraassen (1970, 1980), Patrick Suppes (1967), Joseph Sneed (197 I), Paul Thompson (1983), and Wolfgang Stegmiiller (1976), have advocated what has come to be known as the semantic account of scientific theories. It is presented as an improvement on the positivist account, which is now called the syntactic account of scientific theories.

I believe that the semantic account of scientific theories is better than the syntactic account; however, I propose that it is not simply because it is semantic, rather than syntactic. The improvement, I think, is due to other differences between the two accounts.

Van Fraassen claims that the syntactic approach cannot capture the notion of one theory being “embedded” in another (van Fraassen, 1980, p. 43). In this paper, I define “implantability”, which is the syntactic concept analogous to the semantic concept of “embeddability”. I look at van Fraassen’s definitions of “empirical adequacy” and “empirical equivalence”. I then give syntactic definitions for these concepts, using the notion of implantability.

To conclude, I suggest that the semantic account is an improvement over the syntactic account, not because the former is semantic

Journal of Philosophical Logic 19: 429451, 1990. 0 1990 Kluwer Academic Publishers. Printed in the Netherlands.

430 PETER TURNEY

and the latter is syntactic, but because of differences in how the accounts handle the relation between theories and observations.

EMBEDDABILITY

Van Fraassen defines embedding as follows (1980, p.43):

We say that one structure can be embedded in another, if the first is isomorphic to a part (substructure) of the second. Isomorphism is of course total identity of structure and is a limiting case of embeddability: if two structures are isomorphic then each can be embedded in the other.

He contintiks (1980, p. 44):

This sort of relationship, which is peculiarly semantic, is clearly very important for the comparison and evaluation of theories, and is not accessible to the syntactic approach.

Let us attempt to define the notion of embedding syntactically. The following definitions are loosely based on Bell and Machover (1977, pp. 161-168).

DEFINITION 1. Let S be a structure of the form:

S = (R, U>

The elements of this structure are:

R: a countable sequence of relations. U: a non-empty set of individuals. R = (4, 4, R,, . . .> Ri: a subset of u”“‘; a set of n(i)-tuples of individuals.

The function n maps from whole numbers to whole numbers. Thus, n(i) cannot be zero. Many set theoretical structures have the form of S. We will call a structure with the form of S a system, in order to distinguish S from set theoretical structures in general.

DEFINITION 2. Two systems S = (R, U ) and S’ = (R’, U’ ) are isomorphic when there is a one-to-one functionf, mapping from U onto U’, and a one-to-one function g, mapping from R onto R’, such

EMBEDDABILITY, SYNTAX, AND SEMANTICS 431

that

for all i.

<u,, . . . , utiij) is a member of Ri if and only if

<f(u,>, * . . ,f(u,,&) is a member of g(R,),

DEFINITION 3. Let S = (R, U) be a system. Let U’ be a subset of U. Let Ri be a member of R. Let Rj be defined as follows:

Rj = R, n U”‘ci)

We call RI the restriction of Ri to U’.

DEFINITION 4. Let S = (R, U) be a system. Let S’ = (R’, U’) be defined as follows:

(1)

(2)

U’ is a non-empty subset of U.

The members of R’ are generated by restricting each Ri in R to U’, to give R].

We call S’ a subsystem of S. Note that R and R’ have the same number of members.

DEFINITION 5. A system S is embedduble in a system S’ when the following conditions are met:

(1) There is a subsystem S” of S’.

(2) S is isomorphic to S”.

DEFINITION 6. Let T be a structure of the form:

T = (P, L, K)

The elements of this structure are:

P: a countable set of predicates. Niladic predicates are not permitted.

P = (4, 4, 4, . . .>

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L: a first-order language, constructed from P, together with the standard logical operators, quantifiers, variables, and the identity relation. All variables in any sentence in L must be bound by quantifiers. Since niladic predicates are not permitted and there are no constants, every sentence in L must have quantifiers.

K: a deductively closed, consistent subset of L; the K-true (true, according to K) sentences of the language L.

Many syntactical structures have the form of T. We will call a structure with the form of T a theory. Context should distinguish theories of this sort (theories of first-order logic) from scientific theories.

DEFINITION 7. Let T = (P, L, K) be a theory. Let S = (R, U) be a system. Let A4 be a structure of the form:

A4 = (T, S, I)

Z is a function. The domain of Z is P u L. M is called a model of theory T with system S, when the interpretation I meets the following conditions:

(1) If B is a predicate in P, then Z(B) maps B to a relation R; in R. B must be an n(i)-adic predicate. The function Z must map P one-to-one onto R.

(2) Ifs is a sentence in L, then Z(s) maps s to true or false. The mapping of sentences to truth values is determined by the mapping of predicates to relations. The function Z must follow the usual conventions for a first-order logic.

(3) If k is a sentence in K, then Z(k) is true.

Note that, in condition (3), we do not require that, if Z(k) is true, then k is a member of K. This latter requirement is known as semantic com- pleteness. The requirement of condition (3) is known as soundness.’

