What does "if" mean?

Kazuyoshi KAMIYAMA2017/2/14

CONTENTS

Introduction

Conditionals

The Interpretation of “if” in logic

A philosophical problem of conditionals

Possible worlds approach

The suppositional theory

Remark

References

INTRODUCTION

Edington(2014) is a survey on indicative conditionals. In the survey

she seems to assume that conditional sentences have only one

analysis. She supports the non-propositional analysis. In this slide I

assert that conditional statements have three different uses. In other

words I propose a hybrid analysis of indicative conditionals.

CONDITIONALS

“If p, then q” ,“q if p” are conditional sentences(in short,

“conditionals”).

“The base angle is equal if triangular ABC is an

isosceles triangle.”

“We'll be home by ten if the train is on time.“

“If p, then q”

p：the antecedent

q: the consequent

From linguistic point of view there are two types of conditional

statements.

A. Indicative conditionals

Ex. (*) “You become healthy if you eat an apple every morning.“

A. Subjunctive conditionals (or counterfactual conditionals)

Ex. (**) "If you ate an apple every morning, you would be

healthy."

THE INTERPRETATION OF “IF” IN LOGIC

“If p, the q” (p→q) is true except in the case in

which p is true and q is false.

The “if” interpreted like this is called the “material

implication.”

Material implication assigns truth values to conditional

sentences as follows;

“If Tokyo is the capital of Japan, Tokyo is in East Asia.” true

“If 3>1, then 3>5.” false

“If 3>1, then Tokyo is the capital of Japan.” true

“If the moon is made of cheese, it is made of ketchup.” true

A PHILOSOPHICAL PROBLEMOF CONDITIONALS

Indicative conditional

(*) “You become healthy if you eat an apple.“

Is this sentence true or false?

“You eat an apple“ is an future event. So I cannot say

that it is true at least now. I cannot say it is false either.

The same can be said on "you become healthy."

As far as we interpret “if “ as material implication, (*) is nonsense.

Because p(“you eat an apple”) and q(“you become healthy”) are

not true nor false. “If” cannot assign truth value to the whole

sentence.

Subjunctive conditionals

(**) "If you ate an apple, you would be healthy."

(You did not eat an apple).

If we understand “if” as material implication, (**) is true. Because

“you ate an apple” is false.

But,

(***) "If you ate an apple, you would not be healthy.”

is true for the same reason.

Logician’s interpretation of “if” cannot discriminate (**) from (***).

It seems inappropriate to understand “if” in (*),(**) as the

material implication.

How should it be understood?

(In this slide, we argue about only indicative conditionals.)

POSSIBLE WORLDS APPROACH

(*) "If you touch that wire, you will get an electric shock".

You don't touch it. Was my remark true or false? According tothe non-truth-functionalist, it depends on whether the wire islive or dead, on whether you are insulated, and so forth.

Robert Stalnaker's (1968) account is of this type: consider apossible situation in which you touch the wire, and whichotherwise differs minimally from the actual situation. (*) istrue (false) according to whether or not you get a shock in thatpossible situation. (Edington,2014)

The possible worlds approach assumes the set of

possible worlds. And it uses an ambiguous notion

such as “the possible situation otherwise differs

minimally from the actual situation.”

If you do not want a heavy ontology and want to evade

the ambiguous notion, you need to go to another

approach-the suppositional theory.

THE SUPPOSITIONAL THEORY

Along with the possible worlds approach, “the suppositional

theory”(F.Ramsey(1929), E.Adams (1965) provides one of

two basic answers to the above question. It goes as

follows.

Let us ask what it is to believe, or to be more or less

certain, that B if A -- that John cooked the dinner if

Mary didn't, that you will recover if you have the

operation, and so forth. How do you make such a

judgement? You suppose (assume, hypothesise) that A,

and make a hypothetical judgement about B, under

the supposition that A, in the light of your other beliefs.

(Edgington,2014)

Frank Ramsey(1929) put it like this:

If two people are arguing "If p, will q?" and are both in

doubt as to p, they are adding p hypothetically to their

stock of knowledge, and arguing on that basis about

q; ... they are fixing their degrees of belief in q given p.

(Edgington,2014)

In this interpretation

"If p, then q“ asserts that q is probably true under the supposition

that p.

