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*Conditional Variability*in Calgary Working Papers in Linguistics Number 13 1-13 Sept 23, 1987. University of Calgary [Link] [Info] [List all Publications]

Conditional variability

Downes, Stephen

University of Calgary

Downes, S. (1987). Conditional variability. Calgary Working Papers in Linguistics, 13(Fall), 1-14.

http://hdl.handle.net/1880/51345

journal article

Downloaded from PRISM: https://prism.ucalgary.ca1.0 Introduction

*Scanned from paper and contains numerous OCR errors.*

1.0 Introduction

Conditional statements are presumed to be understood by both

speaker and listener. If the conditional statement is true, all parties

concerned presume that the statement is true for some reason. When a

speaker asserts, "If kangaroos bad no tails they would topple over," then

both speaker and listener know eiactly what the statement means and

whether or not it is true even if there are in fact no kangaroos in the world

without tails. Asserting that conditional statements are understood and

known to be true or false is one thing. Stating eiactJy what it is that is

understood or known is quite another.

In this paper it will be shown that when a conditional statement is

understood or known to be true, a number of implicitly specified variables

are given more or less concrete values. Each of the variables will be defined

and examples will be employed to demonstrate their use in conditional

evaluation. From time to time this analysis in terms of variables will be

contrasted with a 'possible worlds' analysis of conditionals. The purpose of

this paper is not to argue against the possible worlds analysis but rather to

provide an alternative to that analysis.

2.0 Backgound

In logic conditional statements are symbolized (A->B) and are of the

general Corm "If A is the case then B is the case." They are false if and only

if the antecedent. A. is true and the consequent. B. is false.

There are two types or conditional statements: the material

conditional. symbolized as above or sometimes with a 'hook' symbol; and the

strict conditional. which asserts the necessity of the corresponding material

conditional. symbolized variously with the 'fish-hook' or entailment symbol.

Conditional statements were intended to correspond with similar

statements in natural lanauage: the idea was that sentences like "If it rains

I'll get wet" could be represented in formal notation and given truth valuesby deduction from other formalized sentences. · This ambition was never

realized. A larae dass ot ClODditional statements, called variousJy

'subjunctive ClODditionals' or 'counterfactual1' resisted anaJysis into the strict

or material ClODditional form. By 'c:ounterfactuals' I mean the following forms

of conditional statements: statements with false antecedents such as "If

Oswald had not shot Kennedy then he would be alive today", causal

statements and statements which predict into the future such as "If it rains

the river will rise", and subjunctive statements such as "If he had ambition

he would go far."

The failure to anaJyze c:ounterfactuals in terms of material or strict

conditionals has two related causes. Pirst. many c:ounterfactuals, although

true, are not necessarily true. There are some instances in which the

antecedent may be true, the consequent false, and the statement as a whole

true. Second, many laws of inference such as 'strengthening the antecedent'

which are valid for material and strict conditionals are not valid for

c:ounterfactuals.

A recent development in philosophy has been the anaJysis of

c:ounterfactuals not as material or strict conditionals but rather as a distinct

conditional connective with its own rules of inference: the variably strict

conditional (see Lewis 197 3a and Stalnaker 1968 ). This analysis comes with

a price: the truth of a variably strict conditional is determined on the new

analysis not by the state of affairs in the world but rather by the state of

affairs in a possible world. The possible world selected is one in which the

conditional is no longer counterfactual - what was false bas become true,

what was in the future has now occurred - and is selected on the basis of

relevant similarity with the actual world if the c:or'responding conditional is

true in the possible world.

The possible worlds anaJysis of conditional statements has severe

problems. How can we select a possible world on the basis of similarity if at

least part of that similarity might depend on the truth of the very

counterfactuals we are trying to anaJyze? I am not concerned to press that

argument here. Rather, I wish to focus on an alternative. The suggestion is

this: counterfactuals are variable because of implicit variables in the

counterfactual conditional relation. The• variables, if stated eiplicitJy, may

be employed in part to provide the framework of an anaJysis of

counterfactuals which occurs in this world and not in some possible world.

2

•

•3.0 Component Strength

Conditional statements may vary in strength according to the truth

values of the components A (the antecedent) and C (the consequent). I am

not concerned in this paper with how the truth of A and C is established. I

merely wish to indicate that, if determinable, it is determinable in a variety

of manners.