DEFINITION 8. Let T be a theory. Let S be a system, We say that S is compatible with T when there is an interpretation I, such that M = (T, S, Z) is a model of T.


THEOREM 1. Let T and T’ be theories. Suppose S is compatible with T, and s’ is compatible with T’. Suppose S and s’ are isomorphic. Then S’ is compatible with T, and S is compatible with T’.

Proof. Let T and T’ be theories. Suppose S is compatible with T, and s’ is compatible with T’. Suppose S and S’ are isomorphic. We will show that S’ is compatible with T. The same style of argument shows that S is compatible with T’. Let Z be the interpretation which shows that S is compatible with T. Let Sand g be the functions which show that S and S’ are isomorphic. We will construct an interpretation Z’ which shows that S’ is compatible with T. Let B be a predicate in P. Z(B) is a relation in R. g(Z(B)) is a relation in R’. We define Z’(B) as g(Z(B)). Since we have just defined I’ for predicates, the definition of I’ for sentences is determined by the usual rules for first-order logic. Let k be a sentence in K. Consider Z’(k). We know that Z(k) is true for S, so Z’(k) must be true for S’, by the isomorphism between S and S’, and the method of construction of I’. Thus I’ shows that S’ is compatible with T.

DEFINITION 9. Let T be a theory. Let C(T) be the set of all systems which are compatible with T. We call C(T) the compatibility set of T.

DEFINITION 10. Let T = (P, L, K) and T’ = (P’, L’, K’ ) be theories. Let 2 be a function with domain P v L and range P’ u L’. If Z meets the following conditions, then Z is called a translation from T to T’:

(1) If B is an n-adic predicate in P, then Z(B) is an n-adic predicate in P’. Z must be a one-to-one mapping of P onto P’.

(2) Ifs is a sentence in L, then Z(s) is the result of replacing every predicate B in s with Z(B). Z(s) must be a sentence in L’.

(3) If k is a sentence in K, then Z(k) is a sentence in K’.

THEOREM 2. Let T and T’ be theories. Zf there is a translation Z from T to T’, then C(T) contains C(T’). That is, C(T’) is a subset of C(T).

434 PETER TURNEY

Proof. Let T and T’ be theories. Assume there is a translation Z from T to T’. Let S be an arbitrary system in C(Y). We want to show that S is in C(T). Let I’ be an interpretation which shows that S is in C(T’). We will construct an interpretation Z which shows that S is in C(T). Let B be a predicate in P. Z(B) is a predicate in P’. Z’(Z(l3)) is a relation in R’. Define Z(B) as Z’(Z(B)). Since we have just defined Z for predicates, the definition of Z for sentences is determined by the usual rules for first-order logic. Let k be a sentence in K. Consider Z(k). Since k is in K, by the definition of translation Z, Z(k) is in K’. By the definition of interpretation Z’, f’(Z(k)) must be true. By the construction of Z, Z(k) must be true. Thus Z shows that S is compatible with T.

DEFINITION 11. Let T and T’ be theories. If there is a translation Z from T to T’, then T’ is called an extension of T. By Theorem 2, C( T’) is a subset of C(T). An easy way to form an extension of T is to simply expand K.

DEFINITION 12. Let T and T’ be theories. If T is an extension of T’ and T’ is an extension of T, then T and T’ are twins. By Theorem 2, C(T) equals C( T’).2

THEOREM 3. Let T and T’ be theories. Zf C(T) contains C(T’), then there is a translation Z from T to T’.

Proof. Let T = (P, L, K) and T’ = (P’, L’, K’) be theories. Assume C(T) contains C(T’). Let S be a system in C(T’), and thus in C(T). Let Z be an interpretation which shows that S is in C(T), and let I’ be an interpretation which shows that S is in C(T’). Using Z and I’, we can define a function f which maps P one-to-one onto P’. Let B be a predicate in P. Z(B) is a relation R in S. There is a predicate B’ in P’, such that Z’(B’) is also R. We definef(B) as B’. We would like to use f to define Z. Unfortunately, f is not necessarily unique, because Z and I’ are not necessarily unique. For example, suppose there are two relations R, and R, in S, such that the set R, equals the set R,. Suppose Z maps B, to R, and B2 to R,. Then there is an alternative interpretation, in which B, is mapped to R2 and B2 is mapped to R, . To avoid this sort of ambiguity, we shall form a


theory T” = (P”, L”, K” ), which will be an extension of T’. Let P” be equal to P’. Let L” be equal to L’, K” will contain all the sentences in K’, plus some further sentences. Let B, and B, be any pair of predicates in P’. Suppose B, and B, are both n-adic. If the sentence

