Laws: Counterfactual demarcation overruled

HORACIO ABELEDO, EDUARDO H. FLICHMAN AND MARÍA ALICIA PAZOS

LAWS: COUNTERFACTUAL DEMARCATION OVERRULED*

ABSTRACT. A classical view holds that counterfactuals support natural

laws but do no support accidental uniformities, and that this provides

a criterion which demarcates laws from uniformities. In order to

discuss this stand, analyses of the notions of support and counterlegal are

proposed, and some new notions, such as counterfactual associated to a lawlike

statement and counter-lawlike conditional among others, are introduced. Equipped

with this machinery we are able to prove the thesis that when a

counterfactual associated to a lawlike statement (be it a law or a

uniformity) is not supported by it, the counterfactual is a counter-

lawlike conditional. Moreover, since counterfactuals are context-

dependent, it turns out that the same counterfactual may be counter-

lawlike or not, and may be supported by the lawlike statement or not,

depending on the context. As a consequence we are able to refute the

classical view.

1 INTRODUCTION

1

Statements of natural law are often said to support

counterfactuals. If this claim were correct, It could provide an

easy criterion of demarcation between statements of accidental

uniformity and statements of natural law, since uniformities are

said not to support counterfactuals. Hereafter, statements of

natural law and statements of accidental uniformity shall be

called “laws” and “uniformities”, respectively. A provisional and

extremely simple exposition of this approach, which we shall call

"SUP" (support) could be the following one (In sec. 2.3 we will

give a more detailed account).

Let us suppose that the following statement (1) is a lawlike

statement (possible law or uniformity, hereafter "L"), and that

it is true:1

(1) "All S are P."

Let us also consider the following counterfactual:

(2) "If a were an S then it would be a P."

SUP: (1) is a law if and only if (2) is true; and (1) is a

uniformity if and only if (2) is false. This result sets a

demarcation criterion between laws and uniformities.

2

This approach, with which we do not agree and which we shall try to refute,

assumes that between laws and counterfactuals there is an

opposite relation to that between uniformities and

counterfactuals. Of course, this approach needs justification.

But, moreover, its defenders do not state clearly which one of

the following alternatives they endorse:

SUP1. There is a way to determine, given a true L, whether

it is a law or a uniformity, and there is a way to determine the

truth or falsity of a counterfactual, and both ways are

independent.

In this case both criteria should be stated, and according to

them, one should find out why the proposed relation SUP exists.

SUP2. There is a way to establish truth conditions of

counterfactual conditionals, which does not assume, for true L, a

previous demarcation between laws and uniformities. Laws are

defined as those L that support counterfactuals (analogously,

uniformities are defined as those L that do not support

counterfactuals).

SUP3. There is an independent way (i.e. not involving

counterfactuals) to decide, given a true L, whether it is a law

or a uniformity, and this criterion affects somehow the

formulation of the truth conditions of counterfactuals.

3

As a matter of fact, it is not really right to classify SUP3 (as

we have just done) as a case of SUP, because SUP3 needs a previous

demarcation criterion. It is an a posteriori demarcation

criterion and is, thus, qua criterion, useless.

Were alternative SUP1 right, it would provide an excellent

mechanism to differentiate laws from uniformities. This is not

the case of alternatives SUP2 and SUP3, since (in addition to the

previous remark regarding SUP3) both of them assume a

definitional relation. This situation would yield the conclusion

that the intuition or criterion (provided there is one2) that

enables us to differentiate laws from uniformities is the same

one that enables us to assign truth values to the corresponding

counterfactuals. Then, the intuition or the demarcation criterion

is previous to the relation between L and its corresponding

couterfactuals.

In his classical article on counterfactuals, Nelson Goodman

(1947) tries an approach in which laws are used to determine the

truth values of counterfactuals. It may seem at first that he

chooses alternative SUP3, but, as a matter of fact, at least in

this article, there is no independent criterion to decide which L

corresponds to a law and which to a uniformity, nor does he give

any hint about his position regarding a demarcation criterion of

4

the kind SUP. We could think that he does not take a stand

because he is aware of a certain risk of circularity.

Neither does David Lewis (1973, esp. sec. 3.3) take a categorical

stand on SUP. His position regarding the relation between

counterfactuals and laws seems at first to agree with SUP. In his

conception of laws, in which he carefully avoids mentioning

counterfactuals, he claims — by modifying an idea of Frank Ramsey

(1929)3 — that they are the axioms and theorems of true deductive

systems, which optimally combine information content and

simplicity. On the other hand, his counterfactual semantics makes

use of a similarity relation among possible worlds, that does not

appeal, at least explicitly, to laws4. However, in (1979), he is

forced to use pragmatic criteria that often involve appealing to

laws. This brings him near SUP3.

The authors in favor of some of the SUP versions, for example,

Ernest Nagel (1961), Rudolf Carnap (1966) and Carl Hempel (1966),

did not try to establish truth criteria for counterfactuals.

Their main concern was, rather, finding the criteria for the

identification of laws by means of their logical form. However,

they ran into examples of statements having the logical form they

attributed to laws but were clearly uniformities for intuition.

Consequently, they turned to some version of SUP, hoping to find

some non-circular criterion to determine the truth value of

counterfactuals. Their stand may therefore be classified as SUP2.

5

Nicholas Rescher dealt with this problem in his (1964), (1973),

and (1975). In his view, in (1973, ch. V, sec. 2), support of

such counterfactuals is an indispensably essential feature of

laws. According to Rescher, although we assign or "impute"

lawfulness to L, we do it taking into account actual features of

the empirical world. But this nomic "imputation" is, at the same

time, an "imputation" with hypothetical force, this is, with a

couterfactual value. When he speak of support of such

counterfactuals, he refers to what he calls "nomological

counterfactuals", considering those examples of laws that do not

involve the issue of support as "artificial" (1964, ch. 5, sec.

5.2). We will see later that Rescher’s artificial counterfactuals

coincide with a kind of counterlegal (according to our

nomenclature) (sec. 2.4.5, statement 13). Thus, Rescher seems to

take to position SUP3.

In this article, we assume (instrumentally5) Lewis’s semantics

(1973 and 1979) for counterfactuals, and apply some of his (1979)

and other authors' (for example, Flichman 1985) pragmatic

stipulations.6 Notwithstanding, we believe that the conclusions

we obtain in this paper are independent of the semantic analysis

chosen.

In sec. 2 we determine to what counterfactuals a defender of SUP

would wish to assign the relation at issue (counterfactuals

6

associated to an L). This is, what counterfactuals would be (or

not) supported by an L, according to SUP. We also define more

precisely what it means to say that an L supports a

counterfactual, and examine several kinds of counterfactuals

called by us "counter-L", in order to establish which ones will

be of interest in this paper. "Counter-L" must be understood as

"counter-lawlike". We will call the counter-lawlikes

"counterlegals" or "counter-uniforms" according to whether they

relate to laws or to uniformities.

In sec. 3 we show an analytical relation (based on the analyses

of sec. 2) between a certain kind of counterfactuals associated

(positively associated, i.e., p-associated) to an L and the support

of L to those counterfactuals, if certain conditions are

satisfied.

Finally, in sec. 4, in virtue of the results just mentioned, we

deny the possibility of endorsing thesis SUP.

2 WHICH ARE THE COUNTERFACTUALS WE ARE INTERESTED IN?

2.1 COUNTERFACTUALS ASSOCIATED TO AN L .

We would like to characterize first the counterfactuals that are

relevant for the discussion, taking into account the usual

intuition that, at least generally, laws support (or not) certain

counterfactuals associated to them. Therefore, we would surely

7

say that the lawlike statement 7 (oversimplified): "All pieces

of metal, when heated, expand according to equation E", which,

furthermore, is said to be a law, "supports" the counterfactual:

(3) "If this piece of metal were heated, it would expand

according to equation E."

and does not "support" counterfactual

(4) "If this piece of metal were heated, it would not expand

according to equation E."