Let us first consider the antecedent A. The antecedent A may be true

or false. In the latter case, the conditional is called a counterfactual. or more

precisely, contrary-to-fact conditional. In cases where A is true, although

the term ·counterfactual' still misleadingly applies, perhaps the term 'factual

conditional' is more appropriate.

There are also some cases in which the antecedent may be

undetermined or undeterminable; consider, for example:

( 1) If it rains tomorrow the crops will grow.

The antecedent "'it rains tomorrow" is neither true nor false, for tomorrow

has not yet occurred. An antecedent which is a tautology will always be

true; an antecedent which is a contradiction will always be false. Depending

on the semantics chosen, there may be a wide range in between.

Like the antecedent. the consequent may be of varying truth value.

In many cases (and most especially in many of the eiamples we choose to

discuss) the consequent is known to be true or false. Bven if the consequent

is false the conditional itself may be true. This is most clearly demonstrated

by the material conditional: if A is false and C is false then A->C is true.

For our purposes the most interesting cases are those in which the

truth value of the consequent is not known. undetermined, or in some other

way not certainly true and not certainly false. The recognition that the truth

values of the components of conditionals may vary serves almost

immediately to prevent some philosophical errors. For example, Eisenberg

( 1969) argues that all counterfactuals must be eiplicable by the 'conjunction

analysis' as follows:

(2) ((1) (MI->Pl)) & (MZ->Pz) & (-MzJ & (-PzJ

The relevant portions of this analysis for this discussion are the negations

1-Mzl and (-Pz). As Williamson ( 1970) points out. the antecedent and the

consequent need not be false for the statement to qualify as a counterfactual.

Suppose the follovina ezample:

(3) If Wayne bad been here it would have been a good party.

The counterfactual may be true. but someone may respond,

(-4) Wayne was here and it was a good party, but you were in the kitchen all

night. didn't see him. and missed all the fun.

We may thus allow that the truth values of the components may vary. The

components may be absolutely true or false or. depending on the semantics.

anywhere in between.

Let us eiamine a little more precisely the ways in which the truth

values of the components may vary. First, the components of the

counter!actual may vary because the truth of various propositions is

variable in the world. For eiample, something may be 'possibly' true. It

might rain tonight, for eiample. That does not mean that it is true. but it is

also misleading to say that it is false. That "it will rain" is possibly true is a

fact about the world; the truth value of "it will rain" is therefore variable.

Sometimes propositions which are 'possibly' true might have their

truth values fiied more precisely in terms of 'probability'. For eumple,

'This atom of uranium will decompose" is a statement which bas a certain

precise probability of being true. Such a probability value is not arbitrary:

the rate of uranium decomposition just is a probability function.

Second, components of a counterfactual may also vary because of bow

much (or how little) we know about the world. The most common

occurrence of this is in the statement of statistical hypotheses such as 'The

NDP is supported by fourty per cent of Canadians." Variable truth values in

such instances are ezplicitly stated: "samples of this siZe are accurate to five

per cent nineteen times out of twenty."

In other cases the variable quality of our knowledge of some

statement cannot be so precisely measured. Statements like 'Tm reasonably

certain" or "I have little doubt" t1press this. The nature of our determina-

.lion affects this variable: if we see that the car is red we are quite certain

that the car is in fact red; if we are told by a friend that the car is red then

we are less certain.

4.0 Salience

Counterfactual truth may vary as determined by relevant or 'salient'

factors. These factors are best described using an eiample. Consider the

foUowing pair of counterfactuals (from Quine 1960:222):

(5) If Caesar had been in command (in Korea) he would have used the

atom bomb.

(6) If Caesar had been in command he would have used catapults.

By 'context' we mean the situation in which one or another of these

counterfactuals would have been asserted: a political science class. perhaps.

or a history seminar. 'Salience' is determined by context. It refers to those

qualities of Caesar which are the most important to the discussion taking

place. Which or (5) or (6) is true will depend on what quality of Caesar's is

most salient. In this case. if Caesar's primitive knowledge of technology is

most salient. then (6) will be true. If Caesar's ruthlessness is most salient,

then (5) would be true.

On the possible worlds analysis, statements about Caesar's use of the

bomb or catapults are analyzed as above in terms of salience and context.

The possible world selected for reference will be the one which is most

similar to the actual world with respect to these salient qualities. On the

analysis presented in this paper, salience is employed directly in the

determination or truth values for counterfactuals. Salience is presented in

terms of a closely related notion, vagueness.

To show bow this worts, let me consider an eiample.