(Vx,)(Vx,) . . . (Vx,)[B,x,x2 . . . x, = B,x,x, . . . x,]

is in K’, then we do nothing. In this case, we call the predicates B, and B2 synonymous. If the sentence is not in K’, then we add the negation of the sentence to K”. We repeat this procedure for every possible pair of predicates, and then we add whatever sentences we require to make K” deductively closed. Note that T” is consistent, so C(T”) is not empty. Now, suppose S’ is in C(T”). T” is an extension of T’, so s’ is in C(Y), and thus in C(T). Let us redefine Z as the interpretation which shows that S’ is in C(T), and let us redefine I’ as the interpretation which shows that S’ is in C(7”). Using these new interpretations Z and Z’, we redefinef, using the same method as before. This new f is not necessarily unique, but different versions off will only disagree about synonymous predicates. This functionfdefines our translation Z from T to T’. It remains to show that Z satisfies condition (3) of the definition of a translation. We will suppose that Z does not satisfy condition (3) and attempt to derive a contradiction. Since Z does not satisfy condition (3), there is a sentence k in K, but Z(k) is not in K’. We shah define a theory T* which is an extension of T”. We form T* by adding the negation of Z(k) to T”, plus whatever is required to make T* deductively closed. T* is consistent, so C(T*) is not empty. Let S* be in C(T*). Since T* is an extension of T’, S* is in C(T’). However, we see that S* cannot be in C(T), because there is no way to interpret S* so that k is true. This is a contradiction, because C(T) contains C(T’). Thus, there is a translation Z from T to T’.

DEFINITION 13. Let T = (P, L, K) be a theory. Suppose there is a monadic predicate B in P. Let s be a sentence in L. By the definition of a theory, s must have some quantifiers. Let us define Q(s, B) as the result of qualifying all of the quantifiers in s by B. Wherever s says

(Vx) . . .

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Q<s, B) says

(Vx)[Bx 3 . . .]

Wherever s says

(3x) . . .

Q<s, @ says

(3x)[Bx & . . .]

We will ~$1 Q(s, B) the qual$cation of s by B3

DEFINITION 14. Let T = (P, L, K) and T’ = (P’, L’, K’) be theories. When the following conditions are met, T’ is called a theory fragment of T:

(1) P’ is a subset of P. L’ is the language defined by P’. (2) There is a monadic predicate B, which is in P, but is not in P’. (3) If a sentence k is in K’, then Q(k, B) is in K. (4) If B’ is any n-adic predicate in P’, then the sentence

(Vx,)(Vxz) . . . (Vx,)[B’x,x, . . . x, =J [Rx, & Bx2 . . . & BxJ]

is a sentence in K. (5) The sentence

(3x)Bx

is a sentence in K.

DEFINITION 15. Let S = (R, U) and S’ = (R’, U’) be systems. When the following conditions are met, S’ is called a system fragment of s:

(1) U’ is a subset of U.

(2) R’ is a subset of R.

Note that S’ must be a system, so R’ cannot be any arbitrary subset of R. If R’ is any arbitrary subset of R, then the structure (R’, U’) is not necessarily a system, since members of R’ might contain n-tuples


of individuals, such that some individual u is in U, but not in U’. Also, the fact that S’ is a system implies that U’ is not empty.

THEOREM 4. Let T = (P, L, K) and T’ = (P’, L’, K’) be theories. Suppose T’ is a theory fragment of T, with monadic predicate B. Suppose there is a model M = (T, S, I > of T with system S = (R, U >. Then there is a model M’ = (T’, s’, I’ > of T’ with S’ = (R’, VI), where S’ is a system fragment of S.

Proof. Let T = (P, L, K) and T’ = (P’, L’, K’ ) be theories. Suppose T’ is a theory fragment of T, with monadic predicate B. Let S = (R, 17) be a system. Suppose there is a model M = (T, S, I) of T with S. First we define S’ = (R’, 17’). Let U’ be the set of all things in U with the property B. That is, let U’ be Z(B). Condition (5) of the definition of a theory fragment ensures that U’ is not empty. Let R’ be I(P’), the subset of R to which I maps members of P’. Condition (4) of the definition of a theory fragment ensures that R’ is restricted to U’, so S’ is a system fragment of S. Now we define I’. For predicates in P’, I’ is the same as I. For sentences in L’, I’ is determined by the definition of S’ and the definition of I’ for predicates. Let k be a sentence in K’. By condition (3) of the definition of a theory fragment, Q(k, B) is a sentence in K. Since Z is an interpretation of T with S, I(Q(k, B)) is true. By the definition of S’ and I’, I’(k) is true. Thus, we see that M’ = (T’, S’, Z’) is a model of T with S’.

DEFINITION 16. Let T = (P, L, K) and T’ = (P’, L’, K’ ) be theories. Suppose T’ is a theory fragment of T, with monadic predicate B. Let S = (R, U) be a system. Suppose there is a model M = (1”, S, I) of T with S. By Theorem 4, there is a model M’ = (T’, S’, Z’) of T’ with S’, where S’ is a system fragment of S. Let us call M’ a model fragment of M.

THEOREM 5. Let T = (P, L, K) be a theory. Let T’ = (P’, L’, K’) be a theory fragment of T, with monadic predicate B. By condition (3) of the definition of theory fragment, K’ is a subset of L’, such that, if a sentence k is in K’, then Q(k, B) is in K. Suppose that, if a sentence s in L’ is not in K’, then Q(s, B) is not in K. Then every model

438 PETER TURNEY

M’ = (T’, S’, I’ > of T’ is a model fragment of some model M = (T, S, I > of T with some system S.