But we shall say that (3) and (4) are counterfactual "associated"

to . On the other hand, it is clear that there may be

counterfactuals whose truth or falsity does not seem to be

related to any , but instead to concrete circumstances, e.g.:

(5) "If Carnap had been American, Popper would still have been

Austrian."

seems to depend upon particular conditions and not upon laws, or,

more generally, upon any L.

There are other cases, too, in which an L (or, specifically, a

law) seems to have some relation with the counterfactual, but the

truth conditions of this counterfactual do not depend only from

8

the law, but from particular facts, such as one or several

initial conditions. In this case, and in that of example (5) as

well, we would not say that the counterfactual is "associated" to

an L, nor that L "supports" or does not “support" it. For

instance, the appraisal of

(6) If the earth had the mass of the moon, its gravity would be

less."

does not follow only from the law of gravitation, but also from

the mass of both the earth and the moon.

We will try to develop a concise analysis of the idea that a

counterfactual is "associated" to an L, and that an L "supports"

(or does not “support") a counterfactual associated to it. For,

without this analysis, it would be impossible to answer questions

such as: "Do all law statements support counterfactuals?" or "Is

there more than one counterfactual associated to each law?"

We believe that the most important intuitive concept in our

analysis of "counterfactual associated to an L" is that

associated counterfactuals are those in which L determines (not

trivially) the verification of the consequent or its negation, in

all the cases in which the antecedent holds.

9

Hereafter <ctx, w, , CF> shall be an ordered 4-tuple where the

components are, respectively, a certain context, a certain base

world, a certain L true in w and a certain counterfactual.

DEFINITION 1 (ANALYSIS OF THE NOTION OF COUNTERFACTUAL ASSOCIATED TO AN L):

CF is associated to if and only if

(i) The conjunction of with the antecedent of CF entails the consequent or its

negation.

(ii) The consequent of CF or its negation is not inferred from the antecedent alone.

(iii) The antecedent of CF is compatible with . (i.e., does not entail the

negation of the antecedent).

If, in particular, it is the consequent (not its negation) that is entailed from the

conjunction of and CF, we shall say CF is a positive associate of (abbreviated

“is p-associated”). In the opposite case (the negation of the consequent is entailed), we

say CF is a negative associate of (abbreviated “is n-associated”).

In formulas, given a <ctx, w, , CF>, CF (which is: A � C) is

associated to if, and only if:

(I) (. A C) v (. A ~ C)

(II) ~ (A C) . ~ (A ~ C)

(III) ~ ( ~A)

10

Where � stands for the counterfactual conditional connective. A

is the antecedent and C the consequent of CF. The symbol

represents here the strict conditional connective. The other

symbols are those of elementary logic.

We should bear in mind that in many counterfactuals which may

intuitively seem to be associated to an L, the antecedent

includes tacitly statements about concrete circumstances of the base

world (i.e., certain statements about the contextual situation)

that are necessary for the non-trivial deduction of the consequent

(or its negation) from the antecedent and the corresponding L. In

this way, such counterfactuals not only seem but also are

associated to the corresponding L. If these statements are

necessary for the deduction and are not (tacitly or explicitly)

included in the antecedent of the counterfactual, the latter will

not be associated to the corresponding L. We consider a deduction

as not trivial when it does not involve statements (of the contextual

situation) in the antecedent that make it selfcontradictory (for

this would violate restriction (ii)) and that does not involve

statements (of the contextual situation) in the antecedent that

entail the consequent or its negation (for this would also

violate restriction (ii)). All this will be clearly evidenced at

the end of sec. 2.3.

In the present analysis of counterfactuals associated to an L we

have tried to honor the following intuitions: in the first place,

11

as we have seen above regarding (3), (4), (5), and (6), it is

quite intuitive to assume that all L are “associated” only to

certain counterfactuals and that they "support" or not only

certain counterfactuals. In these examples, the corresponding L

"supports" (3), but does not “support" (4), and is “associated”

to both of them. Instead (5) and (6) are not “associated” to any

L, and are obviously not “supported” nor “unsupported” by any L.

We shall analyze the “supporting” relation in sec. 2.2.

Second, since it is not always meaningful to say that a

counterfactual is “supported” or not by an L, it becomes necessary

to find which counterfactuals are “associated” intuitively to an

L. In order to do this, we consider that a condition to be

satisfied is that, given a <ctx, w, , CF>, must ensure the truth

or falsity of the consequent, when the antecedent is true, such

as (i) indicates. This enables us to tell when does support or

does not support its associated counterfactual.

But, besides, our intuition involves the relevance of that L for

the truth or falsity of the consequent. This justifies (II)

since, were (II) not valid, the L in (I) could be substituted

with any other statement, what would imply, among other

consequences, that the counterfactual could be associated to any

L.

12

The third condition can be explained in the following way: let us

suppose that a counterfactual has an antecedent A, that is

incompatible with law . In this case, the conjunction of A and

would be a contradictory statement. Since any formula can be

inferred from a contradiction, whatever the consequent of the

counterfactual, it is inferred from the contradiction. A very

clear case of this is when the antecedent of the counterfactual

is the negation of an L. Let us consider counterfactual ~ � C.

The conjunction of and the antecedent of the counterfactual is

the contradiction . ~, from which any formula follows, such as

C and its negation. So, if we did not introduce our restriction,

taking into account (I), all counterfactuals whose antecedents

were incompatible with A whatever the consequents, would allow

the inference of the consequent and would thus be associated to

.

Some of the consequences of this analysis, which we believe to

agree with commonsensical intuitions, are the following:

A) If an L has the form of a general conditional statement (x)

(Fx Gx)8 all of the counterfactuals that relate F to G or to ~G

for the same instance of the variable, shall be associated

counterfactuals. So, for instance, if “All monkeys have a heart.”

is considered to be an L, the statement “If the fern in my living-

room were a monkey, it would have a heart.” would be an

associated counterfactual.

13

B) The number of counterfactuals associated to an L is boundless:

given a counterfactual associated to an L, the substitution of

the constants instantiating the variables of L with any other

constant (within the reach of the quantifiers), would suffice to

obtain a new associated counterfactual.

C) A counterfactual may be associated to more than one L. For

instance, it may be associated to a fundamental law and to a

derived law.

2.2 SUPPORTING OR NOT A COUNTERFACTUAL

Now we can clarify when one can say that an L “supports” (or not)

a counterfactual. We believe the following is an acceptable

definition:

DEFINITION 2 (ANALYSIS OF THE IDEA OF SUPPORT):

Given a <ctx, w, , CF>, supports CF if and only if CF is p-associated to it and

CF is true. It does not support CF if and only if either CF is p-associated to it and is

false, or it is n-associated to it, whether true or false. Finally, if CF is not associated to it,

it does not make sense to ask whether supports it or not. That is, the category or

relation of being supported by does not apply to counterfactuals that are not

associated to

14

Apparently, we could suppose that a counterfactual p-associated

to an L is supported by that L. However, we shall see in sec.

2.4.3 that this is not always the case.

2.3 THESIS SUP

In virtue of our previous definitions, we state here a more

precise version of thesis SUP:

SUP: a) is a law of w if and only if for any ctx and for any p-associated CF, CF is

true in w ( supports CF). b) is a uniformity of w if and only if a) does not hold. c)

the stance asserted in a) and b) is a criterion of demarcation between laws and

uniformities.

Let us consider now the pair of counterfactuals (7) and (8), in

the following contextual situation:

p is a particle that (in the Newtonian world w) has a non-zero

acceleration a at time t. At the same time t, force f acts upon the

particle. Since the world in question is a Newtonian one,

Newton's second principle is satisfied, enabling us to know the

mass m of p.

(7) "If p had at time t an acceleration 2a, it would be subject

at that time to force f.”

15


at that time to force 2f."

