20th WCP: Erotetic Logic and Explanation by Abnormic Hypotheses

ABSTRACT:

General rules might be false or even known as not applying to all relevant cases, but nevertheless these are the general rules which usually govern our expectations, since they enable us to predict the most "typical" outcomes. When such a prediction fails, however, the need for explanation usually arises. The explanation called for is not simply an explanation of the failure of our prediction: we want to explain why such – and – such outcome took place rather than the expected one. In the simplest case this situation may be schematically described as follows: we have a general rule of the form:

which has proved its usefulness in many cases, but on the other hand, we have found an object a of which the following holds: F₁(a) & ... & F_m(a), R(a) where R(a) describes something that is the case and which is such that the statement

is true. So we ask the question: Why is it the case that R(a)?

How can one find an acceptable answer to this question? A possible way is to formulate and/or apply a relevant abnormic hypothesis which completes the general rule. In most cases a hypothesis of this kind determines a class of possible correct answers to the explanation-seeking why – question within which some selection should be made. The core of the selection procedure consists in performing a series of valid erotetic inferences (i.e., roughly, inferences which have questions as conclusions). If the selection procedure is successful, an acceptable answer to the explanation – seeking question is found, a deductive – nomological explanation of the analyzed departure from a general rule becomes possible and the degree of confirmation of the proposed hypothesis rises. If, however, the selection procedure ends with the rejection of all the relevant possible correct answers to the why – question, the abnormic hypothesis should be modified.

Let us analyze this in detail.

1. What are abnormic hypotheses? In his famous paper "Why – questions" Sylvain Bromberger introduced the concept of abnormic law¹. There are abnormic laws of two kinds: general and special. Here are two examples of general abnormic laws (taken from Bromberger's paper):

(L₁) The level of liquid in a cylindrical container on which a melting object is floating at room temperature will rise unless the object is made of a substance whose density in liquid form is the same or is greater than that of the original liquid at room temperature. If the density is the same, the level will remain the same; if the density is greater, the level will go down.

(L₂) All French nouns form their plural by adding s unless they end in al (except bal, cal, carnaval, etc.) or in eu, or in au, or in ou (except chou, genou, etc.), or x, or z, or s. If and only if they end in al (except bal, etc.) they form their plural by dropping the last syllable and replacing it with aux; if and only if they end in eu or ou or au (except chou, etc.) they form their plural by adding x; if and only if they end in x or z or s they form their plural by adding nothing.

The following are examples of special abnormic laws:

(L₃) The velocity of an object does not change unless the net force on it is not equal to zero.

(L₄) No sample of gas expands unless its temperature is kept constant but its pressure decreases, or its pressure is kept constant but its temperature increases, or its absolute temperature increases by a larger factor than its pressure, or its pressure decreases by a larger factor than its absolute temperature.

By and large, a general abnormic law states that if some initial conditions are met, then a given effect takes place provided that some additional conditions are not met. If, however, any of the additional conditions is met, then the result is different. Moreover, a general abnormic law catalogues all the possible outcomes different from the "main" one and associates with each of them some condition(s) whose satisfaction, together with the satisfaction of the initial conditions, produces the relevant outcome. A special abnormic law, in turn, states that if some initial conditions are met and some additional conditions are not satisfied, the outcome is so-and-so, but if any of the additional conditions is satisfied, the result is just the opposite.

Abnormic laws are regarded as completions of general rules. In particular, (L₁) completes the general rule:

(G₁) The level of liquid in a cylindrical container on which a melting object is floating at room temperature will rise.

whereas (L₂) completes the following:

(G₂) All French nouns form their plural by adding s.

Similarly, the general rules:

(G₃) The velocity of an object does not change.

(G₄) No sample of gas expands.

are completed by (L₃) and (L₄), respectively.

We may say² that a general abnormic law has the following logical form:

where

are distinct one-place predicates.

A special abnormic law has the following logical form:

where

are distinct one-place predicates. The general rule completed by a general or special abnormic law of the form (GAH) or (SAH) has the logical form of (GR).