Proof. Let T = (P, L, K) be a theory. Let T’ = (P’, L’, K’> be a theory fragment of T, with monadic predicate B. Suppose that, if a sentence s in L’ is not in K’, then Q(s, B) is not in K. Let M’ = (T’, S’, I’) be any arbitrary model of T’ with S’ = (R’, U’). Let us assume that there is no model M = (T, S, Z), such that M’ is a model fragment of M, and attempt to arrive at a contradiction. There must be some sentence k in K which does not permit M to exist. To see this, suppose that K consists of nothing but the sentences required by conditions (3), (4), and (5) of the definition of a theory fragment. Then we could easily construct a model M. Define U as equal to U’. Construct R by adding the set u” to R’ for each n-adic predicate in P which is not in P’. This gives us S = (R, U). Construct Z by extend- ing I’ to the predicates in P which are not in P’. Map each n-adic predicate in P which is not in P’ to a set u” in R. The definition of Z for L follows from the definition of Z for P. We see that, if k is in K, then Z(k) is true. Thus M = (T, S, Z) is a model of T with S. This shows that, if there is no model M = (T, S, I), such that M’ is a model fragment of M, then there must be some sentence k in K which does not permit M to exist. What is there about M that is incompatible with k? All we require of M is that M’ be a model fragment of M. It follows that k must say something about S’. Thus, there is a sentence of the form Q(k’, B) in K which is incompatible with S’. But then k’ must be in K’, so M’ cannot be a model of T’. This is a contradiction. Thus, every model M’ = (T’, s’, Z’) of 2”’ is a model fragment of some model M = (T, S, 1) of T with some system S.

THEOREM 6. Let T = (P, L, K) be a theory. Let T’ = (P’, L’, K’ > be a theory fragment of T, with monadic predicate B. Suppose that every model M’ = CT’, s’, I’ > of T’ is a mode/ fragment of some model M = (T, S, I > of T with some system S. Then, if a sentence s in L’ is not in K’, then Q(s, B) is not in K.

Proof. Let T = (P, L, K) be a theory. Let T’ = (P’, L’, K’) be a theory fragment of T, with monadic predicate B. Suppose that every model M’ = (T’, S’, I’) of T’ is a model fragment of some model M = (T, S, Z) of T with some system S. Let us assume that there is


a sentence s in L’, such that Q(s, B) is in K, but s is not in K’. We will attempt to derive a contradiction from this assumption. Since s is not in K’, there is a model M’ = (T’, S’, Z’) in which Z’(S) is false. Suppose M = (T, S, Z) is a model, such that M’ is a model fragment of M. By the definition of a model fragment, Z is based on Z’ in such a way that, since Z’(s) is false, Z(Q(s, B)) must be false. But we have supposed that Q(s, B) is in K. This violates the definition of an interpretation, since, if Q(s, B) is in K, then Z(Q(s, B)) must be true. This is a contradiction. Thus, if a sentence s in L’ is not in K’, then Q(s, B) is not in K.

DEFINITION 17. Let T = (P, L, K) and T’ = (P’, L’, K’ ) be theories. Without loss of generality, we may assume that P and P’ are disjoint sets. If they are not disjoint, then we can replace T with a twin, such that P in the twin is disjoint from P’ in T’. When there is a theory T* = (P*, L*, K*), meeting the following conditions, T* is called the implanting of Tin T’, and T is said to be implantable in T’:

(1) P* contains P and P’. P* contains exactly two more predicates than are contained in the union of P and P’. Let us call these predicates B, and B2. Both B, and B2 are monadic predicates. The identity relation is not counted as a predicate, but as a logical operator, so none of P, P’, or P* contain the identity relation. L* is the first-order language based on P*.

(2) K* is the smallest deductively closed set which satisfies the following: (i) If k is a sentence in K, then the sentence Q(k, B,) is a

sentence in K*. (ii) If k is a sentence in K’, then the sentence Q(k, B2) is a

sentence in K*. (iii) The sentence

is a sentence in K*. (iv) The sentence

WW, x = 4x1 is a sentence in K*.

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(v) If B is any n-adic predicate in P, then the sentence

(Vx,) * * . (Vx,J[Bx, . . . x,

3 [B,x, & . . . & B,x,,]]

is a sentence in K*. (vi) If B’ is any n-adic predicate in P’, then the sentence

(Vx,) . . . (Vx,)[B’x, . . . x,

3 [B,x, & . . . & B2xJ]

is a sentence in K*. (vii) There is a one-to-one function h, mapping P onto P’. For

each n-adic predicate B in P, there is an n-adic predicate B’ in P’, such that B’ equals h(B), and the sentence

(Vx,) . . . (Vx,,)[Bx, . . . x, =I B’x, . . . x,,]

is a sentence in K*. (3) Ifs is a sentence in L but not in K, then the sentence Q(s, B,)

in L* is not in K*.

