peirce on probability and induction

Peirce on Probability and InductionAuthor(s): G. H. MerrillSource: Transactions of the Charles S. Peirce Society, Vol. 11, No. 2 (Spring, 1975), pp. 90-109Published by: Indiana University PressStable URL: http://www.jstor.org/stable/40319731 .

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Peirce on Probability and Induction

G. H. Merrill

My purposes in this paper are three in number: (1) to show as explicitly as I can the properties of so-called

probable arguments allowed by Peirce; (2) to demonstrate the similarities and differences among

these forms of argument; (3) to explicate Peirce's use of such expressions as

"probably," "probable," "probability," etc. That these goals are closely related is obvious and my attainment of each will occur in the context of the attainment of the others.

I choose to concentrate for the most part on three papers of Peirce which were published between 1878 and 1883, and which were intended as essays to be included in Search for a Method (1893). These papers are "Deduction, Induction, and Hypothesis," "The Probability of Induc- tion," and VA Theory of Probable Inference." My method in making very precise the positions and claims of Peirce will be to use wherever possible and fruitful the machinery of the modern theory of sets. This will be used in formulating precise symbolic definitions of concepts which are often vague and confused when found in Peirce's writings. The symbolization will have (at least) two desirable effects: it will make very clear the form of the argument under consideration, and it will force an explicit statement of principles which might otherwise remain hidden beneath the sentences of ordinary language. I shall not in fact make use of a great deal of set theoretic notation and for those who are unfamiliar with that favorite language of logicians I now provide a short and (I hope) intuitively satisfactory dictionary. The following standard symbols shall be employed: Y is the sign of membership, and 'X«Y' is read 'X is a member of Y\ ' <=i is the sign of inclusion, and 'XsY' is read 'X is included in Y' and this is true just in case every member of X is a member of Y. '[x:<£]' is a term and is read 'the set of all x's such that <¿>' or, loosely, 'the set of all things of which <f> is true', or 'the set of <f>Y. ̂ is the sign of equipollence, and 'X^Y' may



Peirce on Probability and Induction 91

be read 'X is equipollent to Y' or 'X has the same number of members as Y\ '[X]: (read: 'singleton X') denotes the set whose only member is X, and 'O' denotes both the empty set and the natural number zero

(the two entities are in fact identical in many systems). I use V to

designate the set of natural numbers (including O). 'X^Y' (read: 'the intersection of X and Yf) is that set whose members are common to X and Y. Aside from the set theoretic notation just introduced I shall feel free to use well-known symbols of the predicate calculus (with identity and description) and elementary arithmetic. With the aid of such symbols I shall introduce certain special symbols by definition. In each such case a reading of the new symbol in English will be

provided.

I. Forms of Probable Inference

In 2.627 Peirce gives an example of each simple form of probable inference.1 These are:

For example, let us begin with this probable deduction in Barbara:

Rule. - Most of the beans in this bag are white, Case. - This handful of beans are from this bag; Hence Result. - Probably, most of this handful of beans

are white.

Now, deny the result, but accept the rule:

Denial of Result. - Few beans of this handful are white, Rule. - Most beans in this bag are white; Hence Denial of Case. - Probably, these beans were taken

from another bag.

This is an hypothetical inference. Next, deny the result, but

accept the case:

Denial of Result. - Few beans of this handful are white, Case. - These beans came from this bag. Hence Denial of Rule. - Probably, few beans in the bag

are white.

This is an induction.



92 G. H. MERRILL

Now a problem arises. The problem is how to interpret the occurrences of 'Probably' in these examples, for Peirce says

We may, therefore, define the probability of a mode of argument as the proportion of cases in which it carries truth with it. (2.650)

But there remains an important point to be cleared up. According to what has been said, the idea of probability essentially belongs to a kind of inference which is repeated indefinitely. An individual inference must be either true or false, and can show no effect of probability; and, therefore, in reference to a single case considered in itself, probability can have no meaning. (2.652)

We have found that every argument derives its force from the general truth of the class of inferences to which it belongs; and that probability is the proportion of arguments carrying truth with them among those of any genus. (2.669)

According to Peirce a form of argument is probable, and not a conclusion or statement. Thus it seems that 'Probably' does not really belong in the above examples at all. Let us for the moment, and temporarily without justification, simply ignore the occurrences of 'Probably' in these examples and proceed to a symbolization.