In a usual context (see note 6) and, as we have said before, in a

Newtonian base world w, in virtue of Newton's second law of

dynamics we would say that (7) is false and (8) true. We would

also say intuitively that (7) is n-associated and (8) is

p-associated to this law, and that the law supports (8) but does

not support (7).

Hereafter, we will often abbreviate "Newton's second law of

dynamics" as, simply, "dynamics".

The analyses suggested in this section agree with our intuitions.

Because on one hand the consequent of (8) and the negation of the

consequent of (7) are inferred from the law together with the

common antecedent of (7) and (8). And, on the other hand, because

in usual contexts we would say of counterfactual (8) that it is

true in the base world, so that it is supported by the law;

whereas (7) is false, so that it is not supported by the law.

Therefore, thesis SUP is satisfied in (8) whereas (7) is not

relevant for thesis SUP, since (7) is not p-associated to the

corresponding L.

Let us observe now that it would not be true that the consequent

of (8) and the negation of the consequent of (7) are inferred

16

from a common antecedent unless we include in the antecedent, at

least tacitly, statements about concrete circumstances: 1) that p

is a particle, 2) that t is an instant, 3) that the mass of p is

m. These are statements of the contextual situation described

above, which we consider part — though tacit — of the antecedent

to the effect of this inference. We do not mean that the tacit

part should be part of the antecedent but, instead, that if it is

not part (tacit or explicit) of the antecedent the

counterfactuals will not be associated to the lawlike statement.

This clarifies what we have said in sec. 2.1 regarding this

issue. Let us observe, as well, that we cannot include among the

tacit (or explicit) statements of the antecedent statements of

the contextual situation, such as "The acceleration of p in t is

a.", or "The force acting upon p at t is f." because they would

trivialize the deduction, as we said in sec. 2.1 (restrictions

(ii) and (iii) of the definition of "counterfactuals associated

to an L" would not be satisfied).

Up to here, thesis SUP seems to work correctly. However, we shall

see that this is not the case for these or other examples in

unusual contexts, even if they are lawlikes usually dealt with as

if they were laws (like, for example, the Newtonian laws of

gravity and dynamics in a Newtonian world). But, before, we must

examine counter-L counterfactuals.

17

2.4 COUNTER-L COUNTERFACTUALS

In the literature about counterfactuals, when the supposition

involved is not just contrary-to-fact but also contrary-to-law,

the counterfactual in question is called a counterlegal. Since we

do not wish to assume there is a previous demarcation, we shall

speak in general of counter-L; in order to interpret this idea

the following analysis could be proposed:

Given a <ctx, w, , CF>, CF is counter-L with respect to if and only if is false

in any of the A-worlds sufficiently similar to w.

However, this is not precise enough, since it includes some un-

genuine counter-L sentences as we shall see in sec. 2.4.2. A more

precise definition (parts of which shall be explained in that

section) is:

DEFINITION 3 (NOTION OF COUNTER -L):

Given a <ctx, w, , CF>, CF is a counter-L respect of if and only if is false in

any of the A-worlds sufficiently similar to w, and ctx is not a usual miraculous

context.

In sec. 2.4.2 we will explain the notion of usual miraculous

context.

18

The A-worlds mentioned in the definition above are, per definition of

A-world, those (similar enough to w) in which the antecedent

holds. If in some of these worlds an L is violated, the

counterfactual "supposes" the negation of that L. In sec. 2.4.5

we will try to clarify why we say some and not all worlds similar

enough to w.

We will examine (in secs. 2.4.1, 2.4.3, and 2.4.4) three kinds of

counterfactuals that are genuine counter-L, and a fourth kind of

alleged counter-L (sec. 2.4.2), which are not genuine. We shall

see that only one out of the three authentic kinds is interesting

for the purpose of our article: that of associated counter-L or

implicit associated counter-L (sec. 2.4.3).

2.4.1 FIRST CASE

DEFINITION 4 (NOTION OF EXPLICIT COUNTER -L):

Given a <ctx, w, , CF>, we shall say of CF that it is an explicit counter-L with

respect to if and only if it is a counter-L with respect to and, besides, its

antecedent is incompatible with

To begin with, let us observe the following examples:

(9) "If the crystal ball had moved in the second half of its

trajectory slowing down while falling in the vacuum, and no other

19

force than its own weight had (noticeably) acted upon it, then it

would not have smashed against the floor."

Contextual situation (w is a Newtonian world): somebody is

carrying out an experiment with a crystal ball located in the

interior and in the upper part of a very tall cylindrical

recipient, in which a vacuum has been made. The ball is held by a

support that, at a given time, is removed by the experimenter.

The ball falls and breaks upon striking the base of the

recipient. So as to simplify the example, we have left only one

force (noticeably) acting upon the ball, namely, the

gravitational attraction of the earth, once the support has been

removed.

In this case, the counterfactual supposes the violation of either

the law of universal gravitation or that of dynamics, according

to the conversational context or according to how the speaker

processes the contextual elements. But, anyway, in any of these

cases it is an explicit counterlegal with respect to the

conjunction of both since the antecedent is incompatible with the

conjunction. As already mentioned (note 1) we do not discuss in

this paper what tipe of statements are to be considered lawlike.

In particular, it could be disputed whether the conjunction of

two L is also an L. What is said here about the conjunction of the

laws shall be of interest only to those who accept this last

principle.

20

(10) "If Newton's law of universal gravitation were not correct,

all out cosmological theories would be wrong."

Here the antecedent states directly the violation of a law.

In this first case of counter-L (explicit counter-L), represented

by examples (9) and (10), the counterfactuals in question, in

spite of being counter-L with respect to some L, are not

p-associated to it because, among other things, condition (iii)

of sec. 2.1 is not met: the antecedent is per se incompatible with

the L, whether conjunctive or not. Therefore, according to our

definition 4, it does not make sense to ask whether the L

supports it or not. Such cases are uninteresting for our

purposes, since thesis SUP, which we discuss here, deals only

with counterfactuals that are associated to the L in question.

It should by clear in our way of defining explicit counterlegals

respect to an L that no change in the conversational context or

in the processing of the contextual elements can modify their

condition of explicit counterlegal with respect to an L.

21

2.4.2 SECOND CASE (COUNTERFACTUALS IN "USUAL MIRACULOUS CONTEXT" OR

ALLEGED COUNTER-L THAT ARE NOT COUNTER-L )

Let us consider once more (7) and (8) in order to suggest among

other things that whenever the usual miraculous contexts (which

we shall explain immediately) are used in a counterfactual

associated to an L, the counterfactual is not a counter-L with

respect to the L. The contextual situation is the following one:

in the (Newtonian) base world w, particle p, mentioned in (7) and

(8), has at time t an acceleration a. Particle p, whose mass is

mp, is located at a distance d from another particle, q, whose

mass is mq, and which attracts p with a force f. The value of f

satisfies the law of gravitation. Moreover, the force that q

exerts on p is the only force (noticeably) acting upon p.


at that time to force f."



For intuition (7) and (8) are not counter-L. We do not wish to

put them down as such in usual contexts. However, in certain contexts

that can be considered as usual (and in the contextual situation

mentioned above) the antecedent, together with certain statements

that follow from the context9, implies a violation of some laws,

22

that are not those laws which (7) and (8) are associated to. We shall call them

"usual miraculous contexts", following partially the terminology

of (Lewis 1979), (Nute 1980) and (Flichman 1985). The statements

in usual miraculous contexts will not be considered counter-L,

for the reasons explained right away.

Miraculous usual contexts: Let us recall that in the (Newtonian) base

world w, particle p has at time t acceleration a. The

hypothetical antecedent of both counterfactuals states an

acceleration 2a for p. Thus, the antecedent entails (together

with statements following from the usual miraculous context) a

change in the acceleration at time t (or at the time immediately

previous) in the A-worlds sufficiently similar to the base world.