Since T* is a theory, T* must be consistent. Clearly T and T’ are theory fragments of T*.4

THEOREM 7. Let T and T’ be theories. Suppose there is an implanting T* of T in T. Then every S in C(T) is embeddable in some S’ in C( T’). Equivalently (by Theorem I ), every S in C(T) is a subsystem of some S’ in C(T’).

Proof. Let T and 7” be theories. Suppose there is an implanting T* of Tin T’. Let A4 = (T, S, Z) be any arbitrary model of T with S. By Theorem 5 and condition (3) of the definition of implanting, there is a model M* = (T*, S*, I*), such that A4 is a model fragment of M*. By Theorem 4, there is a model M’ = (T’, S’, I’), such that M’ is a model fragment of M*. Let S = (R, U) and S’ = (R’, U’). By condition (iv) of the definition of implanting, given the interpretation I*, U is a subset of U’. By condition (vii) of the definition of implanting, R is the restriction of R’ to U. Thus, S is a subsystem of S’.


THEOREM 8. Let T and T’ be theories. Suppose every S in C(T) is embeddable in some s’ in C(T’). Equivalently (by Theorem I), every S in C(T) is a subsystem of some S’ in C(T’). Then there is an implanting T* of T in T’.

Proof. Let T and T’ be theories. Suppose every S in C(T) is embeddable in some S’ in C(T’). Let S be an arbitrary system in C(T). There is a system S’ in C(T’) such that S is a subsystem of S’. Let Z be the interpretation that shows that S is in C(T). Let Z’ be the interpretation that shows that S’ is in C(Y). Let us construct T* as in the definition of implanting. We can easily construct T* to satisfy conditions (1) and (2) of the definition of implanting. It remains to show that we can satisfy condition (3), and that the resulting K* is consistent. To do this, we need to construct a model M* = (T*, S*, Z*) of T* with S* = (R*, U*>. Let U* be U’. Since S is a subsystem of S’, U is a subset of 17’. Let R* be a sequence composed of the memers of R, the members of R’, the set U, and the set U’. Z* agrees with Z for the domain P. Z* agrees with Z’ for the domain P’. I*@,) is U, and I*@,) is U’. This specifies Z* for the domain P*. The definition of I* for L* follows from the definition of Z* for P*. We see that M* is a model for T*, so K* is consistent. By Theorem 6, we can satisfy condition (3) of the definition of implantability. Thus, there is an implanting T* of T in T’.

VAN FRAASSEN’S EXAMPLE

I have some trouble with the geometric example van Fraassen gives, of two theories T, and T,, which are inconsistent with each other, yet every model of T, can be embedded in a model of T2 (1980, pp. 41-44). Van Fraassen presents six axioms:

AO: There is at least one line. Al: For any two lines, there is at most one point that lies on

both. A2: For any two points, there is exactly one line that lies on

both. A3: On every line there lie at least two points. A4: There are only finitely many points. A5: On any line there lie infinitely many points.

442 PETER TURNEY

He then defines three theories: To has axioms Al to A3, T, has axioms Al to A4, and T2 has axioms Al to A3, and AS.

Why is A0 mentioned, when it does not occur in any of the theories, T,,, T,, or T2? Also, notice that Al is an immediate consequence of A2. If two lines m and n both pass through two distinct points P and Q, then by A2, m and n are the same line.

It is difficult to tell whether van Fraassen is interested in affine planes or projective planes (Garner, 1981; Hartshorne, 1967). An afine plane consists of a set of points and a set of lines. A line is a set of points. Two lines are parallel if they are equal, or if they have no points in common. A set of points is collinear if there is a line which contains all of them. The standard axioms for an affine plane are:

Bl: Given two distinct points P and Q, there is one and only one line containing both P and Q.

B2: Given a line m and a point P, not on m, there is one and only one line n, which is parallel to m, and which passes through P.

B3: There exist three non-collinear points.

The standard axioms for a projective plane are:

Cl: Two distinct points P and Q lie on one and only one line. C2: Any two lines meet in at least one point. C3: There exist three non-collinear points. C4: Every line contains at least three points.

Note that, by C2, parallel lines in the projective plane must be equal. To does not entail either a projective plane or an affine plane. To

also does not entail B3, which is true of both projective planes and affine planes. Thus To does not even define what projective planes and affine planes have in common. The axioms of To are provable as theorems about affine planes or projective planes. However, C4 is stronger than A3, and Cl and C2 entail that any two distinct lines meet in exactly one point, which is stronger than Al. This suggests that van Fraassen intended to define affine planes. However, the “Seven Point Geometry” presented by van Fraassen (1980, p. 42) is well-known as the smallest projective plane, and it is not an affine plane, since it violates B2. The Euclidean plane is an affine plane, not


a projective plane. Van Fraassen is incorrect in claiming that the “Seven Point Geometry” can be embedded in the Euclidean plane (1980, p. 43).5

AN ALTERNATIVE EXAMPLE

Let us consider an example of two theories, which are inconsistent with each other, yet every model of one can be embedded in a model of the second. Recall the axioms for an affine plane:

Al: Given two distinct points P and Q, there is one and only one line containing both P and Q.

