Introduction to Formal Logic by moti

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Lecture 10

Vagueness and the sorites argument

Paradoxes and the problem of vagueness

A paradox is a set of assertions each one of which we would consider intuitively
to be true, but which cannot all be true together: that is, they are inconsistent.

The problem of vagueness arises from a paradox.

We will consider the paradox and enumerate the various ways to avoid it.

The assertions that form the paradoxical set are all about a particular argument
called the sorites argument.

The sorites argument

Suppose we have a million tubs, each containing some sand. Tub 1 contains 1
grain of sand; Tub 2 contains 2 grains of sand; and so on.

h1 = Tub 1 contains a heap of sand.
h2 = Tub 2 contains a heap of sand       …and so on.

The argument
h1,000
h1,000  h999
h999  h998
…
h3  h2
h2  h1
Therefore, h1

1) The natural language argument is represented faithfully in our logical
system.
2) The argument is valid (according to our definition of validity).
3) All the premises are true: if we take a single grain from a heap, a heap
remains.
4) The conclusion is false: a single grain of sand is not a heap.

What can we do?

Introduction to Formal Logic      Lecture 10       Richard.Pettigrew@bris.ac.uk
We must deny at least one of the paradoxical assertions since they cannot all be
true.

We cannot deny (2). Crazy to deny (4). Thus, we may deny (1) or (3).

Solution (a): Deny (1).

Claim: The logic we have described so far in this course does not apply to such
propositions as h1, h2, … because they are not bivalent propositions: that is, true
and false are not their only possible truth-values.

Instead, their truth-values may lie anywhere on a spectrum from completely true
to completely false.

We represent completely true by 1 and completely false by 0 and then these
along with the numbers in between constitute the range of the possible truth-
values of the propositions h1, h2, ….

We extend our definitional truth-tables to say what truth-values a compound
proposition gets given the truth-values of its components.

Different ways of doing this give rise to different fuzzy logics.

Then we redefine validity for these new fuzzy logics as follows: In fuzzy logic, an
argument is valid if there is no truth assignment that makes the premises all true
or nearly true and the conclusion false or nearly false.

We then show that, on this definition of validity, the sorites argument is invalid.

Problems with fuzzy logics:

1) The solution requires that propositions that concern quite different subject
matter can be compared as to truth. Is this really possible?
2) What do the fuzzy truth-values mean? How are they measured?

Solution (b): Deny (3)

There are two ways to do this:

A) Claim that at least one premises is false.
B) Claim that at least one premise is not true, but possibly not false either.

Epistemicism takes approach (A).

Epistemicism: At least one of the premises is false, but we cannot know which
one.

Introduction to Formal Logic        Lecture 10        Richard.Pettigrew@bris.ac.uk
Problem with epistemicism:

What determines these sharp boundaries? Not our intentions, which determine
much of the meaning of our other words. So something else. Epistemicists often
appeal to the total usage of the word in the past.

Supervaluationism takes approach (B).

Claim: Given a vague property (e.g. being a heap), there are various acceptable
ways of making this property precise.

Then we introduce a third truth-value called indeterminate and the truth-value of
the proposition Such-and-such is a heap is determined as follows:

The proposition is true if it is true on every acceptable way of making the property
precise.
The proposition is false if it false on every acceptable way of making the property
precise.
The proposition is indeterminate if there is at least one acceptable way on which
it is true and at least one on which it is false.

Just as with fuzzy logic, we must extend the truth-tables to encompass our new
truth-value. However, this time, instead of showing that the sorites argument is
invalid, our new truth-tables show that some of the premises are not true (and
they’re not false either; they’re indeterminate).

Problem with supervaluationism:

According to the supervaluationist, there is a sharp boundary between those
collections of grains of sand that are heaps on all ways of making the concept of
heap precise, and those collections that are heaps only on some such ways.

This seems wrong.

It seems that this boundary should also be vague.

Introduction to Formal Logic       Lecture 10       Richard.Pettigrew@bris.ac.uk

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