(7) If it reaches --40 tonight, Calgary will be the coldest city on the Prairies.

It might reasonably be argued that Calgary is not on the Prairies; rather it is

in the foothills. and so could not be the coldest city on the Prairies no matter

what. Whether or not this counterfactual is true depends on bow 'Prairies' is

defined. Since it is not a precise geographical region its boundaries are

vague. On some accounts, Calgary is on the Prairies, on others it is not. In

fact 'Prairies' refers to not just one geographical area but many, each

differently defined. Some such definitions are not complete definitions; the

eastern border ot the Prairies is not defined at all but the western border is

defined as 'east ot Calgary'.

On Utis ana1ysis we mean by 'sallenc:e' the specification of eiactly

wbich of tbe varying specifications at some vque term will be employed.

What we know at Caesar ii vque at bett. We know that Caesar lived in

ancient Rome and that he wu a brilliant thoup somewhat ruthless tactician.

To assess the pair of counterfactuals above we must define Caesar more

precisely: "Caesar the ancient Roman" or "Caesar the ruthless". If choosing

betwffn either of the two options we may have to consider the truth values

of each proposed definition of Caesar. These may vary just as truth values

for the different components of a counterfactual vary.

5.0 Connective Strength

The strongest form of the conditional connective is necessary

implication. That is, if A is true and known to be true and the connective is

expressed A->C then C must be true and known to be true. Both the material

conditional and the strict conditional are conditional connectives of this form.

If the conditional is true. then if the antecedent is true, the consequent must

be true. Showing one instance in whlch the antecedent is true and the

consequent false shows that the conditional is false.

As discussed in section 2 above, both Lewis and Stalnaker propose a

third type of conditional, tbe variably strict conditional. Tbis conditional is

employed to symbolize what we mean when we use counterfactuals. It

should be dear that the strength at the variably strict conditional does not

tie somewhere betwHD the strength of the material and strict conditionals.

for the strengtb of the latter two is identical. The variably strict conditional

is a form of conditional wbich hu a weaker connective strengtb than either

the material or strict conditional. This difference may be characterized as

follows. If A is the antecedent and C is the consequent and A->B is the

variably strict conditional. then if A is true C mjpt not be. The variably

strict conditional is not necessarily truth or falsity preserving. We may

illustrate tbis using the previously mentioned ruJe at strengthening the

antecedent.

Suppose some conditional statement (A->B) is true. According to the

rule of strengthening the antecedent, if some C is conjoined with the

antecedent A then the resuJtant conditional ((A&C)->BJ remains true. This

law is valid for material and strict conditionals but not valid for variably

strict conditionals. Conjoining some C to the antecedent can change the truth

value of the corresponding conditional. The variably strict conditional may

be more or Jess strong depending on how much or how little nHds to be

added to the antecedent to cause a chqe in truth value.

6

•

•It would be a mistake, I suggest. to suppose that there is only one

type of variably strict conditional. They might be quite strong or they may

have no strength at all. The failure to recognize this latter possibility lies at

the heart of many criticisms of Lewis and Stalnaker. Consider. for eiample,

the following argument proposed by Bennett ( 1974). According to Bennett,

Lewis's analysis fails in the case of the 'accidental' even-if conditional.

Consider the following conditional:

(8) If London is a large city then Jupiter bas twelve moons.

If is the case that Jupiter would have twelve moons whether or not London

were a large city. If London actually is a large city then, on the possible

world account, we should check and see whether Jupiter actually has twelve

moons; if it does the conditional is true. If London is not a large city, then

according to the possible worlds theory we should consult the nearest

possible world in which London is large and count the moons of Jupiter; if

there are twelve then the counterfactual is true. On the possible worlds

story there would in fact be twelve moons since the size of London does not

affect the number of moons possessed by Jupiter.

Bennet argues as follows. While it is consistent to maintain that, in the

nearest possible world, Jupiter has twelve moons. it is also consistent to

maintain that. in the nearest possible world, Jupiter has thirteen moons. The

truth of the accidental conditional is thus, to Bennett, undetermined. Bennett

employs this argument to support the alternative ·regularity' theory of

counterfactuals. But the regularity theory demands that. if there is no

regular relation between the antecedent and the consequent, the

counterfactual is false. But why should we say that? It is not determined

that Jupiter has twelve or thirteen moons given that London is or is not a

large city and so the conditional is neither true nor false.