With the help of a single definition at this point we may symbolize each of these forms of argument and observe the similarity of their form. We therefore define,

Def 1 %(X,Y) - In [p^XnY & q^X & n = p/q] p,qcw

Read: 'The proportion of X's that are Y's\ %(X,Y) is simply the number of X's that are also Y's divided by the total number of X's. The forms of the above examples become

PD 1. %(X,Y)i>k Hyp l.^%(Z,Y)^k Ini 1. ~%(Z,Y)» k 2. ZaX 2. %(X,Y)^k 2. Z^X 3. %(Z,Y)^k 3.^Z^X 3.~%(X,Y)»k

where k is some number (presumably a fraction greater than l/2 but ̂ess than or equal to 1) which we think of as demarcating the line between




"most of the X's" and "only some of the X's". One might easily wonder

why I have interpreted sentences of the form "This handful òf beans are from this bag" in terms of inclusion instead of within a first order

predicate calculus as would seem more natural. I have done so because these arguments are supposed to be analogous in form to Aristotelean

syllogisms (and these are phrased in terms of an inclusion-like relation), and because Peirce indicates that this is how he intends them to be

interpreted. He says this explicitly in 2.710 and gives reasons for his choice. Concerning the above symbolizations we make two observations: that Ind may be reformulated simply by deleting the negation signs, and that if the relation between Z and X is identity, then the three forms are all valid deductive arguments by virtue of Leibniz's Law and since 'X* and 'V are free variables in '%(X,Y)\

I shall now pause in my symbolization of probable argument forms to consider the question of what has happened to the occurrences of

'Probably' in the above forms. Peirce often seems to apply the concept of probability to conclusions of arguments or to facts; for example,

(4) The conclusions of the two modes of inference likewise differ. One is necessary; the other only probable. (2.696)

The general problem of probabilities is, from a given state of

facts, to determine the numerical probability of a possible fact. (2.647)

But elsewhere he says unequivocally that probability is a feature of

forms of inference - 2.649, 2.650, 2.652, 2.669. This apparent inconsistency of usage is mitigated in 2.697:

So long as there are exceptions to the rule that all men

wounded in the liver die, it does not necessarily follow that

because a given man is wounded in the liver he cannot recover.

Still, we know that if we were to reason in that way, we should

be following a mode of inference which would only lead us

wrong, in the long run, once in fifty times; and this is what

we mean when we say that the probability is one out of fifty that the man will recover. To say, then, that a proposition has the probability p means that to infer it to be true would

be to follow an argument such as would carry truth with it in

the ratio of frequency p.



94 G. H. MERRILL

This passage then allows for certain derivative senses of 'probability' and its cognates. It is in one of these senses that 'Probably' is used in the examples of 2.627. What this shows is that we must be very careful to distinguish those occurrences of 'probable' (and its cognates) in which the expression has its technical sense from those - as in 2.627, 2.647, 2.696 - in which it has merely a colloquial or derivative sense. Peirce himself is not always careful to observe this distinction, and his failure to do so leads to certain problems in interpreting the forms of

arguments he offers as valid probable inferences. By symbolizing these

argument forms we will see just which uses of 'probable' are the non- technical ones.