Let us stress that the above schemata only show what is the logical form of abnormic laws. It is not the case that any statement having the form of (GAH) or (SAH) is an abnormic law. Bromberger imposes here some further constraints. The most important are lawlikeness and truth; there are also other requirements (cf. Bromberger (1992), pp. 89 – 90). Yet, we want to speak here about abnormic hypotheses instead of abnormic laws. So we will say that a general abnormic hypothesis is a hypothesis of the form (GAH). Similarly, a special abnormic hypothesis is a hypothesis of the form (SAH). We neither assume nor deny that an abnormic hypothesis is true.

In order to continue we also need the concept of an antonymic predicate of an abnormic hypothesis (law).

The antonymic predicates of (GAH) are E, R₁, R₂, ..., R_k; they are the antonymic predicates of the corresponding general abnormic hypothesis (law). Thus, for example, the antonymic predicates of the law (L₂) are the following: 'Forms the plural by adding s', 'Forms the plural by dropping the last syllable and replacing it with aux', 'Forms the plural by adding x', 'Forms the plural by adding nothing'.

The antonymic predicates of (SAH) (and thus also of the corresponding special abnormic hypothesis or law) are E and E*, where E* is the negation of the predicate E. In what follows, instead of defining the concept of negation of a predicate, we will simply assume that the formula

always holds. The antonymic predicates of (L₃) are: 'The velocity of ... changes', 'The velocity of ... does not change', whereas the antonymic predicates of (L₄) are: 'expands' and 'does not expand'.

2. Answers to why – questions. Why – questions pose a challenge to erotetic logicians, since it is hard to define what counts as a "principal" possible (that is – in terms of different theories – direct, or proper, or conclusive, etc.) answer to a why – question. The (partial) solution to this problem proposed in this paper will be based on (but not identical with) that proposed by Bromberger in his paper "Why – questions."

Bromberger's analysis is restricted to why – questions in the so – called normal form, that is, why – questions which can be put in English in the form of an interrogative sentence which fulfills the following conditions: (a) it begins with the word why; (b) the remainder of the sentence has the (surface) structure of a yes – no question; (c) the sentence contains no parenthetical verbs. The inner question of a why – question in the normal form is the yes – no question expressed by the interrogative sentence which can be obtained from the why – question by deleting the word why; the presupposition of a why – question in the normal form is the sentence which expresses the affirmative answer to its inner question. For example, in the case of the question:

(1) Why is the plural of the French noun cheval chevaux, that is, formed by dropping the last syllable and replacing it with aux?

the inner question is:

(2) Is the plural of the French noun cheval chevaux, that is, formed by dropping the last syllable and replacing it with aux?

whereas the presupposition is:

(3) The plural of the French noun cheval is chevaux, that is, formed by dropping the last syllable and replacing it with aux.

One of the merits of Bromberger's analysis is that it supplements the concept of correct answer to a why – question with a precisely defined meaning³. Correct answers to why-questions are defined as follows:

"b is the correct answer to the why – question whose presupposition is a if and only if (1) there is an abnormic law L (general or special) and a is an instantiation of one of L's antonymic predicates; (2) b is a member of a set of premises that together with L constitute a deductive nomological explanation whose conclusion is a; (3) the remaining premises together with the general rule completed by L constitute a deduction in every respect like a deductive nomological explanation – except for a false lawlike premise and a false conclusion, whose conclusion is a contrary of a; (4) the general rule completed by L cannot be completed into a true abnormic law when the conjunction in its antecedent is replaced by an expression properly entailed by that conjunction (i.e., not also entailing that conjunction.)" (Bromberger 1992, p. 92).

For example, the correct answer to (3) is: (Because) cheval ends in al.

Since the explanans of a deductive – nomological explanation is supposed to consist of truths, each correct answer to a why – question must be true. On the other hand, if a general rule can be completed by more than one abnormic law, the corresponding why – question can have more than one correct answer.

We shall now define a weaker, relativized concept of a possible correct answer to a why – question.

Def.1. A statement B is a possible correct answer to the question "Why is it the case that A?" with respect to a background knowledge K and an abnormic hypothesis H (general or special) iff

We have:

FACT 1. Let R_i

be an antonymic predicate of (GAH) and let

be the predicates associated with R_i in (GAH). Assume also that the following statements belong to K:

Then each sentence of the form P_ij(a)

is a possible correct answer to the question:

with respect to K and the general abnormic hypothesis expressed by (GAH).