The accidental conditional is an e1treme case. There is no strength to

the connection. The conditional is therefore possibly true and possibly false.

nothing more. At the other e1treme are the material and strict conditionals.

The conditional is necessarily true or necessarily false. It is reasonable to

suggest that a range of possibilities lies in between. I will suggest just a few

of them. Natural or physical laws may be one eiample. The laws of nature,

as Hume demonstrated, are not necessary laws. Many such laws, such as

Newton's laws, once considered true, are now generally considered false. We

consider the possibility or failure to be a factor when evaluating currently

accepted laws. It is not a lOBical contradiction to entertain their falsity. The

7variable strength of such laws is sometimes e1pressed in an e1pllcitJy

conditional form: if true. a law. Though weaker than the strict or material

conditional. the conditional which eipresses a law of nature is nonetheless

stronger than an accidental generalization.

A further variation of strength may be the case of non-lawliJce non-

accidental conditionals. The "'dimes in the pocket case" is one such case.

Suppose I put my band in my podet on Canada Day, 1987, and retrieve a

bandfuJ of dimes. It is true that, in Canada, a1J dimes are made of nicteJ. I

could then say:

(9) If I had put my band in my pocket on Canada Day, 1987, a1J the coins I

would bave found would have been made of nicteJ.

This clearly is not necessarily true. It does not even appear to have the

strength of a Jaw of nature. But neither is the conditional an accidental

conditional: there is some sort of connection between placing my hand in my

pocket and touching nic.keJ.

Although it seems clear that different strengths of a conditional

connective are possible. it is not clear how to quantify that variable. What

we want is a syntu which will first alJov for such a range of values and

second determine a syntactic relation between the varying strengthed

conditionals within that range. In the neit section I shalJ outline a syntactic

structure which permits tbis determination.

6.0 The Domain ot the Conditional

What mates a necessary statement necessary? On the Leibnizi.an

thesis a statement is necessary if it is true in all possible worlds. In condi-

tional terms, a conditional is necessarily true (is oC greatest strength) if it is a

universal statement. We have seen that not all conditionals are necessarily

true; there are varying shades ot strength. Therefore universality, a

condition suggested by a number oC analyses and the first conjunct of

Eisenberg's, above, will be sufficient to describe only some small number ot

counterfactuals.

A Jaw of nature is not a necessary statement. On some possible worlds

a law of nature might be different from the laws oC nature in the actual

world. But laws of nature are e1pressed in the general form

(I 0) For a1J 1, if F1 then G1.

8

•

•Both necessary truths and laws oC nature employ the universal quantifier.

Mere use oC the universal quantifier will not be sufficient to distinguish

between the two. A finer distinction is required. Let me suggest the

following.

Consider the size of the domain of the conditional: that is, within what

world, worlds, or parts of worlds a conditional is intended to be true. A

necessary conditional is intended to be true in all possible worlds. A lawlike

statement is intended to be true all over this, the actual, world. We might

say that universality expresses the success rate of a conditional within its

intended domain. The strength of the conditional may therefore be

evaluated according to these two variables: the size of its domain, and its

success rate within that domain. A reduction of the domain or a reduction of

the success rate may weaken the conditional connective. Exactly how this is

to be spelled out is probably a fascinating task and I hope one day to finish

it.

7 .0 Propositions

During the course of this paper I have not clearly distinguished

between counterfactual propositions and counterfactual statements. Let me

accomplish this now. The proposition

( 11 ) Brakeless trains are dangerous.

does not refer only to one train but rather to a large number of trains. It is

expressed counterfactually as follows:

( 12) If any train has no brakes then it is dangerous.

The proposition expressed by ( 12) is intended to correspond with specific

'instances', in this case, specific trains. as follows:

( 13) If train I has no brakes it would be dangerous.

( 14) If train 2 bas no brakes it would be dangerous.

( 15) If train n has no brakes it would be dangerous.

The idea is that if each ct the instances is true then the proposition as a

whole is true. But propositional truth is not an all-or-nothing venture; some

instances may be false while the proposition is true. Suppose, for eumple,

( 16) Train '4489 has no brakes and is not dangerous.

Train 4489 also bas no engine and bas not moved since 1959. Bven tbougb

( 16) is an exception to the general rule that does not mean that the

proposition is false. It is true in most cases.

