Philosophical Devices
Week 4 – Three Kinds of Truth
--------------------------------------------------------------------------------------------------------REGARDING ASSIGNMENT 2, QUESTION 5:
To convert a rational number, represented as a fraction p/q, into a decimal representation, we
use long division to divide p by q, extending p with zeros after the decimal point as
necessary.
Consider the division of p by q. At each step, we are left with a remainder. This remainder
will always be a value between 0 and (q-1) inclusive (if the remainder were larger, it would
mean that the quotient we wrote on top was too small and could have been divided after all).
If the remainder is zero, we’re done and the decimal terminates. If we never get zero as a
remainder, that there are only a finite number of possible remainders means that at some
point, we must get a remainder we’ve already had before. This means the quotient will
repeat: since we are only bringing down zeros from the dividend, the remainder exactly
determines the next digit in the quotient, which determines the next remainder, and so on.
For example, consider the decimal expansion of 3/7:
It should be clear that, after reaching a remainder of 3 for the
second time, the quotient will start repeating – i.e.,
0.428571428571428571…
Therefore, the decimal representation of any rational number will
either terminate, or eventually become periodic. Note also that,
since there are only (q-1) possible non-zero remainders when
dividing by q, the repeating portion of the decimal expansion of a
rational number with a denominator of q can be at most (q-1) digits
long (it could be shorter, but it can’t be longer). So, if you see a
periodic decimal whose repeating portion is 10 digits long, you
know that the rational number it represents must have a
denominator of at least 11 digits, if the repeating part is 12 digits
long, then at least 13, etc…
One more thing to note is that a ‘terminating’ decimal can also be thought of as being
followed by an infinite sequence of zeros. So in this sense, even ‘terminating’ decimals are
actually periodic – just with a really, really boring sequence.
(Bonus challenge: how do we tell the difference between rational numbers whose decimal
expansion terminates and those that repeat?)
--------------------------------------------------------------------------------------------------------A TRIO DISTINCTIONS
Note: these distinctions (as we will presently use them) apply to sentences not things

Analytic Truth: a sentence S is analytically true iff S is true solely in virtue of the
meaning and arrangement of S’s constituent parts
- All triangles have three angles. All Bachelors are unmarried. All stallions are
horses.
Philosophical Devices
Week 4
2
Three Kinds of Truth

Analytic Falsehood: a sentence S is analytically false iff S is false solely in virtue of
the meaning and arrangement of S’s constituent parts
- There are married Bachelors. Bucephalus is a stallion but not a horse. Some
triangles are squares.

Synthetic Truth: A sentence S is synthetically true iff S is true and S’s truth isn’t
guaranteed by the meaning and arrangement of S’s constituent parts
- Hamburg has a port. Some apples are red. Cato was a Roman orator.

Synthetic Falsehood: A sentence S is synthetically false iff S is false and S’s
falsehood isn’t guaranteed by the meaning and arrangement of S’s constituent parts
- Hamburg is in Denmark. Apples are vegetables. Cato praised Cataline.
The important distinction: analytic sentences have their truth value in virtue of meanings
alone, while synthetic sentences also include worldly elements

A priori truth: a sentence S is an a priori truth iff S’s truth can be known prior to or
without any experience
- All bachelors are unmarried. If P, then P. 6464 = 0001000000000000 in binary.

A priori falsehood: a sentence S is an a priori falsehood iff S’s falsity can be known
prior to or without any experience
- Every triangle is also a square. P & P. 1<0.

A posteriori truth: a sentence S is an a posteriori truth iff S’s truth can only be known
as a result of some relevant experience
- My sweater is black. Benjamin is bald. It is cold outside.

A posteriori falsity: a sentence S is an a posteriori falsehood iff S’s falsity can only be
known as the result of some relevant experience
- The sun didn’t come up until 10:30 this morning. Benjamin is in his office right
now.
The important distinction: the only way we can know the truth/falsity of a posteriori sentences
is by making observations, while we can, in principle, come to know the truth/falsity of a
priori sentences without such observations

Necessary truth: A sentence S is necessarily true iff it isn’t possible for S to be false
- If T is a triangle, then T has three sides.

Necessary falsehood: A sentence S is necessarily false iff it isn’t possible for S to be
true
- If T is a triangle, then T is a square.

Contingent truth: A sentence S is contingently true iff S is actually true, but S could
have been false
- Benjamin wears glasses.
Philosophical Devices

Week 4
3
Three Kinds of Truth
Contingent falsehood: A sentence S is contingently false iff S is actually false, but S
could have been true
- Prof. Gähde hates sailing.
The difference: necessarily true/false sentences couldn’t be otherwise – their truth value
remains the same in all possible worlds; contingently true/false sentences could, in some
sense, change their truth value.
--------------------------------------------------------------------------------------------------------OVERLAP BETWEEN THE ANALYTIC/SYNTHETIC & A PRIORI/A POSTERIORI?
Seems all analytic sentences are a priori: if the truth/falsity of a sentence is guaranteed by the
meaning & arrangement of its constituent parts, then knowledge of the meaning &
arrangement will suffice for knowledge of its truth/falsity – no observation is required!
What about vice versa - are all a priori truths analytic?
This amounts to asking whether there are any synthetic a priori truths – are there?




Every event has a cause
Nothing can be red all over and green all over
Newton's laws of mechanics (according to Kant, they are necessary & universal
claims (hence a priori), but are more than mere logical imperatives (hence synthetic))
All triangles contain 180°
Just take the last one – it isn’t even true, if we read it as a claim about triangles in the world!
Of course, we can make it true by talking about idealized triangles which satisfy Euclid’s
axioms & definitions, but this renders it analytic.
Similarly with Newton’s laws of mechanics!
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GROUP ASSIGNMENT
I.
Examples of the above kinds:
II.
Say whether the following are analytic, synthetic, a priori, a posteriori, necessary,
contingent, or some combination thereof:
(1)
Vixens are female foxes
(2)
Coffee contains caffeine
Philosophical Devices
III.
Week 4
(3)
Blood transports oxygen
(4)
Energy is always conserved
(5)
Bicycles have two wheels
(6)
All whales are mammals
(7)
Marilyn Monroe is Norma Jean Baker
(8)
The inventor of bifocals invented bifocals
(9)
The metre stick is a metre long
(10)
I am here now
4
Three Kinds of Truth
Distinguish two readings of ‘Every philosopher loves a distinction’.