Clearly the use of 'probably' in the examples of 2.627 is a very loose use of the term. In fact I shall argue that in these examples and in similar examples and statements of valid inference forms, the occurrences of 'probable' and its cognates are formally superfluous.2 It has already been shown that Peirce held strongly to the position that probability is a feature of an argument form (Cf., 2.650, 2.652, 2.669, 2.697). In "Notes on Ampliative Reasoning" he says,

Every argument or inference professes to conform to a

general method or type of reasoning which method, it is held, has one kind of virtue or another in producing truth. In order to be valid the argument or inference must really pursue the method it professes to pursue, and furthermore, that method must have the kind of truth -producing virtue which it is

supposed to have. For example, an induction may conform to the formula of induction; but it may be conceived, and often is conceived, that induction lends a probability to its conclusion. Now that is not the way in which induction leads to the truth. It lends no definite probability to its conclusion. It is nonsense to talk of the probability of a law, as if we could pick uni- verses out of a grab-bag and find in what proportion of the law held good. Therefore, such an induction is not valid; for it does not do what it professes to do, namely, to make its

conclusion probable. But yet if it had only professed to do what induction does (namely, to commence a proceeding which must in the long run approximate to the truth), which is

infinitely more to the purpose than what it professes, it would

have been valid. (2.780)



Pence on Probability and Induction 95

The arguments of 2.627 are then examples of probable inferences and it is in this sense that their conclusions are probable or that we may say that their conclusions are probably true. Suppose, though, that the occurrence of 'probably' in the conclusion of the first example is not

superfluous - i.e. suppose it to have some formal significance. Now Peirce would say that the argument

I. (a) Rule: Most of the beans in this bag are white.

(b) Case: This handful of beans are from this bag. Hence (c) Result: Most of this handful of beans are white.

is a valid probable deduction since reasoning of this form would for the most part (i.e. to some degree of probability greater than l/2) "carry truth" with it. This is to say that the conclusion is probable in the strict sense. Now if 'probably' does - as assumed - have a formal

significance, it surely must be the case that

I. (d) Probably, most of this handful of beans are white.

means that the sentence I.(c) is probable in the strict sense (i.e., that

I.(c) is the conclusion of a valid probable inference). But I establishes this very result. That is, from I. (a) and I.(b) we infer a probable conclusion I.(c). But I.(d) says of this probable conclusion that it is

probable, and this assertion (I.(d)) is not - in the above context -

probable, but necessarily true. Thus if I is a valid probable deduction

(as Peirce would claim) and 'Probably' does have formal significance in the example of 2.627, then this example is not one of probable inference but of (explicative) syllogistic. Peirce himself points out this

result for a similar example in a footnote to 2.720. Thus Peirce's use

of 'probable' and its cognates within purported probable inferences is

merely a rhetorical device and is formally superfluous. In "A Theory of Probable Inference" eight valid forms of probable

inference are given. Certain similarities are evident and others become

obvious upon symbolization of the forms. Peirce begins with the follow-

ing example of "the simplest kind of probable inference":

About two per cent of persons wounded in the liver recover,

This man has been wounded in the liver;

Therefore, there are two chances out of a hundred that he

will recover.



96 G. H. MERRILL

In explaining the difference between the form of this argument and that of syllogism Peirce says something which is very confusing to the reader since it is, according to Peirce's own principles, apparently false. He says,

. . . the probable argument may approximate indefinitely to demonstration as the ratio named in the first premiss approaches to unity or to zero. (2.694)

Now how are we to interpret this? He seems to be saying that if we

replace 'two per cent' with 'ninety per cent' in the above example we somehow get an argument that is more valid. But he must not be

interpreted in this way. Validity is a feature of the form of an argument, and by tampering with the ratio named in the example we in no way change the form. What Peirce must be taken to mean here becomes clear when we examine his formalized version of this argument form. This is,

Form II

Simple Probable Deduction

The proportion p of the M's are P's; S is an M; It follows, with probability p, that S is a P.

Again, 'It follows, with probability p, that' is logically superfluous here, just as in

Every A is a B C is an A Therefore, C is a B

'Therefore' is not part of the conclusion. The conclusion in Form II is not 'It follows, with probability p, that S is a P', but rather 'S is a P\ Now taking this as the conclusion we see immediately that Form II will

"carry truth" to its conclusion in a greater ratio if p is increased in the

premiss. And this is what Peirce means when he says that the argument may approximate to demonstration as the ratio named in the first

premise approaches unity or zero. The difference between Form II and

syllogism is not the form of what is inferred but the degree of trust-




worthiness of the conclusion given the truth of the premises. Peirce makes this point in 2.649 and 2.696.