A2: Given a line m and a point P, not on m, there is one and only one line n, which is parallel to m, and which passes through P.

A3: There exist three non-collinear points.

These axioms might be extended in two different ways:

A4: There are only finitely many points. A5: On any line there lie infinitely many points.

Consider two theories, T = (P, L, K) and T’ = (P, L, K'). These theories share the language L, which talks about points and lines. The language L has no constants. It has the following predicates:

B,: x is a point. B2: x is a line. B,: x lies on y.

Of course, L has the identity relation, “x equals y”. We could introduce other predicates, but any other relevant predicates can be expressed using these basic predicates. For example,

“Line x is parallel to line y.”

can be expressed as

“Line x equals line y, or it is not the case that there is a point z such that z lies on x and z lies on y.”

K consists of the deductive closure of Al to A4. K' consists of the deductive closure of Al to A3, and A5. Every system S compatible

444 PETER TURNEY

with T can be embedded in some system S’ compatible with T’. How- ever, T and T’ are inconsistent, because A4 and A5 are contradictory. We wish to show that T is implantable in T’.

Our first task is to find a twin of T, such that the twin of T shares no predicates with T’. To do this, we replace the predicates

If,: x is a point. B,: x is a line. B,: x lies on y.

with the predicates

B4: x is a point*. B,: x is a line*. B6: x lies* on y.

For convenience, we will refer to this twin of T as T = (P, L, K). Thus, P is the set of predicates B4, B,, and Be; L is the language defined by P; K is the deductive closure of the sentences that result when Al to A4 are transformed by replacing B, to B, with B4 to B6. We leave T’ = (P’, L’, K’ ) the same: P’ is the set of predicates B, , B,, and B3; L’ is the language defined by P’; R is the deductive closure of Al to A3 and A5.

Now we define T* as in the definition of implantability. P* contains P and P’ and two more monadic predicates, B, and $. L* is the language defined by P*. K* is the deductive closure of the following axioms. By (v) of the definition of implantability:

D 1: All points* have property B, . D2: All lines* have property B7. D3: For all x and all y, if x lies* on y, then both x and y have

property B,.

Keeping Dl to D3 in mind, (i) of the definition of implantability gives us:

D4: Given two distinct points* P and Q, there is one and only one line* containing* both P and Q.

D5: Given a line* m and a point* P, not on m, there is one and only one line* n, which is parallel* to m, and which passes* through P.


D6: There exist three non-collinear* points*. D7: There are only finitely many points*.

By (vi):

D8: All points have property BB. D9: All lines have property &.

DlO: For all x and all y, if x lies on y, then both x and y have property $ .

Keeping D8 to DlO in mind, (ii) of the definition of implantability gives us:

Dl 1: Given two distinct points P and Q, there is one and only one line containing both P and Q.

D12: Given a line m and a point P, not on m, there is one and only one line n, which is parallel to m, and which passes through P.

D13: There exist three non-collinear points. D14: On any line there lie infinitely many points.

We already satisfy condition (iii) of the definition of implantability, as a consequence of Dl and D6. By (iv), we have:

D15: Everything with property B, has property Bs.

By (vii), we have:

D16: Every point* is a point. D 17: Every line* is a line. D18: For all x and all y, if x lies* on y, then x lies on y.

This completes the axioms for K*. K* is consistent. There is no conflict between D7 and D14.

Although every point* is a point, it does not follow that every point is a point*. Thus we may have finitely many points*, but infinitely many points.

It only remains to verify condition (3) of the definition of implantability: If s is a sentence in L but not in K, then the sentence Q(s, B,) in L* is not in K*. The concern here is that some sentence introduced by condition (ii) may have some implications expressible in L. For

446 PETER TURNEY

example, if K’ had a sentence which said

El: There are less than 100 points.

then K* would have a sentence which says the same. Since every point* is a point, it would follow that

E2: There are less than 100 points*.

This sentence would be in K*, since K* is deductively closed. Then there would be an s in L+ but not in K, namely E2, such that Q(s, B,), namely E2 again, is in K*. However, we do not have this problem, because D14 has no implications for the number of points*. Since A5 is the only place where K* diverges from K, condition (3) is satisfied.

Although T and T’ are inconsistent, T is implantable in T’. Contrary to van Fraassen, embeddability has a syntactic analogue.

EMPIRICAL ADEQUACY AND EMPIRICAL EQUIVALENCE

Of course, van Fraassen does not define embeddability merely for the sake of defining embeddability. The concept of embeddability is essen- tial for van Fraassen’s definitions of empirical adequacy and empirical equivalence. We will use the concepts introduced above to present van Fraassen’s definitions, and to give corresponding syntactic definitions.

