The Logic of `If' -- Or How to Philosophically Eliminate Conditional Relations

Sorites (Σωρίτης), ISSN 1135-1349
http://www.sorites.org
Issue #19 -- December 2007. Pp. 51-57
The Logic of `If' Or How to Philosophically Eliminate Conditional Relations
Copyright © by Rani Lill Anjum and Sorites
The Logic of `If'
Or How to Philosophically Eliminate Conditional Relations
by Rani Lill Anjum

[Abstract]

Introduction

In this paper I present some of Robert N. McLaughlin's critique of a truth functional approach to conditionals as it appears in his book On the Logic of Ordinary Conditionals. Based on his criticism I argue that the basic principles of logic together amount to epistemological and metaphysical implications that can only be accepted from a logical atomist perspective. Attempts to account for conditional relations within this philosophical framework will necessarily fail. I thus argue that it is not truth functionality as such that is the problem, but the philosophical foundation of modern logic.

1. Six principles of logic

In classical logic before Frege, there were three basic principles of logic that Copi calls «The three laws of thought»:Foot note 1

.The principle of identity: If a statement is true, then it is true.
The principle of contradiction: No statement can be both true and false.
The principle of excluded middle: A statement is either true or false.

With the introduction of Fregean logic, three new principles were added. This made modern formal logic purely extensional:

The principle of composition: The meaning of a complex expression is determined by its structure and the meanings of its constituents.
The principle of truth functionality: The truth-value of a complex expression is determined by its structure and the truth of its constituents.
The principle of form: The logic of a complex expression is determined by its (syntactic) form, not its (semantic) content.

2. The material conditional

Although Fregean logic has some obvious advantages over Aristotelian logic, being formally more advanced and flexible, some problems follow from the introduction of the three extensional principles. For instance, where one within classical logic can distinguish between the two statements `If Socrates is a man, then he is mortal' and `If Socrates is mortal, then he is a man' by judging the first as true and the second as false, Fregean propositional logic will make both these statements come out as true, since Socrates was both a man and mortal.

This is commonly thought to be an unfortunate consequence of applying the principle of truth functionality to conditionals. Interpreted truth functionally, the only combination that makes a conditional `if p then q' false is when the antecedent `p' is true and the consequent `q' is false at the same time, something that renders the conditional true for all other combinations:

A conditional is true whenever its antecedent is false, which means that for instance the inference `not-p, therefore, (if p then q) and (if p then not-q)' is valid.
A conditional is true whenever its consequent is true, which means that for instance `q, therefore, (if p then q) and (if not-p then q)' is a valid inference.
A conditional is true whenever both its antecedent and consequent are true, so that `p and q, therefore, (if p then q) and (if q then p)' is a valid inference.
A conditional is true whenever both its antecedent and consequent are false, which means that the inference `not-p and not-q, therefore, (if p then q) and (if q then p)' is valid.

McLaughlin argues that truth functionality represents a dead-end for understanding conditionals. According to him, a truth functional logic cannot be a satisfactory model of a language that includes conditionals:

: The truth functional calculus of propositions is one of the foundation stones in modern logic. It is so eminently reasonable, and its principles perform so important a function in all branches of standard logic, that doing logic without it is unthinkable. And yet, it is a fact that the calculus is only satisfactory as a logical model of a language in which conditionals do not exist. Even such humble assertions as `If Smith works hard, then he will get a raise' fall outside the range of statements that can serve as interpretations of the forms of truth functional logic. This would not be an embarrassment if conditionals were marginal characters on the linguistic stage. But our conviction that this is not so, that they are important actors with major roles to play, is evidenced by the many attempts that have been made to find models that properly express their logic.Foot note 2

3. Three alternatives to the material conditional

Now many logicians have tried, as McLaughlin notes, to search for alternative and more satisfactory models for conditionals in order to avoid the problems faced by the truth functional account of conditionals.

