Monotonicity

96 CHAPTER 4. THE CLASSICAL METATHEORY OF LANGUAGE havegP r(q_{(g,P r)}(J⁰))⊆gP r(q_{(g,P r)}(I)). This reduces the second case to the first

case.

Conversely, assume that conjecture is wrong. We show that under this assumption,qg_{P r} is not upward normal.

Lemma 51 If conjecture 49 is wrong, then(q_{(g,P r)}g, P r◦q_{(g,P r)})is not upward normal.

Proof. By assumption, we have finite languagesI, J such that I ⊆ J ⊆ gP r(I), and there is noJ⁰ ⊇J such thatgP r(J⁰)⊆gP r(I). That means that for allJ⁰ ⊇J, we havegP r(q_{(g,P r)}(J⁰))6⊆gP r(q_{(g,P r)}(I)), and hence we have g_{P r}◦q_{(g,P r)}(J⁰)6⊆g_{P r}◦q_{(g,P r)}(I). Therefore, upward normality fails.

Of course, these last results do not refer to any particular properties of (g, P r), so they can be easily seen to obtain in much more generality.

4.6. PROPERTIES OF PRE-THEORIES II 97 Proof. TakeI :={a, bac};I⁰ =I∪ {dae}. This works as counterexample

for both.

The main point why P rfails to be monotonic is that it is quite restrictive:

the criteria are implicational, and in this sense they do not only refer to what has to be in the language, but also implicitly to what must not be in a certain finite language in order to allow for an analogy. The same holds for the simple P1:

Lemma 56 P1,(g, P1) are not monotonic.

Proof. Recall that we havew~ ≈^P1_I ~v ifw~ v~v andw~ ≤I~v. So just consider I:={a, bac}, whereP1(I) ={(a, bac)}; andI⁰=I∪ {xbacy}, whereP1(I⁰) = {(bac, xbacy),(a, abacy)}. Consequently, we haveh◦gP1(I) ={bⁿacⁿ:n∈N};

h◦gP1(I⁰) ={xⁿbacyⁿ :n∈N} ∪ {(xb)ⁿa(cy)ⁿ:n∈N}.

But what kind of analogical maps/pre-theories are monotonic, and how do we construct them? We will first look at analogical maps. As it turns out, there is an easy and reliable way to construct them. Given any analogical mapP, we can use apowerset construction to immediately get a monotonic extension ofP:

Definition 57 The powerset extension ℘(P) of an analogical mapP is defined by℘(P)(I) :={(~x, ~y) : (~x, ~y)∈P(J)for someJ ⊆I}.

IfP is an analogical map, then so is℘(P), and moreover:

Lemma 58 If P is an analogical map, then℘(P)is an analogical map with the same domain and range, and℘(P)is monotonic.

The proof is an easy exercise. Now we can ask: is this construction a reasonable one, is it desirable from a (meta-)linguistic point of view? This question is of course hard to answer. Of course, monotonicity is desirable in some sense; as a matter of fact it solves one of the main problems we stated in the beginning: in view of the fact we can only observe fragments of the observable language, we are unsure about our “language”, but with a monotonic analogical map, we can at least give a partial answer, in that we know: a statement of the formw~ is part of our ”language”will never be falsified by new evidence (still we remain unsure about statements of the form: w~ is not part of our

”language”, see the discussion on negative evidence).

But whereas before, our problem was that an analogical map such as P ris probably rather too restrictive, we might now think that we are too liberal. In particular, there is no evidence which might make a certain analogy illegitimate, which is to say for the linguist that there is no evidence which can make a certain projection implausible. I am not too sure whether this is desirable. But note that this is a fault which is intrinsic toany monotonic analogical map.

We will not settle the question on whether monotonicity is necessary or even desirable for pre-theories; this will remain, as many issues, a matter of taste. However, what we can show is that, given that the problem of monotonic pre-theories is that they are rather too liberal, the powerset construction is an optimal solution in a strong sense:

Definition 59 Given two analogical mapsP, P⁰, say that P is smaller than P⁰ (in symbolsP ≤P⁰), if for all finite languagesI,P(I)⊆P⁰(I).

98 CHAPTER 4. THE CLASSICAL METATHEORY OF LANGUAGE Theorem 60 Given an analogical map P, ℘(P) is the smallest monotonic analogical map which is larger thanP.