We symbolize Form II as

SPD %(M,P) = p ScM

Hence, S c P

and we note that SPD has probability p We must also add the restriction (which Peirce suppresses) that S is drawn at random from among the M's.

Complex Probable Deduction is the next argument form considered

by Peirce. The form he gives is.

Form III Complex Probable Deduction

Among all sets of n M's, the proportion q consist each of m P's and of n-m not-P's,

S\ S", S"\ etc.; form a set of n objects drawn at random from among the M's;

Hence, the probability is q that among S', S", S"', etc. there are m P's and n-m not-P's. (2.698)

I now introduce two additional definitions which allow us to symbolize Complex Probable Deduction:

Def 2 [X;n] = [Y: YsX&n »Y]

(read: 'the set of all n-membered sets of XV)

[X;n] consists of just those sets which have exactly n members and each of whose (n) members is an X.

Def 3 [X|m,n] = [Y: Y^m & YaX^ii]

(read: 'the set of m-membered sets each of which consists of n X's and m-n non-X's)

Thus every set in [X|m,n] has n members which are X's and m-n members which are not.

We now symbolize Complex Probable Deduction as,



98 G. H. MERRILL

CPD %([M;n], [P|n,m]) = q S c [M;n]

Hence, S € [P|n,m]

We note that CPD has probability q. We must add the restriction to CPD that the members of S (S in our formulation corresponds to the set whose members are S', S", S'", etc. in Peirce's Form III) constitute a random sample of M, and we shall examine random samples at a later point.

Now our symbolization begins to prove its worth. For not only is the symbolic version of CPD not very complex, but it has the same form as SPD. Indeed CPD results from SPD by substitution of the expressions '[M;n]', '[P|n,m]', and 'q' for 'M1, 'F, and y respectively. Thus CPD is an unnecessary addition to the basic valid forms of probable inference since it is a substitution instance of SPD. Peirce realized that these two forms were very similar. He in fact said that the same

principle (that of probable deduction) applied to both (2.698). Yet it is quite clear that he still thought of them as distinct forms and the reason he thought of them in this way seems to be that in SPD S is

thought of as being a single individual, while in CPD a set of such individuals is involved. But even in his Form III the emphasis is on individuals. The single occurrence of 'set' is easily eliminated by re-

placing the second premise with "S\ S", S'", etc.; are n objects drawn at random from among the M's." Thus it might easily seem that we need two distinct forms: one to deal with the case of a single individual and one to deal with the case of a number of individuals. This is just what Peirce's Form II and Form III do. Yet a thorough -going formalization shows that there is no difference in form, and since in specifying valid inference forms we are not interested in whether an inference is about apples or numbers or colors or sets we may easily eliminate CPD from the basic forms of inference.

Statistical Deduction takes the form,

Form IV Statistical Deduction

The proportion r of the M's are P's, S', S", S'", etc. are a numerous set taken at random from

among the M's;




Hence, probably and approximately, the proportion r of the S's are P's. (2.700)

Or in our symbols,

SD %(M,P) = r

Hence, %(S,P) = r

with the proviso that S is a "numerous" random sample of M and that the form is probable and approximate. To say that SD is probable and

approximate is to say that it is probable (to a certain degree) that the ratio of S's that are P's is somewhere in a certain interval. Let us say that this interval is from r - k to r-f k. This is in fact a simplification since the interval need not be symmetric about r. As k increases the

probability that the ratio in question is in the interval increases also. Peirce gives a method to compute the probability that the ratio falls in an interval given the upper and lower bounds of that interval but this need not concern us here. SD would conform more closely to the earlier forms of probable inference if it were restated as,

SD* %(M,P) = r

Hence, %(S,P) = r±k

where k is as above. We need not say in this case that SD* is

approximate, but only that is has probability q where q is computed according to Peirce's method and is dependent on k.