One tradition, represented by Adams and Edgington,Foot note 3 argues that the abovementioned problems with a truth functional interpretation of conditionals indicate that natural language conditionals do not have truth conditions at all, but rather probability conditions. Adams' thesis is that the conditional probability a subject assigns to `q' given `p' is defined as equal to the probability she assigns to `p & q' divided on the probability she assigns to `p', thereby preventing the consequent from being less probable than the antecedent.

Another tradition, influenced by Adams and Grice, and represented by Jackson,Foot note 4 claims that a conditional has truth conditions according to the material conditional, but assertion conditions according to whether the conditional can be asserted or believed. One then distinguishes between some truth functional content of a conditional on the one hand and its communicated pragmatic content, which is therefore claimed not to be relevant for the truth of the conditional, on the other. In defining assertability as the conditional being robust with respect to the truth of its antecedent, Jackson uses Adams' thesis for conditional probability of conditionals. This means that the assertability of a conditional is still regarded as calculable from the antecedent and the consequent.

A third tradition that is widely accepted is the possible-worlds theory of conditionals, represented by LewisFoot note 5 and StalnakerFoot note 6 They give an account of conditionals according to which a conditional is true (in the actual world) if in the closest possible world where `p' is true, `q' is true as well. In this approach, one avoids that a conditional is true whenever its antecedent is false. However, in every possible world, the material conditional is claimed to hold as a model for conditionals, so that in a possible or an actual world, all true statements entail each other:

: According to Stalnaker if `p' is true in our world, then our world is the closest accessible world containing p and, because the conditional is proved by its confirmation, the conditional relation holds between p and all its world's happenings. If this stone is on the ground and the sun shines, then we are bound to accept both `If the sun shines, then the stone is on the ground' and `If the stone is on the ground, then the sun shines'.Foot note 7

4. Calculability and reduction

In all the approaches sketched above, the conditional is treated as a function of the value of its antecedent and consequent. Hence one generally expects to be able to calculate the truth, probability or assertability of `If p then q' from the truth, probability or assertability of `p' and `q' as such. McLaughlin argues against such a reductionist approach to conditionals:

: Consider, for instance, what is usually taken as the strongest evidence for the truth of `If p then q', the joint fulfillment of `p' and `q'. This by no means furnishes proof of `If p then q'... The truth of `If p then q' cannot be extracted, as a matter of logical necessity, from the truths of `p' and `q'... Even some who reject the truth functional model of the singular conditional propose systems in which the truth of the conditional can be equated with the joint occurrence, in certain possible worlds, of the events mentioned in the `if' clause and that mentioned in the `then' clause.Foot note 8

McLaughlin here points to the fact that a conditional `If p then q' is in general defined to follow from `p and q'. However, even though it is true that grass is green and snow is white, this is not sufficient for me to infer that `If grass is green, then snow is white', nor that `If snow is not white, then grass is not green'. Moreover, even if I start out with a conditional, for instance `If Norway joins EU, then the interest rate will fall', I cannot from the fulfilment of both the antecedent and the consequent infer the truth of the conditional. It might for instance be the case that the interest rate falls for other reasons than Norway joining EU.

So even though many logicians acknowledge the problems with the material conditional and try to solve them by finding alternative ways of representing conditionals in a logical system, all the alternatives seem to have the character of being pure ad hoc solutions to problems related to the material conditional. Also, all the approaches are attempts to include conditionals in extensional logical systems. Adams, Edgington, Grice, Jackson, Stalnaker and Lewis offer alternative solutions that are more intricate than the material conditional approach; however, all the different approaches basically depend on the material conditional.

There seems to be a common craving for a pure formalism including calculability and extensionality. This is what Sören Stenlund refers to as a «calculus conception of language». Characteristic for this view is that one considers language to be in principle a calculus, or a formal system that can be isolated from the situations in which it is used.Foot note 9 As long as one is not willing to give up on this conception of language, is it then at all possible to find a satisfactory approach to natural language? I think not.