There is (at least) an L of the base world that forbids this

duplication (from a to 2a) which, for that reason, we shall call

(a la Lewis) a “miracle”. This means simply that there is at least

an L of the base world which does not hold in those A-worlds. That

is why the antecedent implies (together with statements following

from the context) a violation of at least an L. But let us

observe that — provided we consider (7) and (8) as counter-L

which, as we shall see, is not the case in a usual context — they

would be counter-L with respect to the violated L but, at the

same time, they would be counterfactuals associated to dynamics,

which is not violated in that context as we have seen above (end

of sec. 2.3). Therefore, as in the case of explicit counter-L,

they are not — in a usual miraculous context — counter-L

23

associated to the L, with respect to which they would allegedly be

counter-L. We shall see immediately that they are not counter-L

at all.

Let us see now why Newton's dynamics is not violated in this

(miraculous) usual context. The change of a (at t or immediately

* This article has been produced under the framework of the research

team directed by Eduardo H. Flichman and co-directed by Horacio Abeledo

in the Ciclo Básico Común (University of Buenos Aires), on a grant from

UBACYT (University of Buenos Aires, Division of Science and

Technology), and during the stay of one of the authors (Flichman) as a

Visiting Fellow at the Center for Philosophy of Science of the

University of Pittsburgh. Translated into English by Sandra Girón.1 We will not discuss here whether laws and uniformities have the same

logical form, nor whether they have the logical form presented in (1).2 In (Flichman 1990 and 1995) one of the authors of this paper proposes

a demarcation criterion (independent from the subject of

counterfactuals) in the epistemological level, which he extends to the

ontological level in (1992).3 In his (1929) Ramsey rejects his own approach, which he had developed

in an earlier unpublished work written one year before, in 1928. For

more references, see the first footnote in (1973, sec. 3.3). 4 Briefly, according to Lewis’s semantic analysis in (1973 and 1979), a

counterfactual A C ("if A were the case, then C would be the

case") is true (when uttered in a possible world called by us here

"base world") if in all possible worlds sufficiently similar to it in

which A is true (sufficiently similar A-worlds), C is true too. As may

be seen, the similarity relation to be used in this analysis is24

before t) may be imputed according to the deployed context, among

other possibilities, to a violation of the law of universal

gravitation or to a violation of dynamics. We are going to

discuss both contexts. The first one (A) is usual miraculous. The

second one (B) is not (it is a forced miraculous context). We will

considered primitive. Lewis points out that he refers to a relation of

global comparative similarity between worlds, but he is aware of the

vagueness and dependency upon the context (ambiguity) that is to be

found in this relation. In (1979) he tries to establish pragmatic

criteria, so as to reduce the ambiguity, at least for examples of

certain specific kinds. We use the expressions "sufficiently similar"

or "similar enough" instead of "most similar", in order to take into

account of the possible existence of cases in which the limit

assumption fails. The limit assumption failure, whose plausibility we

do not deny (nor affirm), corresponds to the possible cases of an

infinite succesion of A-worlds progressively similar to the base world,

without having an A-world (or worlds) most similar (to the base world).5 We do not share his modal realism. On the contrary, we consider

possible worlds as conceptual construals.6 We consider the context deployed by a speaker for a counterfactual to

be originated by the speaker’s processing of three contextual elements:

1. the contextual situation: this is the portion of reality (expressed

in statements) relevant for the analysis of the counterfactual, i.e.,

data corresponding to the base world. 2. the text: this is the

counterfactual itself. Here the consequent must be included, since a

speaker may deploy different contexts for two counterfactuals with a

common antecedent and different consequents. 3. conversational context.

The speaker processes these three contextual elements, deploying a25

study it later in this section and in 2.4.3. There exist usual

non-miraculous contexts — we are following here partially the

terminology of (Swain 1978), (Nute 1980) and (Flichman 1980) —

that are very different from context (B), because they are usual

contexts (we shall see that (B) is not usual), and, in such

contexts, in the A-worlds sufficiently similar to w, no law is

violated.

similarity ordering of possible worlds, with respect to the base world.

This is the expression of the context. The conversational context

enables the speakers to agree so as to deploy the same context during

the conversation (in order to dissolve the ambiguity, given the

counterfactual’s context-dependence). Without a conversational context,

there are counterfactuals whose context is deployed in the same way by

all or many speakers. We call here this context "usual context".7 "L" is the abbreviation of "lawlike" or "lawlike statement", whereas

"" stands for a specific L.8 As a matter of fact this statement form, which we have chosen because

of its simplicity, would correspond only to extremely simple lawlike

statements, that are not usually relevant in scientific theories. If we

had decided to deal only with this statement form, a possible

definition — instead of ours — could have been: a counterfactual is p-

associated to an L if and only if either: 1) it is equal to that L,

except in that the symbol has in the counterfactual the place

has in the L, or 2) it is a logical consequence of a counterfactual of

the form specified in 1). However, this formulation would have assumed

that all L have a conditional form. This, in turn, is not quite clear.

So then, the present definition admits the possibility of non-

conditional lawlike statements having p-associated counterfactuals.26

Always considering the same contextual situation we can deploy,

among others, contexts (A) and (B) conversationally interpreted:

Miraculous usual context (A), for (8):


at that time to force 2f.”

(8A) "If p had at time t an acceleration 2a (since dynamics holds) it

would be subject at that time to force 2f, (hence q would be acting on p

with force 2f at that instant and, therefore, the law of universal gravitation would be

failing, since p and q are at a distance d, the mass of p is mp, the mass of q is mq,

and, if the law of gravitation were satisfied, the force upon p should be f, instead of

2f).”

In this context, (8) is true and (7) false. It is a miraculous

kind of usual (A) context.

We could deploy many similar usual miraculous contexts where

dynamics is defended and the acceleration change is imputed to

other reasons (existence of other kinds of forces, etc.) always

involving L-violations. In all these contexts, which relate to9 By "statements that follow from the context" we mean those statements

true in the A-worlds similar enough to the base world. Let us recall

that the context is expressed by the ordering of worlds according to

its similarity to the base world. These statements are those resulting

from the processing of all the contexual elements (see note 6).27

the same contextual situation, the allegedly counter-L

counterfactual would be counter-L with respect ot an L to which

it is not associated.

But, let us recall once more that the counterfactuals are

context-dependent. Surprising though it may at first seem, there

is at least one context in which (7) is true and (8) false! It is

namely context (B), which is not a usual miraculous one, although

it involves L-violations in the A-worlds sufficiently similar to

w. It is not usual (but is) miraculous, because it is a forced

context, since it requires a hard and difficult intuitive

"gestaltic switch" (i.e., a strong and convincing conversational

context) in order to be understood. This makes each of our two

examples a genuine counter-L with respect to dynamics to which,

besides, both are associated. It is a miraculous unusual (forced)

context.

Conversational interpretation of forced miraculous context (B),

for (7):

(7B) "If p had at time t an acceleration 2a (since q is at distance d of

p and both masses are mq and mp, respectively, and q attracts p with force f according

to the law of gravitation) it would be subject at that time to force f

(and dynamics would hence be failing)."

In context (B), (7) is true whereas (8) is false. It is clearly a

forced context; and as already said both are counter-L with28

respect to dynamics, also associated to dynamics. Such cases

shall be examined in sec. 2.4.3. Miraculous usual context (A),

instead, which interests us here, is not going to be considered

as a counter-L because of the reasons we expose right away.

It is also possible to deploy, in the same contextual situation,

usual non miraculous contexts, as we have said some paragraphs

above, such that in the A-worlds sufficiently similar to w, no L

is violated, as can be seen in (Swain 1978), (Nute 1980) and

(Flichman 1985). But we are not interested in these contexts now,

because the corresponding counterfactuals cannot be mistaken for

counter-L.