Proof. Take an analogical mapQwhich is larger thanP and monotonic, and an arbitrary finite languageI. As Q is larger than P, we know that for everyJ ⊆I,Q(J)⊇P(J). Furthermore, asQis monotonic, we know that for everyJ⊆I,Q(J)⊆Q(I). Therefore, assumea∈℘(P)(I). Thena∈P(J) for someJ ⊆I. Therefore,a∈Q(J); and by monotonicity,a∈Q(I). Therefore,

for any finite languageI,Q(I)⊇℘(P)(I).

So the powerset construction is the smallest monotonic extension for any given analogical map. So if one thinks that a given analogical map is linguistically justified, and one thinks that monotonicity is necessary/desirable, then one just has to use the powerset construction. Given two analogical maps P, P⁰, we denoteextensional equality byP≡P⁰, by which we mean that for all finite I, we haveP(I) =P⁰(I).

Corollary 61 Given a monotonic analogical mapP, we have℘(P)≡P. Actually, this follows immediately from the last lemma; for the sake of exposition, we will give another proof.

Proof. Assume thatP is monotonic. Then for any finite languageI,J ⊆I, we haveP(J)⊆P(I). Therefore, we haveS

J⊆IP(J)⊆P(I). Conversely, as

I⊆I, equality follows.

Now that we have seen that there are very satisfying solutions for providing monotonic analogical maps, we will see whether this can be transferred to pre-theories. Obviously, for any pre-theory (f, P), we can construct a monotonic pre-theory (f, ℘(P)). So for the projection we obtain the following equality:

f℘(P)(I) =f℘(P)(I)(I) =f^S

J⊆IP(J)(I) (4.22)

But it turns out thatf_℘(P)isnot the smallest projection which is larger than fP and monotonic. We can show that it is possible to extendfP to a monotonic map using a powerset construction in at least one smaller way:

℘(f)P(I) := [

J⊆I

fP(J) (4.23)

This actually gives us an implicit definition of a pre-theory (℘(f), P). It is easy to see that this pre-theory is monotonic. We can also easily show that:

Lemma 62 1. For all finite languagesI, pre-theoriesP,S

J⊆IfP(J)⊆f_℘(P)(I)(I).

Furthermore,

2. there exist finite languages I and pre-theories(f, P)such that S

J⊆IfP(J)( f℘(P)(I)(I).

Proof. 1. ⊆: Assume that for some J ⊆I,fP(J) =L. AsP(J)⊆℘(P)(I), and J ⊆ I, we have L ⊆ f_℘(P)(I)(I). As this holds for all J ⊆ I, the claim follows.

2. (: Take (g, P r) andI1={axbyc, ax1xx2byc, axby1yy2c}. Then we have S

J⊆I1gP r(J) = {a(x1)ⁿx(x2)ⁿbyc : n ∈ N0} ∪ {axb(y1)ⁿy(y2)ⁿc : n ∈ N0}, whereasg℘(P r)(I₁)(I1) = {a(x1)ⁿx(x2)ⁿb(y1)^my(y2)^mc : n, m ∈ N0}, which is

clearly a superset.

4.6. PROPERTIES OF PRE-THEORIES II 99 So which definition is preferable? As the problem with monotonicity is that it is pre-theories become rather too permissive on occasions, we would opt for the smaller. This is confirmed by the following result:

Theorem 63 Given a pre-theory(f, P), projectionfP, then℘(f)P is the smallest projection which is 1. monotonic and 2. larger than fP.

The proof is standard set-theoretic and almost identical to the one of theorem 59, and therefore omitted. So (℘(f), P) should be preferable to (f, ℘(P)). I have the impression though that (f, ℘(P)) is much more elegant in its definition and intuitive in its application.

We should at this point remark that there is an important property regarding the interaction of upward normality and monotonicity:

Lemma 64 Let (f, P) be a pre-theory which is upward normal and (weakly) monotonic. Then we have for all finite languages I, J, ifI⊆J ⊆fP(I), then fP(I) =fP(J).

Proof. Assume I⊆J ⊆fP(I), and fP(I)6=fP(J). By (weak) monotonicity, we have fP(I)(fP(J). But also, for allJ⁰ ⊇J, we havefP(J)⊆fP(J⁰). This

contradicts upward normality.

So we get an immediate corollary:

Corollary 65 If I⊆J ⊆g2P2(I), theng2P2(I) =g2P2(J).

So pre-theories being upward normal and monotonous are very predictable.