First, we have van Fraassen’s semantic definitions: A set of experimental observations or measurement reports many be thought of as a set of systems. When a set of systems is used to represent a set of observations, we will call the set of systems a set of appearances. A scientific theory about a set of appearances is also a set of systems. When a set of systems is used to represent a scientific theory, we will call the set of systems a set of models. A scientific theory also designates subsystems of the set of models as candidates for representing appearances. These designated subsystems are called the empirical substructures of the models. A scientific theory is empirically adequate if it has some model such that all appearances are isomorphic to empirical substructures of that model (van Fraassen 1980, pp. 45, 64). Two scientific theories are empirically equivalent, if, for any model of one theory, there is a model of the other theory, such that the empirical

EMBEDDABILITY, SYNTAX, AND SEMkNTICS 447

substructure of the one model is isomorphic to the empirical substructure of the other model (van Fraassen 1980, p. 46).

I can only give the bare bones of van Fraassen’s definitions here. For more detail and explanation, the reader should consult van Fraassen’s (1980) work. The fundamental idea is that observations (appearances) are embedded in scientific theories (models).

I will now give syntactic definitions for all of the terms introduced in the preceding paragraph: A set of experimental observations or measurement reports may be thought of as a set of theories. When a set of theories is used to represent a set of observations, we will call the set of theories a set of appearances*. A scientific theory about a set of appearances* is also a set of theories. When a set of theories is used to represent a scientific theory, we will call the set of theories a set of models*. A scientific theory also designates theory fragments of the set of models* as candidates for representing appearances*. These designated theory fragments must be implantable in their corresponding models*. These designated theory fragments are called the empirical substructures* of the models*. A scientific theory is empirically adequate* if it has some model* such that all appearances* are twins of the empirical substructures* of that model*. Two scientific theories are empirically equivalent* if, for any model* of one theory, there is a model* of the other theory, such that the empirical substructure* of the one model* is a twin of the empirical substructure* of the.other model*.

We see that it is possible to syntactically define empirical adequacy and empirical equivalence. The syntactic definitions correspond exactly to the semantic definitions. Observations (appearances or appearances*) are embedded (or implanted) in scientific theories (models or models*).

CORRESPONDENCE RULES

In the positivist account, the relation between theories and observations is handled with correspondence rules. There is a language L, which contains two classes of terms, observational predicates and observational constants V,, and theoretical predicates VT. These two classes of terms define three classes of sentences in L. A sentence may

448 PETER TURNEY

contain only terms from V,, in which case it is an observational sentence. A theoretical sentence contains only terms from VT. The third class of sentences contains terms from both V, and Vr. The correspondence rules are a special subset of this third class of sentences.

The exact form of the correspondence rules was a matter of great debate, when positivism was at its height. For an excellent and thorough treatment of the positivist account of scientific theories and the debate over correspondence rules, I recommend Suppe’s The Structure of ScientiJic Theories (1974).

Correspondence rules were supposed to enable theoretical statements to be deduced from observational statements, and vice versa. They were intended to be the link between a theory and a collection of observations.

A common criticism of the positivist account is that there is, in the real world, no sharp distinction between theoretical terms and observational terms. Furthermore, it is easy to form a sentence which uses only observational terms, yet makes an assertion which cannot be observationally tested. Van Fraassen repeats these criticisms, but his main point is that the positivist account cannot give good definitions for empirical equivalence and empirical adequacy (1980, pp. 53-56).

CONCLUSIONS

I agree with van Fraassen, that the positivist account does not yield good definitions for empirical equivalence and empirical adequacy. However, I do not agree with his claim that this is because the positivist account is syntactic, while his account is semantic. I believe that the key difference between the accounts lies in how they deal with the relation between theories and observations. I believe that van Fraassen’s account is indeed an improvement on the positivist account, but this improvement has nothing to do with the distinction between syntax and semantics. I think that there is a symmetry between syntax and semantics, which makes it unlikely that one is better than the other for an account of science.

The definition of embeddability is admittedly shorter than the definition of implantability. This is because the syntactic definition is more explicit than the semantic definition. There is a trade-off


between being explicit and being brief, which may make a semantic approach more appropriate for some applications, but a syntactic approach will be preferred for other applications.

I suggest that the positivist approach should not be called the syntactic approach, because a syntactic approach based on implantability would have more in common with van Fraassen’s approach than with the positivist approach. Similarly, the semantic approach should be called by a more appropriate name, perhaps the embeddivist approach?

Van Fraassen argues that theory and observation cannot be dis- tinguished by syntactic methods. He makes the distinction by semantic methods. We see now that there is a syntactic method, which is equivalent to his semantic method. The moral is this: The relevant distinction here is not between syntax and semantics. It is between positivism and embeddivism. It is between two ways of linking theory and observation: Correspondence rules versus embedding/implanting.

ACKNOWLEDGEMENTS

Thanks to Prof. Paul Thompson for advice, instruction, and con- structive criticism. Thanks to Prof. Alasdair Urquhart for help with the logic. Thanks to the Social Sciences and Humanities Research Council of Canada for financial support. (Award 456-88-0169.) Thanks to two anonymous referees of the Journal of Philosophical Logic for valuable comments.