The main reason why we do not wish to include among counter-L

counterfactuals in usual miraculous contexts (that are, together

with the usual non miraculous contexts mentioned above, the usual

contexts which we normally use for counterfactuals), is that all

common "normal" counterfactuals in usual miraculous contexts,

(except those with true antecedent) would be considered as

counter-L.10 On the other hand, if they were considered counter-L,

they would be in the same situation as explicit counter-L,

namely, they would not be counter-L associated to the L, with

respect to which they are counter-L. Therefore, they would be

irrelevant (just as explicit counter-L are) for the problem at

issue, namely, thesis SUP. We therefore stipulate that such

counterfactuals are not counter-L. Still, it must remain clear

29

that, should we put them down as counter-L, this would not

generate any problems for our further argument: we would be

dealing with the same case as that of explicit counter-L.

It must remain clear that, when we refer to counterfactuals in

usual miraculous worlds, we assume deterministic worlds, for the

sake of simplicity. But the final result (of our not considering

them as counter-L, or of our considering them as counter-L) does

not vary if we extend the discussion the a not deterministic

world, because of the same reasons we have pointed out.

2.4.3 THIRD CASE

A very special kind is that of a counter-L with respect to an L

and associated to the same L. This we shall abbreviate

"associated counter-L".

10 Under the usual non-miraculous contexts studied in (Swain 1978),

(Nute 1980) and Flichman 1985) are not counter-L. This generates

problems to anyone who wishes to consider counterfactuals in a usual

miraculous context as counter-L. For both kinds of contexts (usual

miraculous and usual not miraculous) are equivalent as to their truth

values, they are usual and are not easily distinguished by intuition.

It would not be intuitive to say that a given counterfactual under

usual miraculous context is counter-L and under a usual non-miraculous

context is not.30

DEFINITION 5 (NOTION OF ASSOCIATED COUNTER -L, OR IMPLICIT ASSOCIATED COUNTER -L):

Given a < ctx, w, , CF>, CF is an associated counter-L with respect to if and

only if it is a counter-L with respect to and a counterfactual associated to .

Let us come back to counterfactuals (7) and (8):


at that time to force f."



As we have seen in sec. 2.4.2 ; in a usual context - for example,

of the kind (A) - they are non-counter-L counterfactuals, n- and

p-associated, respectively, to dynamics. (8) is different from

(7) in that the law supports (8), whereas it does not support

(7).

Let us examine our statement (7), always in the same contextual

situation, but now in a context of the kind (B): not usual,

forced, and conversationally interpreted in sec. 2.4.2. Let us

express it once more:

(7B) "If p had at time t an acceleration 2a (since q is at distance d of

p and both masses are mq and mp, respectively, and q attracts p according to the law

31

of gravitation, with force f), it would be subject at that time to force

f (therefore, dynamics would be failing)."

While we deploy this context, (7) becomes automatically true and

(8) false. In this forced context, which is obviously not usual,

(7) and (8) are openly counter-L (in this case, counterlegals).

In this context, the worlds sufficiently similar to w violate

dynamics. And this case is clearly different from the first one,

of sec. 2.4.1 (explicit counterfactuals), and from the second

one, sec. 2.4.2 (counterfactuals of miraculous contexts or

alleged CF that are not counter-L): (7) and (8) — in context (B) — are

counterfactuals associated to an L (in this case, a law) that is violated in worlds

sufficiently similar to the base world, and are counter-L with respect to the same L, with

respect to which they are associated counterfactuals, while in the first case and in the

second case this did not occur.

We will also call the associated counter-L "associated implicit

counter-L (Flichman 1985, ch. IV) since their property of being

counter-L follows from the context (and not only from the

contextual situation tacitly or explicitly included in the

antecedent — see note 9). It is true that the violation of an L

is partially implicit in the antecedent since the worlds involved

by the context are A-worlds sufficiently similar to the base

world. But it is the context that states which these A-worlds

are. If we remember thesis SUP, it is clear that these are the

counter-L we are interested in here (particularly, the

32

p-associated ones) since we are only interested in the

p-associated counterfactuals, and some of them, as we have seen,

are counter-L (implicit p-associated counter-L)

2.4.4 FOURTH CASE (IMPLICIT COUNTER-L NOT ASSOCIATED TO THE

CORRESPONDING L )

We shall see in an example that there are implicit counter-L not

associated to the corresponding L. In order to do so, let us go

back to our example (9) of sec. 2.4.1 in the same contextual

situation.


trajectory, slowing down while falling into the vacuum, and no

other force than its own weight had (noticeably) acted upon it,

then it would not have broken by striking the floor."

We have seen that this is an explicit counter-L, with respect to

the conjunction of the L (in this case, the laws) of gravitation

and dynamics. We have also seen that, regardless of the fact that

the antecedent is incompatible with the mentioned conjunction,

the deployed context may vary. This leads to the assumption that

either the law of gravitation or dynamics is satisfied (in the

A-worlds sufficiently similar to the base world).

33

Another example that illustrates this point is the following, in

the same contextual situation as (9):


trajectory slowing down while falling in the vacuum, and no other

force than its own weight had (noticeably) acted upon it, then

the law of gravitation would have failed.

In context (A), which is a usual context, (11) can be

conversationally interpreted as:

(11’) "If the crystal ball had moved in the second half of

its trajectory slowing down while falling in the vacuum, and no


then, (since dynamics is satisfied) the law of gravitation would have

failed."

(11’) and, hence, (11) are true in context (A).

In context (B) — a forced context — (11) can be conversationally

interpreted as:

(11") "If the crystal ball had moved in the second half of

its trajectory slowing down while falling in the vacuum, and no


34

then, (since the law of gravitation is satisfied, dynamics fails, so that it is not

true that) the law of gravitation would have failed.

In this forced context it is clear that (11") without the main

parenthesis — that is, (11) — is false.

In both contexts, (11) is an explicit counter-L with respect to

the conjunction of both laws. But the interesting point is that

it is an implicit counter-L with respect to the law of

gravitation in (A) and an implicit counter-L with respect to

dynamics in (B). However, it is important to observe that (11) is

neither associated to the law of universal gravitation nor to

dynamics. Therefore, in any of both contexts, it is an implicit

counter-L not associated to the law with respect to which it is

implicit. This is what we wanted to show.

We do not mean to say that there are not any examples of implicit

counter-L that are associated to the L with respect to which they

are implicit, and that are, at the same time, explicit counter-L

with respect to another L. But such a case would only be a mixed

counter-L, corresponding simultaneously to the first and third

case. We shall be interested in it only because it belongs to the

third case. However, our example suggests the existence of a

fourth case:

35

DEFINITION 6 (NOTION OF IMPLICIT COUNTER -L, NOT ASSOCIATED TO THE CORRESPONDING L):

Given a <ctx, w, , CF>, where CF is not associated to , we shall say that CF is

a non-associated implicit counter-L if and only if it is a counter-L with respect to

and is not an explicit counter-L with respect to (the antecedent is compatible with

).

We are not interested in this last case of counter-L, because

they are not associated to the L regarding to which they are

counter-L.

It is interesting to observe that, in the case of statement (11),

the violation of a law is directly expressed in the consequent.

If we examine (11) in (B), i.e., if we examine (11"), the

counter-L is a non-associated implicit counter-L with respect to

dynamics, whereas the consequent states the violation of the law

of gravitation. This shows that the consequent does not ascribe

the condition of counter-L to a counterfactual. (Except in the

case of a true counterfactual, as we shall see in Consequence 3,

sec. 2.4.5).

2.4.5 THREE CONSEQUENCES AND A PRECISION

Consequence 1: A counterfactual with an antecedent that is

incompatible with an L (whether conjunctive or not) shall be a

counter-L with respect to that L. This is the case of explicit

36

counter-L. Such cases, in which the counterfactual is not

associated to the L, are not interesting for the purpose of our

thesis.

Consequence 2: Even though the antecedent of a counterfactual CF is

not by itself incompatible with an L, CF shall be a counter-L

with respect to that L in a certain context, provided that, for

the world ordering deployed in that context, in some of the

A-worlds similar enough to the base world, that L is violated.