In other words, if the presupposition of a why – question says that an object a has the property R_i, where R_i is an antonymic predicate of a general abnormic hypothesis such that R_i does not occur in the general rule completed by it (that is, roughly, the presupposition expresses a departure from the general rule), and the background knowledge K contains the statements (I) and (II), then each sentence which says that the object a has the property P_ij, where P_ij is a predicate associated with R_i in the general abnormic hypothesis, is a possible correct answer to the why – question with respect to K and the hypothesis. In the case of special abnormic hypotheses we have:

FACT 2. Assume that the following statements belong to K:

Then each sentence of the form P_j(a),

is a possible correct answer to the question:

with respect to K and the special abnormic hypothesis expressed by (SAH).

It seems that the above consequences comply with intuitions. Note that a possible correct answer to a why – question need not be true (but of course can be). We also do not assume that an item of a background knowledge must consist of truths. We require that the relevant abnormic hypothesis must be consistent with the relevant pieces of a background knowledge, but, for the sake of generality, we do not impose further conditions here. Let us finally observe that any possible correct answer to (#) which has the form of P_ij(a), the corresponding general abnormic hypothesis, and the statements (I) and (II) constitute the explanans of a potential deductive – nomological explanation whose explanandum is R_i(a). Similarly, any possible correct answer to (##) of the form P_j(a), the relevant special abnormic hypothesis, and the statements (I) and (III) constitute the explanans of a potential deductive – nomological explanation whose explanandum is E*(a). In both cases an answer to the why – question is the necessary premise which makes the (potential) deductive – nomological explanation possible.

3. Explaining departures from a general rule. Establishing what sentences can play the role of possible correct answers to a given why – question is one thing; finding an acceptable answer to this question is another.

Let us come back to the beginning of this paper. The analyzed cognitive situation was as follows: although all the initial conditions of a general rule were fulfilled by an object a, the outcome that took place was R(a), where R(a) is defferent from E(a) predicted by the rule (and was such that

is true). So we ask the question:

How can we find an acceptable answer to (4)? There is no recipe that can be applied in all cases. But the following advice can always be given: look for an abnormic hypothesis which has R among its antonymic predicates and which completes the general rule. If there is no such a hypothesis, try to infer it in a legitimate way from your knowledge.

There are cases in which we can deduce a certain possible correct answer to the why – question directly from the relevant general abnormic hypothesis and the initial conditions. For example, if we have a general abnormic hypothesis in which R is associated with exactly one predicate P in the corresponding biconditional, i.e. the hypotesis contains the clause

then the sentence P(a) is logically entailed by the general abnormic hypothesis on the basis of the premises (I) and R(a). On the other hand, P(a) is a possible correct answer to (4) with respect to the general abnormic hypothesis and the background knowledge which contains, int. al, the statement (I) and the statement

(cf. Fact 1).

In many cases, however, no possible correct answer to (4) is entailed by the set made up of the general abnormic hypothesis, the statement (I) and the presupposition of (4); for conciseness, let us designate this set by U. But if R is an antonymic predicate of the general abnormic hypothesis which is associated in it with at least two distinct predicates, that is, R is identical with a certain R_i such that the general abnormic hypothesis contains the clause:

where s>1, then the set U logically entails the following disjunction of possible correct answers to (4):

On the basis of (6) we arrive at the following disjunctive question evoked by it:

From the question (7) on the basis of the premise (6) we can go to the following implied question:

If the affirmative answer to (8) is justified, an acceptable answer to the initial why-question (4) is found and the procedure is terminated. If, however, the negative answer to (8) is justified, then from the question (7) on the basis of the premise (6) and the negative answer to (8) we go to the implied question:

Again, if the affirmative answer to (9) is justified, the procedure is terminated since the success has been achieved; if the negative answer to (9) is justified, we go from (7) on the basis of (6) and the negative answers to (8) and (9) to the implied question:

And so on until an affirmative answer to a certain consecutive yes – no question will be justified. In each case the next step (if any) is determined by the outcome(s) of the previous step(s) in the above – indicated manner. Let us observe that if the sentence (6) is true, then at least one of the consecutive yes – no questions must have a true affirmative answer. Yet, the sentence (6) is a consequence of a hypothesis and thus need not be true. If the procedure described above ends with the negative answers to all the consecutive yes – no questions, the initial abnormic hypothesis needs revision. On the other hand, if an affirmative answer is justified, the degree of confirmation of the abnormic hypothesis rises.