Peirce introduces the form of induction by saying,

The principle of statistical deduction is that these two proportions - namely, that of the P's among the M's, and that of the P's among the S's - are probably and approximately equal. If, then, this principle justifies our inferring the value of the second proportion from the known value of the first, it

equally justifies our inferring the value of the first from that of the second, if the first is unknown but the second has been

observed. We thus obtain the following form of inference:



100 G. H. MERRILL

Form V Induction

S', S", S'", etc. form a numerous set taken at random from among the M's,

S', S", S'", etc. are found to be - the proportion p of them - P's;

Hence, probably and approximately the same proportion, p, of the M's are P's. (2.702)

Our symbolization is,

IND SsM or alternatively, IND* S^M %(S,P) =P %(S,P) =P

Hence, %(M,P) = p Hence, %(M,P) = P ± k

where again we have the "probably and approximately" proviso, and k is as before.

Peirce observes that both SD and IND "depend upon the same principle of equality of ratios, so that their validity is the same." (2.703) The principle alluded to is (informally)

If a whole contains a certian ratio of, say, P's then a randomly selected part of that whole will contain the same ratio of P's (approximately). And conversely.

It is easily seen that there are really two principles here - one being the converse of the other - but I do not wish at this point to critisize this oversight. Peirce goes on to make the important observation,

Yet the nature of the probability in the two cases is very different. In the statistical deduction, we know that among the whole body of M's the proportion of P's is p; we say, then, that the S's being random drawings of M's are probably P's in about the same proportion - and though this may happen not to be so, yet at any rate, on continuing the drawing sufficiently, our prediction of the ratio will be vindicated at last. On the other hand, in induction we say that the proportion p of the sample being P's, probably there is about the same proportion in the whole lot, or at least, if this happens




not to be so, then on continuing the drawings the inference will be, not vindicated as in the other case, but modified so as to become true. (2.703)

This is to say that the major distinction between statistical deduction and induction is the effect that new evidence has on the conclusion. In statistical deduction new evidence does not affect the conclusion at all for we are quite sure (from our knowledge concerning the whole) that in the long run our prediction will be correct. With induction, on the other hand, our knowledge of the whole is gained piecemeal through an examination of its parts. Thus new evidence in this context forces us to revise our prediction. It is with precisely this epistemological point in mind that Peirce makes the distinction between the validity and the strength of an argument, and we shall closely examine this distinction at a later point.

Peirce next introduces certain forms which he considers to be modifi- cations of earlier forms. These appear to be considerably more compli- cated inferences, and depend upon the notion of r-likeness:

We often speak of one thing being very much like another, and thus apply a vague quantity to resemblance. Even if

qualities are not subject to exact numeration, we may conceive them to be approximately measurable. We may then measure resemblance by a scale of numbers from zero up to unity. To

say that S has a 1 -likeness to a P will mean that it has every character of a P, and consequently is a P. To say that it has a 0-likeness will imply total dissimilarity. We shall then be

able to reason as follows:

Form II (bis) Simple probable deduction in depth

Every M has the simple mark P, The S's have an r-likeness to the M's; Hence, the probability is r, that every S is P. (2.704)

Peirce has not made the concept of r-likeness very precise here and in

fact a truly precise account would have to take place within a formal

language and the predicate '. . . has an r-likeness to ' would be

relativized to that language. Such a task is beyond the scope of this



102 G. H. MERRILL

paper, but with the aid of two additional definitions we can give the notion of r-likeness a high degree of precision.

Def4 P=[K: PsK &V (K'S. K & K* ̂ 0=> 3 pcK')] K' p€P

(read: 'the set of marks of (a) P') The set of marks (or properties, or features, etc.) of a P (or, alternatively, of a member of the set P) is the set of all those sets which contain (as members) every member of P and which do not contain any "superfluous" members. For example, the set of marks of a red rose is the set whose members are: the set of flowers, the set of plants, the set of red things, the set of things with thorns, etc. But since no roses are cobblestones, the set whose members are all either red things or cobblestones is not in the set of marks of a red rose. This is obviously a purely extensional interpretation of "mark", but it is in keeping with the Aristotelian character of the inferences.