Statements (among them, the antecedent), that follow from the

context, are, together, incompatible with that L. But we are only

interested in the case where the counterfactual is associated to the L. Otherwise, we

would be dealing with a non-counter-L counterfactual in a usual

miraculous context (sec. 2.4.2), or with a non-associated

implicit counter-L (sec. 2.4.4). The only case we are interested

in is that of associated implicit counter-L (in particular,

p-associated).

Consequence 3: Another interesting consequence arises by analyzing

counterfactuals with a consequent contradictory to an L. A true

counterfactual with a consequent incompatible with L shall be a

counter-L, not because its consequent is by itself incompatible

with L, but because the counterfactual is true, since the

consequent is then true in the A-worlds sufficiently similar to

the base world. In these worlds, L is therefore a false

37

statement. Thus, one of the statemens that follow from the

context is the negation of the L.

On the other hand, if the counterfactual with incompatible

consequent is false, it may not be a counter-L, since, even

though its consequent is incompatible with the L, the consequent

does not hold in the sufficiently similar worlds, so that it is

not necessary that an L be violated. Case (11) is interesting in

context (B), i.e., (11"), for it is a counter-L with respect to

an L that is not denied by the consequent.

There may be cases of true counterfactuals with a consequent

incompatible with an L, like the counter-L of example (11), studied

in sec. 2.4.4 in a context — (A) in that case — that makes it

true. Another example: under forced context (B) in the contextual

situation of (7) and (8) it is true that:

(12) "If p had at time t an acceleration 2a, dynamics would be

false."

This is the case of the counterfactuals Rescher calls

"artificial", as we pointed out in sec. 1. Let us show now one of

Rescher's examples in (1964). The contextual situation is the

following one: = "All lions have a tail" is an accepted law

(true in w, according to our terminology). Besides, "Julius

Caesar was not a lion" and "Julius Caesar did not have a tail"

38

are statements about known facts (true statements in w, according

to our terminology).

(13) "If Julius Caesar had been a lion, there would have been a

lion without a tail (because Julius Caesar did not have a tail)".

The deployed context, indicated clearly by the conversational

parenthesis at the end of (13), is of the kind (B) of sec. 2.4.2,

i.e., forced miraculous.

But the only kind of relevant counter-L for our problem is that

of the implicit p-associated counter-L. Both (12), and (13) are

not: they are implicit n-associated counter-L.11 Consequently,

such counter-L are not interesting for our thesis.

One precision: Finally, we shall try to explain why our definition

of counter-L involves as a necessary condition that the L is

violated in some (and not necessarily all) of the A-worlds sufficiently similar to w.

Consider an example in which the contextual situation is the

following: p is a proton with, obviously, positive charge. On the

11 No counterfactual with a consequent incompatible with an L could be

p-associated to the L. Suppose is a lawlike statement from which the

negation of the consequent C is inferred. Therefore, . A ~ C. But

also, . A C, since it is p-associated. Both statements can only be

true if is incompatible with A. But then it is not associated because

of condition (II) of sec. 2.1 (Definition 1).39

other hand, let us suppose that : “All electrons have a negative

charge.” is a true L in w (let us suppose, for the sake of our

argument, that having a negative charge is not a part of the

definition of "electron"). We suppose also that is a law.12

(14) "If p were an electron, p would have a negative charge."

Now we are going to deploy, in the same contextual situation, a

context analogous to (A) of sec. 2.4.2, i.e., usual miraculous.

In the A-worlds sufficiently similar to w, p has a negative

charge. The counterfactual is true. It is not a counter-L. It is

p-associated to , and, because it is true, it is supported by .

The conversational interpretation of this context is the

following one:

(14’) "If p were an electron (since all electrons have a negative

charge), p would have a negative charge.”

Now, let us deploy, in the same contextual situation, a forced

miraculous context of kind (B) of sec. 2.4.2. In the A-worlds

sufficiently similar to w, p has a positive charge (the datum of

electrical charge is privileged, in detriment of ). The

counterfactual turns out to be false. In this context (14) is a12 Obviously, this is a base world without positrons (for, if electrons

did not have per definition a negative charge, and if positrons did exist,

they would be electrons with a positive charge, so that the L would not

be a law and, moreover, it would be false.40

counterfactual in a forced context13, p-associated to . It is an

associated implicit counter-L. Since it is false, it is not

supported by . The conversational interpretation of this context

is:

(14") "If p were an electron (since p has a positive charge, therefore,

an electron with positive charge would exist, so that it would not be true that), p

would have a negative charge."

Removing the conversational part between parentheses in (14") —

but keeping it in mind — we see that (14) is false in this

context.

Let us deploy now, always in the same contextual situation, a

context of a new kind, which we call (C): the A-worlds

sufficiently similar to w are classified in two very different

kinds. In some (at least, in one of them), p has a negative

charge. In the other ones (at least, in one of them), p has a

positive charge.14 The conversational interpretation of this

context is the following:

(14’’’) "If p were an electron (since, on one hand, all electrons havenegative charge, so that p might have had a negative charge; and, on the other hand, p

has a positive charge, so that an electron with a positive charge might have existed

13 Due to its extreme simplicity this example may seem not forced, or

not so forced for intuition.41

and p might have had a positive charge; we are not able to assert categorically that) p

would have had a negative charge."

In this (for many of us) very bizarre and forced context (C), the

counterfactual (14), which is p-associated to , is obviously not

supported by , since (14) is false. That (14) is false is clear

when we remove (even though we keep it in mind) the

conversational part in brackets.

This is the crucial point in order to decide to consider that

(14), in context (C) and in the same contextual situation, is an

implicit associated (would) counter-L. The reason for this is

intuitive rather than methodological, since here the intuitions

are very weak so as to decide whether it is a counter-L or not.

Indeed, if we postulated that (14) in the given contextual

situation and in context (C) is not a counter-L, we would have

found a counterexample for thesis SUP. Since a true L, that is a

14 Both kinds of worlds are, in a certain sense, "tied". By this term we

mean there are series of A-worlds progressively more similar to the

base world, such that, either: 1) there is at least one world that is

the most similar to the base world, in which p has a positive charge

and, therefore, fails; and there is at least another world, that is

also the most similar to the base world, in which holds and

therefore p has a negative charge; or 2) the limit assumption fails

and, for every A-world of the series, of the first kind, however

similar it may be, there will always be a more similar world of the

second kind and viceversa.42

law in w, would not be supporting its p-associated counterfactual.

We do not care if this example is considered too farfetched to

qualify as a counterexample. We do not need such kind of

counterexamples to show that SUP is not satisfied. That is why we

have included these cases among (associated implicit) counter-L.

3 OUR THESIS

Departing from Lewis’s semantic analysis (see note 4), hereafter,

"Lewis", and some other definitions given above, let us prove the

following thesis:

Thesis: Given a <ctw, w, , CF>, where CF is p-associated to , if CF is not a

counter-L with respect to then supports CF.

It is clear that, since we suppose that CF is p-associated to ,

should it be a counter-L, it would necessarily be an associated

implicit counter-L.

Let us remember the following definitions:

DEFINITION 1 (ANALYSIS OF THE NOTION OF COUNTERFACTUAL ASSOCIATED TO AN L):

CF is associated to if and only if

(i) The conjunction of with the antecedent of CF entails the consequent or its

negation.

43

(ii) The consequent of CF or its negation is not inferred from the antecedent alone.

(iii) The antecedent of CF is compatible with . (i.e., does not entail the

negation of the antecedent).

If, in particular, it is the consequent (not its negation) that is entailed from the

conjunction of and CF, we shall say CF is a positive associate of (abbreviated

“is p-associated”). In the opposite case (the negation of the consequent is entailed), we

say CF is a negative associate of (abbreviated “is n-associated”).

