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.
Recommended