We may now define 'r-likeness' as follows:

Def 5 S has an r-likeness to P 2 %(p£) = r (alternatively: an S has an r-likeness to a P)

Thus S has an r-likeness to P just in case the proportion of marks of P which are also marks of S is equal to r. If all of the marks of P are marks of S, then r is 1; and if no marks of P are marks of S, then r is 0. Notice that the relation of having an r-likeness to is neither

symmetric nor transitive, though if r = 1 it is reflextive. Even if

%(P,3) - 1 this does not mean that P and S are identical, for in addition to having all the marks of P, S may have other marks. P and S are identical just in case %(P,S) - 1 and %(3J) = 1. Form II {bis) now becomes, ^

SPDD PcM %(M,S)=r

Hence, P c S

Here 'P c M' may be read 'P is a mark of (every) M\ Again our formalization helps us to see that this form of inference is really not a new one. It is the very same form as SPD - we merely substitute 'M', '"S\ 'F and Y for 'M\ 'P\ 'S' and y respectively and SPD yields SPDD. While Peirce saw a similarity between these forms it is unlikely that he thought it to be so great.




Corresponding to Statistical Deduction we next have,

Form IV {bis) Statistical deduction in depth

Every M has, for example, the numerous marks P', P", P"\ etc.,

S has an r-likeness to the M's; Hence, probably and approximately, S has the proportion r

of the marks P\ P", P", etc. (2.705)

which we symbolize as,

SDD PSM or alternatively, SDD* PSM %(ti$) =r %(M,3) =r

Hence, %(P,3) = r Hence, %(P|5) = r ± k

Where SDD carries the "probably and approximately" proviso, and k is as before. Again, this is seen to be a substitution instance of an earlier form - SD (or SD*).

Corresponding to induction is,

Form V (bis) Hypothesis

M has, for example, the numerous marks P\ P", P"\ etc. S has the proportion r of the Marks P\ P", P"\ etc.; Hence, probably and approximately, S has an r-likeness

to M. (2.706)

which appears in our symbols as,

HYP P£M or alternatively, HYP* PSLM %(P;3)=r %(P,S) =r

Hence, %(El,S) = r Hence, %(M,S) = r ± k

where k is as before. This is an instance of IND (or 1ND*). We have so far shown that the various forms of probable inference

which Peirce sets forth in "A Theory of Probable Inference" reduce to three: SPD, SD, and IND. However, a very close relation between SPD and SD may be observed if we reformulate SPD as,



104 G. H. MERRILL

SPD' %(M,P) = p [S]SM

Hence, [S]SP

Now the two premises of SD andSPD are of the same form and only the conclusions differ in form. We may express the fact that S is P in three different ways:

(a)S«P (b) [S]SP (c) %([S],P) = 1

As a result of Defl we have the following interesting theorem:

Thm %([X],Y) = r & r>0 z) %([X],Y) = 1

This theorem says that if any proportion (greater than zero) of a set with only one member is included in another set then the whole singleton set is so included. It is a direct consequence of this theorem that SPD is an instance of SD. To see this we need only examine the form,

SPD'' %(M,P) = p [S]ftM

Hence, %([S],P) = p

It might at first be thought that this results in an inconsistency, for from the sentences '0 <p <l\ *9f(M,P) = p', '[S]sM' we may infer ^([S]>p) = P and p<1' But from '0<p<r and the above theorem together with this conclusion we may infer %([S],P) = 1', and hence 'p = 1' which contradicts 'p <1\ But we must remember that since SPD" is an instance of SD it is subject to the "probably and approximately" restriction, and it is the "approximately" in this that saves us from the contradiction. This is evident in considering the form,

SPD"* %(M,P) = p [S]SM

Hence, %([S],P) = P±k

which is obviously an instance of SD*. And now we have shown that in "A Theory of Probable Inference" only two basic forms of inference are offered. The diagram below briefly illustrates the relations among the various forms which Peirce sets forth. Each form below is an instance of the one immediately above it.



Pence on Probability and Induction 105

SD (SD*) IND (IND*) / \ I

SPD (SPD*) SDD (SDD*) HYP (HYP*) /(SPD"*)\

CPD* SPDD

II. Hypothesis and Induction

Before going on to examine Peirce's distinction between the forms of

Hypothesis and Induction we must first briefly examine some notions which have not yet been made suffciently clear, viz. those of randomness, validity and strength.