NOTES

’ This definition of M is somewhat unusual. Here, the interpretation I connects the predicates in P to the relations in R, and different models may have different interpretations. The usual approach in model theory is to assume that Z is tixed. That is, assume that there is a standard interpretation Z for all models (Bell and Machover, 1977, pp. 162-164). In the context of a philosophical discussion of scientific theories, this assumption seems too strong. For example, suppose we wish to assert that two different theories use different words to describe the same structure. Here, we can say, “Predicate B in theory T with interpretation Z refers to the same thing to which predicate B’ in theory T’ with interpretation I’ refers. That is, Z(B) equals Z’(R).” When Z is tixed for all models, this assertion is difficult to make. * De Bouvere (1965) defines a syntactic relationship Between theories that is similar to Being twins. For comparison, here is his definition. A theory T’ = (P’, L.‘, K’) with

450 PETER TURNEY

predicates B, and B2 is a dej?nitional extension of a theory T = (P. L, K) with predicate B, ifl there is an explicit definition of B2 from B, , say k, such that K’ is the deductive closure of K u {k}. Two theories T with predicate B, and T’ with predicate B, are synonymous iff there is a theory T* with predicates B, and B2, such that T* is a definitional extension of T, and also of T’. De Bouvere also gives an equivalent semantic definition of this concept. (I have translated de Bouvere’s notation into the notation used in the rest of this paper.) Schoenfield (1967, p. 67) calls synonymous theories weakly equivalent. 3 Rabin (1977, pp. 612-613) defines a notion similar to this idea of qualified quan- tification. For comparison, here is his definition. Let T = (P, L, K) be a theory. Let s be a sentence in L. Let d(x) be a formula with a free variable x, where the formula d(x) is composed of predicates in P. This formula d(x) plays the role of the monadic predicate B in Definition 13. The relarivization of s to d(x) is constructed in the same manner that Q(s, B) is constructed. (I have translated Rabin’s notation into the notation used in the rest of this paper.) Schoenfield (1967, pp. 61-65) also uses this device. 4 Note that the definition of implantability is purely syntactic. Interpretations and models are not mentioned at all, either implicitly or explicitly. The only reason that interpretations and models are mentioned before and after this definition is that I want to prove that implantability corresponds to embeddability. Otherwise, I could dispense entirely with any mention of semantic notions. 5 The fact that we can draw a diagram of the “Seven Point Geometry” on a Euclidean plane does not imply that we can embed it in a plane. Van Fraassen’s diagram is merely a diagram, not an embedding. In particular, the circle which joins D, E, and F does not map to a line in the plane. In fact, since affine planes contain only lines and points, the circle which joints D, E, and F does not even exist. Even van Fraassen’s axions only mention lines and points. Remember that we are dealing with a highly abstract geometry. For example, we have not introduced any metric for our space.

It is easy to verify that the Euclidean plane satisfies the axioms for an affine plane, but the “Seven Point Geometry” violates the axiom, “Given a line m and a point P, not on m, there is one and only one line n, which is parallel to m, and which passes through P.” In the ‘Seven Point Geometry”, if P is not on m, then there is no line which is parallel to m, and which passes through P.

REFERENCES

Bell, John and Mohse l&hover (1977), A Course in Mathematical Logic. New York: North-Holland.

de Bouvbe, Karel (1965) “Synonymous theories”. In The Theory of Models. Edited by J. W. Addison, Leon Henkin, and Alfred Tarski. New York: North-Holland. Pages 402423.

Garner, L. E. (1981), An Outline of Projective Geometry. New York: North-Holland. Hartshome, R. (1967), Foundations of Projective Geometry. New York: W. A. Benjamin. Rabin, Michael 0. (1977), “Decidable theories”. In Handbook of Mathematical Logic.

Edited by Jon Barwise. New York: North-Holland. Pages 595-629. Schoenfield, Joseph R. (1967), Mathematical Logic. London: Addison-Wesley. Sneed, Joseph (1971), The Logical Structure of Mathematical Physics. Dordrecht:

D. Reidel.


Stegmiiller, Wolfgang (1976), The Structure and Dynamics of Theories. New York: Springer-Verlag.

Suppe, F. (1974), The Structure of Scientijc Theories. Illinois: University of Illinois Press.

Suppes, Patrick (1967), “What is a scientific theory ?“. In Philosophy of Science Today. Edited by Sidney Morgenbesser. New York: Basic Books. Pages 55-67.

Thompson, Paul (1983), “The structure of evolutionary theory: a semantic approach”, Studies in the History and Philosophy of Science 14: 215-229.

van Fraassen, Bas C. (1970), “On the extension of Beth’s semantics of physical theories”, Philosophy of Science 37: 325-339.

van Fraassen, Bas C. (1980), The ScientiJic Image. Oxford: Clarendon Press.

Department of Philosophy, University of Toronto, Toronto, Ontario M5S lA1, Canada.

embeddability, syntax, and semantics in accounts of scientific theories

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