5. A (truth) functional world

Let us now assume that conditionals can in fact be adequately accounted for within an extensional, truth functional, syntactic, compositional approach. What would be the consequences? According to McLaughlin, a truth functional logic can only be a model of a language addressed to a certain kind of world. So given that the truth functional logic is a satisfactory model of natural language, then natural language must have properties according to the six basic principles of modern propositional logic, as referred above. Moreover, given that natural language is truth functional, so that a truth functional logic is in fact an adequate tool for deciding truth and validity of our statements and arguments; then natural language must be a language of a world that has its structure according to these principles.

: The propositional calculus is well designed to serve as the logic of a language addressed to a world of which all one had to say was that certain events have or have not occurred or will or will not occur. And this arrangement would be acceptable if we really thought it the case that the world is a collection of discrete or unrelated events, the occurrence or nonoccurrence of which can be expressed by simple assertions and their denials. Because if reality consists wholly of elementary facts that can, in appropriate circumstances, be determined by observation to obtain or not, and if the sentence letters of the calculus are interpreted by statements or propositions that express these facts, then if the facts are known, all the truths about the world that are capable of rational representation can be embraced in a long conjunction of simple or atomic propositions and their negations of the form `p.q.-r.-s, ...' each element of which states the presence or absence of a fact.Foot note 10

This picture of the world addressed by a truth functional language seems to fit neatly into the philosophical picture of logical atomism. Within this tradition, the world is taken to consist of atomic facts that are described by simple statements. These atomic facts form part of a structure that corresponds to the truth functional structure of language. What McLaughlin points out is that the basic principles of logic, taken together, involve a reduction, but also dissolution, of conditional relations as such. To reduce the conditional form `If p then q' to a negated truth functional conjunction `¬(p & ¬q)' or to a truth functional disjunction `¬p ∨ q' eliminates the very notion of a conditional relation.

But this is to trivialize an important region of speech. The relations between events are just as significant to us as the question of their occurrence. We want to know the circumstances under which things happen -- and this is so whether the subject matter is particles of light, cells in the body, commercial responses to movements of prices on the open marked, or the behavior or feelings of animals and humans. To ground a claim that `if p then q' is true on the fact that `p' is false or `q' true misses the point of what the conditional is meant to say. A conditional, when applied to the experienced world, attempts to report a dependency between events, a dependency that obtains as a result of physical connection, social convention, or human purposes; this objective is not fulfilled by the fact alone that our observations show that the assertion has not been disconfirmed.Foot note 11

Most of our interactions with the world and each other depend on us handling conditionals in a totally different way from the observing of occurrence and non-occurrence of singular events. According to Grice, however, «no one would be interested in knowing that a particular relation (truth functional or otherwise) holds between two propositions without being interested in the truth-value of at least one of the propositions concerned, unless his interest were of an academic or theoretical kind».Foot note 12 But is this really so? If someone tells me that `If you eat that mushroom, you'll die', I am not first of all interested in the truth or probability of the antecedent or consequent. I am interested in the conditional relation.

6. Treating conditionals extensionally

Assuming that our interest in the world is primarily an interest in the occurrence or non-occurrence of single events, so that our interest in relations between events is reduced to a question about whether or not we observe `p' and `not-q', most of our conditionals would be useless and senseless:

: We know that there are countless assertions that we regard as being true or false but that employ concepts that do not take observable instances. There are no observable properties through which such concepts as pressure, energy, force, gravity, and time are immediately instantiated.Foot note 13

Take for instance the following statement: `If a body is not subject to any net external force, it will continue in a uniform movement or stay at rest.' We know that no bodies can be unmoved by forces. According to our truth functional approach, any conditional that cannot be directly tested must be true, since it then cannot be proved false. This means that the following conditional will also come out as true: `If a body is not subject to any net external force, it will search for the nearest cat to kill it.' As we saw above, the Stalnaker/Lewis tradition wants to reduce this problem by referring to the closest accessible possible worlds in which the antecedent `p' is true. So if the consequent `q' is true in all the possible worlds where `p' is true, the conditional `if p then q' is (necessarily) true. However, such a theory does not seem to get the job done for conditionals.