An accepted answer to the question (4) is the necessary premise which makes possible a deductive – nomological explanation of the analyzed departure from the general rule on the basis of the relevant abnormic hypothesis and the premises (I) and (II).

The considerations presented so far pertained to general abnormic hypotheses. The situation is similar in the case of special abnormic hypotheses.

4. Validity of erotetic steps. The procedure described above involved erotetic arguments (e – arguments for short), that is, arguments which have questions as conclusions, whereas the premises are declarative sentences and/or questions. But are they valid e – arguments? The answer depends on the meaning of the term "validity." It can be shown that they are valid in the sense of inferential erotetic logic.⁴ The question (7) is evoked by the sentence (6) (in the technical sense of the term "evocation"; cf. Appendix below) and thus the transition from (6) to (7) is a valid e – argument, since in the case of e – arguments which have declaratives as premises and questions as conclusions validity is defined in terms of evocation. Similarly, the consecutive yes – no questions which occur in the procedure described above are implied (also in the technical sense of the word) by the initial question (7) and the corresponding declarative premise(s). On the other hand, validity of an e – argument in which the conclusion is a question, whereas the set of premises consists of a question and a set of declaratives, is defined in terms of (erotetic) implication. For details, see Appendix below.

APPENDIX

Let L be a formalized language which consists of two parts: assertoric and erotetic. The assertoric part of L is a first – order language. As far as the assertoric part of L is concerned, the concepts of term, atomic well – formed formula, (declarative) well – formed formula (d – wff for short), freedom and bondage of variables, etc., are defined as usual; by sentences of L we mean d – wffs of L without free variables. Questions are the meaningful expressions of the erotetic part of L; at this moment we do not decide, however, what is the particular form of questions of L. Yet, we assume that to each question Q of L there is assigned an at least two – element set dQ of direct answers, which are sentences of L. We also assume that the assertoric part of L is supplemented with a standard model – theoretical semantics with the concepts of interpretation, satisfaction, truth, etc., defined in the usual way, and that the class of all the interpretations of L includes a non – empty subclass (not necessarily a proper subclass) of normal interpretations. Then we introduce the concept of multiple – conclusion entailment:

Evocation is defined by:

Thus X evokes Q just in case Q is sound (has a true direct answer) in each normal interpretation of L in which all the d-wffs in X are true, but no direct answer to Q is entailed by X. Roughly, X evokes Q iff the truth of the elements of X guarantees the soundness of Q and Q is not logically redundant with respect to X.

An e – argument of the first kind is an ordered pair <X, Q>, where X is a non – empty and finite set of d – wffs and Q is a question. We say that an erotetic argument <X, Q> is valid iff E (X, Q).

Erotetic implication is defined by:

Roughly: Q implies Q₁ on the basis of X just in case Q₁ is sound if Q is sound and X consists of truths, and Q₁ is in each case cognitively useful with respect to Q on the basis of X, that is, each direct answer to Q₁, if it is true and if all the d-wffs in X are true, narrows down the class within which a true direct answer to Q can be found.

An e – argument of the second kind is an ordered triple <Q, X, Q₁>, where Q, Q₁ are questions and X is a finite set of d – wffs. An e – argument <Q, X, Q₁> is valid iff
Im(Q, X, Q₁).

Let us now be more specific. Let L* be a language of the considered kind which has the language of Classical Predicate Calculus as its assertoric part. Assume also that the vocabulary of L* contains the signs: ?, {, }, which are erotetic constants of L*. A question of L* is an expression of L* of the form ?{A₁, ..., A_n}, where A₁, ..., A_n (n > 1) are nonequiform (i.e. syntactically different) sentences of L*. If ?{A₁, ..., A_n} is a question, then the sentences A₁, ..., A_n are the direct answers to this question. Suppose that each interpretation of L* is normal (and thus entailment amounts to logical entailment). Let B₁, ..., B_n, where n > 1, be nonequiform (syntactically different) atomic sentences of L*. One can easily verify the following:

where i < n.

Thus we can say that the e – arguments involved in the procedure described above are valid erotetic arguments⁵.