Stephen Downes

Knowledge, Learning, Community

Calgary Working Papers in Linguistics, Mar 23, 2023

This Journal Article published as 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

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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.

Mentions

- , Mar 23, 2023
, - Conditional variability, Mar 23, 2023


Stephen Downes Stephen Downes, Casselman, Canada
stephen@downes.ca

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