Peirce has little to say about randomness; for example,

... the sample should be drawn at random and inde-

pendently from the whole lot sampled. That is to say, the

sample must be taken according to a precept or method which, being applied over and over again indefinitely, would in the

long run result in the drawing of any one set of instances as often as any other set of the same number. (2.726)

A random sample is one taken according to a method that

would, in the long run, draw any one object as often as

any other. (2.731)

This is, as far as it goes, self-explanatory and unfortunately quite imprecise, but it is only in recent years that the notion of randomness has been given a high degree of precision.5 It is clear, however, that Peirce saw - though he seems to have never stated this explicitly -

that whether or not a sample could be considered random depended upon the epistemic state of the sampler. That Peirce realized this and

thereby anticipated a modern interpretation of randomness is particularly evident in 2.677, 2.696, 2.727, and 2.735.

The distinction between validity and strength is essential to the logic of probable inference. An argument is valid or not solely in virtue of its form. It is the form alone, and not the content or truth of the

premises, which determines the validity of an argument. Thus one tests the validity of a probable inference by seeing if it conforms to one of the several forms examined above.



106 G. H. MERRILL

Strength, however, is a different matter:

Validity must not be confounded with strength. For an argument may be perfectly valid and yet excessively weak. I wish to know whether a given coin is so accurately made that it will turn up heads and tails in approximately equal proportions. I therefore pitch it five times and note the results, say three heads and two tails; and from this I conclude that the coin is approximately correct in its form. Now this is a valid induction; but it is contemptibly weak ... It is doubtful whether the idea of strength can be made less vague. But we may say that an induction from more instances is, other things being equal, stronger than an induction from fewer instances. Of

probable deductions the more probable conclusion is the

stronger. (2.780)

Thus, for example, an argument conforming to the schema SPD gains in strength if p is increased, but its validity is of course unchanged. An argument of form CPD will be stronger if a larger sample is taken, and this is true also of SD and IND. Simply stated then, validity is

dependent upon form and strength is dependent upon either the amount of evidence gathered or a certain ratio (intuitively, the probability of the argument) embodied in the evidence. Hence validity is (as it should be) a syntactic property and strength (much like soundness) is a semantic one.

Peirce takes great pains to establish a distinction between what he calls hypothesis and induction. The basic distinction is this:

In induction, the instances drawn at random are numerable

things; in hypothesis they are characters, which are not

capable of strict enumeration but have to be otherwise estimated. (2.709)

But Peirce was quite aware that these two varieties of inference, at least as they were set forth in "A Theory of Probable Inference" could not be distinguished by an appeal to their logical forms, for indeed the forms are identical. Thus he observes,

If I am permitted the extended sense which I have given to the word "induction," this argument [Form V (bis)] is




simply an induction respecting qualities instead of respecting things. (2.706)

We have already seen that the form of Hypothesis (HYP) is simply an instance of the form of Induction (IND) and so no distinction

may be made on logical (i.e., formal) grounds. Nonetheless Peirce

attempts a distinction, and it is an interesting one. Peirce seemed to desire - even in the light of the above comments -

to claim that Hypothesis and Induction differed formally. In fact, notice that on p. 92 above the forms Hyp and Ind do differ, and notice further that Hyp does not correspond to HYP. Peirce has started out in

"Deduction, Induction, and Hypothesis" to explore three distinct forms of probable inference. In "A Theory of Probable Inference" he appears to have believed that he was exploring the same three forms, but

throughout Peirce's writings on such matters there is a confusion between Hypothesis and Abduction which differ in form. I do not wish to deal with the various problems concerning abduction here, but we

may at least note that the form of abduction is,

ABD %(S,P) = r

%(M,P) =r Hence, S^M

Some of the problem in "A Theory of Probable Inference" can be attributed to Peirce's attempt to force some of the properties of abduction into the form of Hypothesis. This conclusion helps to explain some of Peirce's claims such as,