In other words, the conditional probability of p given that q is high,

i.e.,

P(p/q) is high (for example, P(p/q) > 0.9).

This analysis seems natural. But, let us consider the

following conditional statement;

"If the 9th planet exists in the solar system, that will

draw an elliptic orbit."

The “X” is the 9th planet in the solar system.

All planets in the solar system draw elliptic orbits.

The two sentences logically implies;

The “X” draws an elliptic orbits.

In this case the following analysis is natural.

"If p, then q“ asserts that the conjunction of p and the statements

that we have accepted S implies that q. (p⋀ S ⊨ q)

* In this case a conditional statement is a shortened forms of

logical implication.

So let me revise the above analysis:

"If p, then q" says the following:

1) the conjunction of p and the statements that we have

accepted implies that q is true, or

2) the conditional probability of q given p is high( for example,

P(q/p) > 0.9).

"If the 9th planet exists in the solar system, that will draw an

elliptic orbit." is an example of 1) case.

"We'll be home by ten if the train is on time“ is in the 2) case.

2) case uses the notion of probability. Probability is interpreted

in several ways. Basic interpretations are subjective one and

objective one. In the former case probability means the “degree

of belief” and in the latter case it means a physical propensity of

an event.

Note: Objective probability is interpreted as a physical

propensity, or disposition, or tendency of a given type of

physical situation to yield an outcome of a certain kind, or to

yield a long run relative frequency of such an outcome. (the

propensity interpretation of probability, See “Propensity

probability” in Wikipedia)

Let us consider "We'll be home by ten if the train is on time.“

“We’ll be home by ten” could mean the objective probability

that “we are home by ten” happens is high.

So let us revise our analysis more.

"If p, then q" says the following:

1) the conjunction of p and the statements that we have

accepted implies that q is true, or

2) the conditional probability of q given p is high( for example,

P(q/p) > 0.9), where probability means objective one, or

3) the conditional probability of q given p is high( for example,

P(q/p) > 0.9), where probability means subjective one

(degree of belief).

In the cases 1) and 2), "If p, then q“ expresses a proposition.

Therefore we can give the truth condition to the conditional,

which goes as follows;

Case 1)

"If p, then q" is true if and only if there exists a set of statements S

that we have accepted as true and the conjunction of p and S

implies q.

Case2)

"If p, then q" is true if and only if the conditional probability of q

given p is high( for example, P(q/p) > 0.9), where probability

means objective one.

In the case 3) we cannot give a truth condition, because in this

case "If p, then q" does not express a proposition, in other words,

it does not claim truth.

REMARK

The problem of compounds of conditionals (Edington,2014)

“If A, then if B, C"

"It's not the case that if A, B

"Either we'll have fish, if John arrives, or we'll have leftovers, if he

doesn't"

"If (B if A), C"

If conditionals do not express propositions with truth conditions, we have no account of the

behavior of compound sentences with conditionals as parts (Lewis (1976)). If we understand as

above that conditionals express propositions, it is not the problem for the suppositional theory. In

other words, if you use compound sentences, “if” means that in the case 1) or case 2) (if-sentence

states a proposition).

REFERENCES

Adams, E. W.(1965): “A Logic of Conditionals,” Inquiry, 8, 166–97.Edgington,D. (2014): ”Indicative Conditionals,“Stanford Encyclopedia of Philosophy

(http://plato.stanford.edu/entries/conditionals/)“Interpretations of Probability,”

(https://plato.stanford.edu/entries/probability-interpret/)

Lewis, D.(1976): “Probabilities of Conditionals and Conditional Probabilities,”,

Philosophical Review, 85: 297–315

“Propensity probability”(https://en.wikipedia.org/wiki/Propensity_probability)

Ramsey, F. P. (1926): “Truth and Probability” in The Foundations of Mathematics and

other Logical Essays, 156-198.

–––(1929): “General Propositions and Causality” in Ramsey 1990, pp. 145–63.

–––(1990): Philosophical Papers, ed. by D. H. Mellor. Cambridge University Press.

Stalnaker, R.(1968): “A Theory of Conditionals” in Studies in Logical Theory,

American Philosophical Quarterly (Monograph Series, 2), Oxford: Blackwell,

98–112.