A proposition is a statement that corresponds to more than one

instance. Since not all instances need be true for the proposition to be true

the strength ct a proposition may vary. Propositional variability may be

quantified according to the domain of the proposition and the success rate

(proportion ct true instances) within that domain.

a. Causal CouatertactuaJ1

A great number ct the counterfactual propositions we assert every

day are causal propositions. By that I mean the assertion that some A causes

some B to occur. Causal propositions, lite other propositions, correspond to a

set of instances. If we assert that A causes B then we assert that A1 causes

82, and so on.

There remains a problem to be resolved. Suppose you heat some

water. The water boils: that is, little bubbles form and steam rises. The

cause of the water boiling is the heat: the symptoms are tbe steam and

bubbles. We could say, quite acturateJy, that the heat caused the bubbles

and the steam. But now it is equally possible to say that, if there are

bubbles, then there will be steam; that is, that the bubbles cause the steam

to rise. The relation between the heat and the steam is quite different from

the relation between the bubbles and the steam; the nrst is a causal relation,

the second an apparently accidental relation.

At the same time, however, the strengths ct the two conditionals will

be the same. That is, the domain in both cases will be the same (the system

desaibed above). The universality will be the same. The truth values ct

each instance ct this proposition will be the same. Yet typicaJJy we assert

that the causal relation is stronger than the accidental relation. The

distinction between the causal conditional and the accidental conditional is

contained in the idea of 'causal dependency'. The idea is that the steam and

the bubbles depend on the heat, and not each other. in order to occur. A

10relation of dependency is an asymmetric relation. That is. if A depends on B

then B does not depend on A. Accordingly we test for dependency by testing

pairs of counterfactuals: (A->B) and (B->A). But both (A->B) and (B->A) will

be true in e11ctly the same instances even in relations of dependency.

We have to consider the contraries of both: (-A->-B) and (-B->-A)

(Lewis 1973b). If a relation of dependency eiists then in some instances

where the effect B is not present the cause A will be present and yet in very

few instances where the cause A is not present will the effect B be present.

The causal proposition is therefore a complei proposition which depends on

the truth values of four corresponding counterfactual propositions. More

formally if

(17)A->B

is a causal proposition then the four corresponding propositions will be

(18) A->B

(19) B->A

(20) -A->-B

(21) -8->-A

each of which will be given a truth value which corresponds to the number

of instances in which it is true.

Lewis ( 1973b) eipresses this theory within the conteit of a possible

worlds analysis of counterfactuals but it is not necessary to refer to a

possible world to establish the variably strict truth of each of the

propositions in question. We therefore retain the strength of Lewis's

proposal while omitting the weakness.

9.0 Summary

In this paper it has been shown that a number of variables are

implicitly given conaete values when a conditional statement or proposition

is asserted.

First, the antecedent and the consequent of the conditional may

have varying truth values depending on how certain they are in the world

and how well they are known.

Second, features of the world which are

relevant to the evaluation of the conditional which are more or less vaguely

defined will be defined precisely.

Third, the strength of the conditional

connective will vary depending on its intended domain and its intended

success rate within that domain.

Fourth, conditional propositions which

correspond to sets of instances will vary with respect to the number of

instances over which the conditional is intended to be true.

Fifth, some

11conditional propositions will correspond to sets of several other conditional

propositions and will be evaluated with respect to the truth value of each of

the other conditional propositions.

GiVen a clear specification of each of these variables it is possible to

state euctly what is understood when a conditional statement is understood.

In addition, such a clear specification of the variables will specify euctly

what must be true for the c::oriditional to be true. It should be understood

that conditional truth is not an all-or-nothing venture and that some

conditionals will be partly true or even have no truth value at all. depending

on the variables. None of these variables requires reference to some

possible world for specification. Therefore the analysis proposed in this

paper provides a viable alternative to the possible worlds analysis.

References

Bennett, Jonathon. 1974. Counterfactuals and possible worlds. Canadian Journal of Philosopht 4:381-402.

Eisenberg, J.A. 1969. The logical form of counterfactuals. Dialogue 7:568-583.

Lewis, David K. I 973a. Counterfactuals~ Harvard University Press.

Lewis, David K. 1973b. Causation. The Journal ofPhilosophy 70:556-567.

Quine, W.V.O. 1960.Word and Object. MIT Press.

Stalnaker, Robert. 1968. A theory of conditionals. In N. Rescher (ed.), Studies in Logical Theory. Basil Blackwell.

Williamson, Colwyn. 1970. Analysing Counterfactuals. Dialogue. 8:310-314.