DEFINITION 2 (ANALYSIS OF THE IDEA OF SUPPORT):

Given a <ctx, w, , CF>, supports CF if and only if CF is p-associated to it and

CF is true. It does not support CF if and only if either CF is p-associated to it and is

false, or it is n-associated to it, whether true or false. Finally, if CF is not associated to it,

it does not make sense to ask whether supports it or not. That is, the category or

relation of being supported by does not apply to counterfactuals that are not

associated to .

DEFINITION 3 (NOTION OF COUNTER -L):

Given a <ctx, w, , CF>, CF is a counter-L respect of if and only if is false in

any of the A-worlds sufficiently similar to w, and ctx is not a usual miraculous context.

44

DEFINITION 5 (NOTION OF ASSOCIATED COUNTER -L, OR IMPLICIT ASSOCIATED COUNTER - L):

Given a < ctx, w, , CF>, CF is an associated counter-L with respect to if and

only if it is a counter-L with respect to and a counterfactual associated to .

Let us now prove our thesis:

1) Let us suppose a <ctx, w, , CF>, where CF is A � C. [hypothesis]

2) Let us suppose that CF is p-associated to [hypothesis]

3) Let us suppose that does not support CF. [hypothesis]

4) Thus, CF is false. [from 1), 2), 3) and definition 2]

5) Therefore, at least in some of the A-worlds sufficiently

similar to w, C is false. This is, ~C is true in at least one of

the A-worlds sufficiently similar to w. [from Lewis and 4)]

6) Moreover, A C. [from 2) and def. 1]

7) Thus, by modus tollens, ~(A), is inferred, in at least one of

the A-worlds most similar to w. [from 5), 6) and modus tollens]

8) But then the falsity of the conjunction is due to the falsity

of , for A is not false in such worlds. Therefore, is false in

some of the A-worlds most similar to w. [from 7), Lewis and

disjunctive syllogism]

9) Therefore, CF is a counter-L with respect to [from 2), 8),

and def. 3]

10) Therefore, if CF is p-associated to , and if does not

support it, then the counterfactual is a counter-L with respect

to [from 1) to 9) and deduction metatheorem]

45

11) Or, what is the same, if CF is p-associated to , and if it

is not a counter-L with respect to , then supports it. [from

10) by transposition]

Several obvious logical steps have been left out.

On the other hand, as we said above, from 2) and from definitions

3 and 5 it follows that CF is an implicit counter-L with respect

to so that "counter-L with respect to " could have been

substituted with "implicit associated counter-L with respect to

" in 9), 10) and 11).

In his (1994) John W. Carroll uses a principle that is in some

sense similar to our thesis. However, he does not apply it to

lawlike statements but only to laws, so that he does not arrive

to the conclusions we will put forth in sec. 4. Carroll’s

principle says that if P is physically possible (so that the

counterfactual: "If P had occurred, then Q would have occurred" is

not a counterlegal), and if P Q is physically necessary

(i.e., if "Whenever P occurs, Q occurs as well" is a law), then

the counterfactual "If P had occurred, then Q would have

occurred" is true. On the other hand, a) Carroll deals only with

explicit counterlegals. He arrives thus to different results from

ours, for we show that an implicit counter-L (particularly, an

implicit counterlegal) may not be supported. b) He refers to

counterlegals with respect to any law (i.e. those in which the

46

antecedent is not physically possible). But he cannot study cases

such as being associated to an L (or, particularly, a law) or

being a counter-L (or, particularly, a counterlegal) with respect

to another one. c) In Lewis’s semantics Carroll’s principle would

be satisfied only provided we suppose that the worlds where the

laws of the base world hold (i.e. the physically possible

worlds), are more similar to the base world than the others, and

provided we suppose that the context does not alter that

situation, or, at least, that there is one privileged context

with that ordering which is the only relevant one for the problem

of support. These are neither Lewis’s assumptions nor ours.

4 CONCLUSIONS: INVALIDITY OF SUP

In all of our examples, where the hypotheses of our proof were

satisfied, its conclusions were obviously satisfied as well. But,

moreover, in all of our examples the reciprocal holds, even

though we have not proved it (nor we know whether it can be

proven in general). In our examples, the basic hypotheses 1) and

2) are satisfied, and the associated counterfactuals either are

implicit counter-L with respect to an L, not supported by that L,

or else are supported but are not counter-L. A counterexample for

the reciprocal would be a case of an implicit counter-L

associated to an L that is supported by the L. We do not know

whether it is possible to prove the reciprocal. We do not know if

such counterfactuals do exist, but, if they did exist, we would

47

not wish to say that the counterfactual is supported by the L,

for it would seem that it is not the truth of L that determines

that both C and (therefore) the counterfactual are true. So that,

should such examples exist, we would have to reformulate the

definition of "support" so as to exclude them. If we do so, the

counterexample for the reciprocal would disappear.

Our thesis shall enable us to show the invalidity of SUP.

On one hand, given a <ctx, w, , CF> such that CF is p-associated to

and regardless of whether is a law or a uniformity, supports CF, but

provided that CF is not an implicit counter-L associated to . This

is our thesis. We have presented several examples. On the other

hand, we have shown examples in which the counterfactuals

p-associated to a true L in w, in certain contexts, in which they

are not implicit counter-L associated to , are not supported by

.

Does a lawlike support counterfactuals? Indeed, it does support

some counterfactuals, namely, those p-associated to it, which,

moreover, given the corresponding context, are not counter-L.

This assertion is indeed true, since it is analytic; but its

analyticity makes it trivial.

The same can be said regarding any L we wish to examine in order

to determine whether it is a law or a uniformity. There will be

48

certain counterfactuals, p-associated to , that may be true in

certain contexts (in which they are not counter-L), and false in

other contexts (in which they are counter-L).

We have not proven that laws support their (p-associated)

counterfactuals that are not counter-L and that uniformities do

not. On the contrary, what we have proven is that any true L of a

theory, regardless of its being a law or a uniformity (provided

that some other criterion — that has nothing to do with

counterfactuals — enables us to recognize it as such) supports

the counterfactuals that are p-associated to it and that are not

(in the given context) counter-L.

So that every law supports its p-associated counterfactuals that are not counterlegals in

the given context, and every uniformity supports its p-associated counterfactuals that

are not counteruniforms in the given context.

If, under the influence of some intuitive (or intuition-

generating) criterion that is totally independent of the subject

of counterfactuals, we suppose that L is a law (like in example

(8)), then, regarding a usual context, the law supports the

counterfactual CF it is p-associated to: CF is true and not

counterlegal. In the same example, instead, regarding the forced

context we studied from sec. 2.4.2 on, the same law does not

support CF: CF is false and counterlegal (implicit associated).

The point is that the fact that the first context is usual,

49

whereas the second one is forced, is determined, precisely,

because of our previous knowledge, and, consequently, our

previous intuition, that L is a law. The same intuition considers L as a

law and the context as usual. It is this intuition that makes us

consider that the law and not its violation holds in the A-worlds

similar enough to the base world. That is why it is so difficult

to place ourselves in the forced context and consider that the

law is violated in the similar enough A-worlds, thus making the

counterfactual a counterlegal. This result is intimately

connected to the pragmatic rules used by the speakers (implicitly

or explicitly) while they process the context of a counterfactual

(See note 6).

However, the inverse situation, regarding uniformities, does not

necessarily have to occur. If we have a criterion, independent

from the subject of counterfactuals, allowing us to consider a

true L as a uniformity, such a criterion generates an intuition

(or rather is generated by an intuition) that helps us fix the

usual context. This does not mean that the context necessarily

locates the violation of the uniformity in the A-worlds

sufficiently similar to the base world, for the greatest

intuitive weakness involved by considering L as a uniformity,

does not necessarily generate the violation of L in the A-worlds

similar enough to the base world (see note 6). Therefore, in case

we are dealing with a counterfactual that is p-associated to an

L, which in turn we consider as a uniformity, there are two

50

possible situations: there will be examples in which the

counterfactual is false in a usual context, whereas it is true in

a forced context; and there will be examples, too, in which the

inverse occurs. We are going to examine examples for both

situations.