: In helping us to impose order on the world, they (conditionals) help us to determine our own actions. Consequently, we are not concerned to know, when `p' is false, whether `q' is true in the closest (but by definition unactual) world in which `p' is true. (How could one find out such a thing?) We want to know what would happen, what conclusion one might draw, if the event expressed by `p' is actualized by its occurrence.Foot note 14

Extensional logic does not only represent a reduction and downgrading of conditionals. It seems to dissolve the whole idea of conditional relations. If anything, we should have learned from Hume that we cannot reduce a conditional relation to the direct observation of joint occurrence or constant conjunction of two distinct events. But this seems to be an unavoidable consequence of treating conditionals extensionally.

7. Concluding remarks

We have seen how difficult it is to cut loose from the basic principles of logic, including functionality and calculability. McLaughlin makes clear that our models of language must be consistent with our models of the world and the relation between language and the world. Put differently, he makes clear that it would be inconsistent to maintain a set of logical principles and at the same time deny their epistemological and metaphysical implications. What we need, then, is a metaphysics that allows conditional relations as basic and primitive. Only then will we have a basis for developing an adequate logic, modelling a language, addressing a world that we can recognise as ours. This is not a world of unrelated events and accidental regularities, but a world of potentials, connections and dependencies.

Acknowledgments

: Thanks to Johan Arnt Myrstad for constructive comments and countless discussions about conditionals.

References

Adams, Ernest W.: The Logic of Conditionals, Dordrecht, Reidel 1975.
Copi, Irving: Introduction to Logic, New York, Macmillan 1968.
Edgington, Dorothy: `Do Conditionals Have Truth-Conditions?', in Jackson (ed.) 1991, pp. 176-201.
Edgington, Dorothy: `On Conditionals', Mind 104.414., April 1995, pp. 235-329.
Grice, Paul: Studies in the Way of Words, Cambridge, Harvard University Press 1989.
Jackson, Frank: Conditionals, Oxford, Blackwell 1987.
Jackson, Frank (ed.): Conditionals, Oxford, Oxford University Press 1991.
Jackson, Frank: `On Assertion and Indicative Conditionals' in Jackson (ed.) 1991.
Lewis, David: Counterfactuals, Oxford, Blackwell 1973.
McLaughlin, Robert N.: On the Logic of Ordinary Conditionals, Albany N.Y, State University of New York Press 1990.
Stalnaker, Robert: `A Theory of Conditionals', Studies in Logical Theory: American philosophical quarterly monograph, Oxford, Blackwell 1968, pp. 98-112.
Stenlund, Sören: Language and Philosophical Problems, London, Routledge 1990.

Rani Lill Anjum
Department of Philosophy
University of Tromsø
Norway
rani [dot] anjum [at] sv [dot] uit [dot] no

[Foot Note 1] Copi (1968), p. 244.

[Foot Note 2] McLaughlin (1990), p. 1.

[Foot Note 3] Adams (1975), Edgington (1991) and (1995).

[Foot Note 4] Jackson (1987) and (1991).

[Foot Note 5] Lewis (1973).

[Foot Note 6] Stalnaker (1968).

[Foot Note 7] McLaughlin (1990), p. 31.

[Foot Note 8] McLaughlin (1990), pp. 8-9.

[Foot Note 9] See Stenlund (1990) for a detailed account of the calculus conception of language.

[Foot Note 10] McLaughlin (1990), p. 2.

[Foot Note 11] McLaughlin (1990), p. 6.

[Foot Note 12] Grice (1989), p. 61.

[Foot Note 13] McLaughlin (1990), pp. 6-7.

[Foot Note 14] McLaughlin (1990), p. 35.