Deduction proceeds from Rule and Case to Result; it is the formula of Volition. Induction proceeds from Case and Result to Rule; it is the formula of the formation of a habit or general conception - a process which, psychologically as well as logically, depends on the repetition of instances or sensations. Hypothesis proceeds from Rule and Result to

Case; it is the formula of the acquirement of secondary sensa- tion - a process by which a confused concentration of predi- cates is brought into order under a synthesizing predicate. (2.712)

What is said here concerning Hypothesis is not true of the form HYP, but it is true of the form ABD. Peirce later saw this confusion in his



108 G. H. MERRILL

writings (Cf., 2.102) and withdrew certain claims concerning hypothetic inferences which he made in "A Theory of Probable Inference." He still felt, however, that there was an important difference between what he called hypothetic inference and inductive inference. The difference

may be described as a metaphysical or ontological one or - more

precisely - as a semantical one. Peirce thought of induction as being "about" individuals and the classes to which they belong. But hypothesis is concerned with individuals and their "marks". That is, hypothesis is inference involving not merely classes, but properties or

qualities. Thus Peirce says, This kind of argument, however, as it actually occurs, differs

very much from induction, owing to the impossibility of

simply counting qualities as individual things are counted. Characters have to be weighed rather than counted. (2.706)

I find his comment that characters must be weighed rather than counted to be totally uninformative, but it seems that his main point is that it is very hard to say just which or how many qualities a given object has. While this is true, it seems to be a point to be made against all of his forms of inference and not only against hypothesis.

I think the clue to this attempted distinction is in the change from locutions such as,

S. M is (a) P. to the locution,

S\ M has the mark P. Now S is easily interpreted in a purely extensional way in terms of

membership or inclusion. But the relation of having may give us pause. The switch is from an Aristotelean locution to a Platonic one. Indi- viduals are in classes (or wholes), but they have properties or participate in Forms. Thus in Form II (bis)y Form IV (bis), and Form V (bis) I believe Peirce was thinking in terms of properties (in a non-extensional sense) and hence not of classes as he had previously. In any event, no distinction may be made between Hypothesis and Induction as forms of inference.

III. Against Peirce's Interpretation

It is apparent that Peirce's approach to probability and his careful consideration of the various valid forms of probable inference has a

great deal to recommend it. But the interpretation of probability is




unfortunately an imposing structure which is built on sand. It is

shaky in its very foundations. Consider Peirce's most explicit statement

concerning the probability of an argument:

We may, therefore, define the probability of a mode of argument as the proportion of cases in which it carries truth with it. (2.650)

But just what does this mean? Do we consider only the actual cases in which the form is applied? This cannot be, for in that case the

probability of the form would change as we increased the number of actual cases. We must then consider all possible cases in which we could or might apply the form. Thus "probability" is to be defined with the aid of a subjunctive conditional. This in itself introduces

many problems. But the real difficulty when we consider possible applications is in arriving at the proportion of cases mentioned. The number of cases is obviously not finite, possibly even uncountable, and so we cannot intelligibly speak of a proportion under these circumstances. We might of course talk of the proportion as the number of cases

approaches a certain limit, but we are not really helped by this. Un-

fortunately the process of assigning a probability to a mode of argument in the manner Peirce requires empirical investigation. We must somehow look at each (possible) case and see whether or not it has a true

conclusion. Such a method simply cannot be applied successfully to an

infinite set nor to a set of merely possible cases.

Loyola University of Chicago

NOTES 1. Decimal expressions of this sort refer to the numbered sections in Collected

Papers of Charles Sanders Peirce, ed. by C. Hartshorne and P. Weiss, Belknap Press, I960.

2. This is not terribly surprising when we remember that Peirce was often

writing for an audience which could not be expected to be very sophisticated logically or philosophically. Hence it is not unlikely that certain occurrences of

probable' were added simply to emphasize the type of inference involved.

3. For a very good informal discussion of a modern characterization of randomness see pp. 79-81 of Henry E. Kyburg's Probability and Inductive Logic, Mac-

millan, 1970.



peirce on probability and induction

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