Analyticity, Necessity, and the a priori Ayer, ch 4

1
Analyticity, Necessity, and the a prioriAyer, ch
4

• September 17, 2008

2
The Moral for Philosophy of Science

• Statements in science are not (usually)
  deductively proven by observation statements.
• We need to understand inductive inference if we
  are to understand the logic of science.
• Ayers Weak Verifiability Principle can be seen
  as a (failed) attempt at this.

3
Examples of necessary truths

• The sum of the interior angles of a triangle
  equals 180.
• 7 5 12
• Either p or not-p.
• All Fs are F.

4
Why are necessary truths problematic for Ayer?

• a. Through observation we can never establish
  that a statement is necessarily true.
• b. Through observation we can never be completely
  certain of a statement.
• c. Only statements that are connected to
  observation have factual content.
  (empiricism/positivism).
• d. Statements of mathematics and logic appear to
  be necessary, certain, and to have factual
  content.

5
Ayer is responding to this argument

• 1. Statements of mathematics/logic are necessary
  and certain.
• 2. Statements of mathematics/logic have factual
  content.
• 3. If empiricism/positivism is true, then no
  necessary and certain statements have factual
  content.
• --------------------------------------------------
  -------------
• 4. Thus, empiricism/positivism is false.

6
Ayer is responding to this argument

• 1. Statements of mathematics/logic are necessary
  and certain.
• 2. Statements of mathematics/logic have factual
  content.
• 3. If empiricism/positivism is true, then no
  necessary and certain statements have factual
  content.
• --------------------------------------------------
  -------------
• 4. Thus, empiricism/positivism is false.

7
Ayer is responding to this argument

• 1. Statements of mathematics/logic are necessary
  and certain.
• 2. Statements of mathematics/logic have factual
  content.
• 3. If empiricism/positivism is true, then no
  necessary and certain statements have factual
  content.
• --------------------------------------------------
  -------------
• 4. Thus, empiricism/positivism is false.

8
J.S. Mills Account

• Mathematical and logical truths are merely highly
  confirmed empirical generalizations.

Ayers Argument (p. 75) 1. If mathematical
propositions were empirical generalizations then
we would not always preserve their validity. 2.
But we do always preserve their
validity.----------------------------------------
------------------------------------3. Thus,
mathematical principles are not empirical
generalizations.
9
Ayers Account
Mathematical/logical claims are certain and
necessary, but they have no factual content.

• How can we know mathematical/logical claims with
  certainty?
• Why are mathematical/logical statements
  necessary?
• Why do mathematical/logical claims seem to have
  factual content?

10

• How can we know mathematical/logical claims with
  certainty?
• Why are mathematical/logical statements
  necessary?
• Why do mathematical/logical claims seem to have
  factual content?

Ayers Answer p. 79, 85
11

• How can we know mathematical/logical claims with
  certainty?
• Why are mathematical/logical statements
  necessary?
• Why do mathematical/logical claims seem to have
  factual content?

Ayers Answer p. 78
12
Ayer Mathematical/logical statements are
CERTAIN and NECESSARY because they are analytic.
13
THE BIG QUESTION

• Does analyticity explain why necessary truths are
  necessary and how a priori knowledge is possible?

14
ANALYTICITY EXPLAINS A PRIORI KNOWLEDGE

• Analytic sentences are true in virtue of the
  meaning of terms.
• We decide what our terms mean, so we know what
  our terms mean.
• Thus, we can know the truth of analytic sentences
  just from knowing what our terms mean. And we can
  be certain of this, since their truth follows
  from the meaning of the terms.

15
ANALYTICITY EXPLAINS NECESSITY

• Analytic sentences are true in virtue of the
  meaning of the terms.
• Thus, analytic sentences do not depend on the
  world for their truth. Rather, they depend on
  what we stipulate our terms to mean.
• Thus, they are true no matter how the world is,
  and so they are necessary.

16
ANALYTICITY DOESNT EXPLAIN A PRIORI KNOWLEDGE

• We dont stipulate the meanings of each
  math/logic statement one by one.
• Instead, we stipulate the meaning of relatively
  few terms and schemas and then draw out the
  consequences.
• But by consequences, we mean logical
  consequences.
• But then it is our knowledge of the meaning of a
  few terms and schemas and our knowledge of
  logical consequences that explains our knowledge
  of logical statements.
• But then analyticity alone doesnt explain a
  priori knowledge of math/logic. Knowledge of
  meaning and knowledge of logical consequence
  explains our knowledge of math/logic.