So that the same criterion of demarcation, and, consequently, the

same demarcatory intuition induced by (or, inducing) it, makes us

consider an L as a law, on one hand, and makes us select a usual

context in which the counterfactual p-associated to that L is

true (i.e. the L supports it), on the other hand. Whereas the same

intuition induced by (or inducing) the same criterion makes us

consider another L as a uniformity, on one hand, and select a

(usual) context in which the counterfactual p-associated to this

L is either false (the L does not support it) or true (the L does

support it).

Up to this point we have studied examples of (alleged) laws. Now

we shall study (alleged) uniformities:

(15) "All members of the Board of Directors of the Boca Juniors

Club of Buenos Aires during the period 1993-1995 are bald during

the whole period."15

15 The same example has been used in (Flichman 1995) with a different

purpose.51

Of course, we do not believe that being bald during that time is

a part of the definiens of being a member of the Board of Directors

of the Boca Juniors Club of Buenos Aires during the period 1993-

1995. On the other hand, we consider (in the contextual situation

of the example) that each member is bald because of independent

reasons, that do not have anything to do with the fact of being

elected for this position. Also, John is neither bald nor a

member.

(16) "If John had been a member of the Board of Directors of the

Boca Juniors Club of Buenos Aires during the period 1993-1995,

then John would have been bald."

The base world w is a world that is similar to the effective

world (ours), except that in w (15) holds. On the other hand (16)

is clearly false in a usual context.

Since we firmly believe that (15) is not a law, but a uniformity,

that is, since we have a criterion inducing (or induced by) our

belief, we are led to fix a context for counterfactual (16) —

that is p-associated to (15) — which makes (16) false. This is

because, in this usual context, in the A-worlds sufficiently

similar to the base world, there is privilege in favor of certain

properties of an individual (namely, that John is not bald) and

in detriment of the uniformity (15), which is then less relevant

(than the individual's properties) for the similarity among

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worlds (see note 6). Therefore, (15) does not support (16) in

this (usual) context.

It is clear that, had we assumed, instead, that (15) is a law, we

would have completed the context so that the privilege relation

would be inverted. In such a case, (16) would be true in a usual

context, as in the following analogous situation: in a usual

context it is true that, if this radiation traveling in the

vacuum with speed smaller than c had been light, it would have

had speed c (where c is the speed of light in the vacuum).

Let us observe that, under the firm belief that it is a

uniformity, (16) is an implicit associated counter-L in a usual

context, and is thus an implicit associated counteruniform. In

the case of laws, we studied cases of implicit associated

counterlegals — (7) and (8) – but in those cases the context was

forced.

Instead, (16) is true in a forced context, that does not make it

a counter-L (i.e., counteruniform). This is, conversationally

interpreted.

(16') "If John had been a member of the Board of Directors of

the Boca Juniors Club of Buenos Aires during the period 1993-

1995, then (since all members of the Board of Directors of the Boca Juniors Club of

53

Buenos Aires during the period 1993-1995 are bald) John would have been

bald."

(16'), which is p-associated to (15), turns out to be true in

this context: (15) supports (16').

In this case, the situation is inverse to that of laws. Laws do

support their p-associated counterfactuals in a usual context, in

which the latter are neither counter-L nor counterlegals.

For example, (8) is p-associated to dynamics, it is supported by

it, and is true in a usual context (it is not a counterlegal). On

the other hand, (8) is not supported by the law, is false in a

forced context (it is an implicit associated counterlegal).

Instead, (16) is p-associated to the uniformity (15), and is

supported by it — is true in a forced context, as (16') – that

is, it is not a counteruniform. Whereas (16) is not supported by

(15), it is false in a usual context (it is an implicit

associated counteruniform).

Nevertheless, as we have said before and shall illustrate

immediately, the example (15)-(16), which we have just examined,

is not general for all uniformities. Let us examine a variant of

it; namely, the same statements (15) and (16), but in a different

contextual situation:

54

All members of the Board of Directors of the Boca Juniors Club of

Buenos Aires during the period 1993-1995 had hair before taking

over, but the swearing-in ceremony was held the day before the

new authorities took over, in a room containing radiation; this

radiation causes baldness after an hour for at least two years.

Therefore, from the moment the new members took over until the

end of their functions, all of them were bald.16

In this particular contextual situation, statement (15) is still

a uniformity, for it is a greatly localized statement, with

extremely few determined and finite instances. The statement

asserting that all people with hair exposed to such radiation

experience hair loss after an hour and for at least two years,

can be considered a law. But we are not dealing with such a

statement but with (15). Let us note that it is accidental that

the swearing-in ceremony took place in a room that had been

accidentaly irradiated. Nobody would say that (15) is a law, in

this contextual situation. But it is indeed related to the law

that we have just mentioned. It is clear that, in this case, for

the counterfactual (16), statement (15) is not going to be

violated in the A-worlds similar enough to the base world in a

16 This variant of the example has also been discussed in (Flichman

1995), where it is held that it is not (15) which supports the

counterfactual. Flichman, co-author of this paper, has changed his mind

about the matter. Nevertheless, the analysis of the problem is not a

subject of the present article. 55

usual context. The law stating the effect of radiation is

stronger for intuition than the fact that the members of the

Board of Directors had hair before swearing-in (see note 6).

Therefore, (15) supports (16) in a usual context, and ceases to

support it only in a forced context. (16), in a usual context, is

true and is not a counteruniform; it is false and a

counteruniform only in a forced context.

We have shown both kinds of examples for uniformities. In one of

them, the behavior of the uniformity is opposite to that of laws.

In the other, uniformities behave just as laws do.

This explains why usually (i.e. in usual contexts), laws support

their p-associated counterfactual17. But this also explains why it

cannot be said that uniformities have the opposite behavior.

Therefore, it becomes clear that thesis SUP is not valid.

Moreover, if the situation of uniformities had been opposite to

that of laws in all the cases, such a situation would not have

qualified as a criterion of demarcation. It would have been a a

posteriori criterion (and, therefore, useless qua criterion). It would

have worked only in usual contexts, once we know whether it is a

law or a uniformity, i.e., once we have previously applied

another criterion of demarcation.17 This would bring us to a position similar to alternative SUP3 (at

least, if we claim that there is an independent criterion to

distinguish laws from uniformities), but only in the case of usual

contexts.56

Therefore, the possibility of using the truth or falsity of a

counterfactual that is p-associated to a true L, in order to know

whether it is a law or a uniformity, is discarded. The authors of

this paper are not in agreement as to the existence of some law-

uniformity demarcation criterion (see note 2 in relation to this

point). But certainly, should a criterion exist, it is not

criterion SUP.

NOTES

REFERENCES

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York.

Carroll, J. W.: 1994, Laws of Nature, Cambridge University Press,

Cambridge / New York / Melbourne.

Flichman, E. H.: 1985, La causación: ¿Ultimo reducto del antropomorfismo? -

Estudio crítico del análisis contrafáctico de la causación entre eventos, presented at

SADAF (Sociedad Argentina de Análisis Filosófico). Photocopy.

57

Flichman, E. H.: 1990, ‘A Crucial Distinction: Initial Data and

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Nagel, E.: 1961, The Structure of Science: Problems in the Logic of Scientific

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Nute, D.: 1980, Topics in Conditional Logic, Reidel, Dordrecht /

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Ramsey, F. P.: 1929, ‘General Propositions and Causality’, in his

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Rescher, N.: 1964, Hypothetical Reasoning, North Holland, Amsterdam.

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Ciclo Básico Común (University of Buenos Aires)

Eduardo H. Flichman

Santos Dumont 2475 - 12 - B

(1426) Buenos Aires - Argentina

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Laws: Counterfactual demarcation overruled

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