Transcript Knowledge - University of San Diego Home Pages

Knowledge
Concepts
• Knowledge (“knowing that__”) as justified true
belief
• Truth value
• Belief
• Justification
• Counterexample
• Sorites Paradox
• Mathematical Induction
What’s the point of
this discussion?
• We confuse truth with notions like belief, knowledge and
justification.
• That makes us reluctant to accept the account of truth value that
figures in classical logic.
• If we get clear about what knowledge is--and isn’t--then the
claims we make about truth value won’t seem that crazy.
• We will also use this discussion as an excuse to talk about
some other important concepts along the way.
IS THIS
SO?And if so,
what does it
mean?
Noble Lie
Propositional Knowledge
• Propositional knowledge is knowing that as distinct
from…
• Knowing who or
• Knowing how
x knows that P
Knowledge as Justified True Belief
(the “JTB” account of knowledge)
• What is justification?
• What is truth?
• What is belief?
Truth
Correspondence with
reality
Truth Value
• There are just two truth values: true and false (‘bivalence’)
• Truth value does not admit of degree
• Truth value is not relative to persons, places, times,
cultures or circumstances
How do we know? We stipulate that this is how we’ll
understand truth value! We idealize…
Idealization
• Idealization is the process by which scientific models
assume facts about the phenomenon being modeled
that are not strictly true. Often these assumptions are
used to make models easier to understand or solve.
• Examples of idealization
– In geometry, we assume that lines have no
thickness.
– In physics people will often solve for Newtonian
systems without friction.
– In economic models individuals are assumed to be
maximally rational self-interested choosers.
Maps Idealize
Bonini’s Paradox
Defending bivalence
• Is our idealized notion of truth value close enough to the
messy real world idea of truth and falsity?
• To make the case that it is, we’ll consider some apparent
counterexamples
– Where truth value seems to be a matter of degree
– Where truth value seems to be relative
• And respond to them.
• First…how to respond to putative counterexamples…
Counterexample
• A case that shows a general claim to be false
• E.g. claim: for all numbers a, b, x, if a > b then ax >
bx. True?
• NO! The case in which x = 0 is a counterexample!
• And there are lots more.
Rebutting apparent
counterexamples
• But not everything that looks
like a counterexample really is one
• E.g. claim: All monkeys have tails.
• Apparent counterexample: Chimpanzees don’t have
tails.
• NOT A COUNTEREXAMPLE! Chimps aren’t
monkeys--they’re apes.
Defending our idealized
account of truth value
• We’ll consider apparent counterexamples to our claims about
truth value which purport to show that:
– Some propositions have truth values that are ‘between true
and false
– Some propositions are neither true nor false
– The truth value of some propositions is relative to persons,
places, cultures, etc.
• We’ll respond to these counterexamples in various ways in
order to show that our account of truth value isn’t completely off
the wall.
Bivalence: ‘2-valuedness
• Claim: there are just two truth-values, true and false-nothing else, nothing in between, no almost-true or
almost-false.
• Apparent Counterexamples:
– Conjunctions
– Vagueness
‘There’s some truth in all religions.’
Counterexample???
Buddhism: Four Noble Truths
 Dukkha: Suffering exists: (Suffering is real and almost
universal. Suffering has many causes: loss, sickness, pain,
failure, the impermanence of pleasure.)
 Samudaya: There is a cause for suffering. (It is the desire to
have and control things. It can take many forms: craving of
sensual pleasures; the desire for fame; the desire to avoid
unpleasant sensations, like fear, anger or jealousy.)
 Nirodha: There is an end to suffering. (Suffering ceases with
the final liberation of Nirvana (a.k.a. Nibbana). The mind
experiences complete freedom, liberation and non-attachment.
It lets go of any desire or craving.)
 Magga: In order to end suffering, you must follow the Eightfold
Path.
The Eight-Fold Path
 1) Samma ditthi Right Understanding of the Four Noble Truths
 2) Samma sankappa: Right thinking; following the right path in life
 3) Samma vaca: Right speech:
 No lying
 No criticism,
 No condemning, gossip
 No harsh language
 4) Samma kammanta Right conduct by following the Five Precepts
 5) Samma ajiva: Right livelihood; support yourself without harming others
6) Samma vayama Right Effort: promote good thoughts; conquer evil thoughts
 7) Samma sati Right Mindfulness
 8) Samma samadhi Right Concentration: Meditate to achieve a higher state of
consciousness
Buddhism: Five Precepts
 Do not kill.
 Do not steal. This is generally interpreted as including the
avoidance of fraud and economic exploitation.
 Do not lie. This is sometimes interpreted as including name
calling, gossip, etc.
 Do not misuse sex. For monks and nuns, this means any
departure from complete celibacy. For the laity, adultery is
forbidden, along with any sexual harassment or exploitation,
including that within marriage.
 Do not consume alcohol or other drugs.
Nicene Creed (abridged)
 I believe in one God the Father Almighty…
 And in one Lord Jesus Christ, the only-begotten Son of God…
 Who for us men, and for our salvation came down from
heaven…
 And was crucified also for us under Pontius Pilate….
 the third day he rose again
 I believe in the Holy Ghost…
 Who proceedeth from the Father and the Son…
 I look for the Resurrection of the dead…
 And the life of the world to come.
Conjunctions
We pledge that, in logic class, we will not talk about
the ‘degree of truth’ in large-scale doctrinal packages
We resolve that we will treat them as
conjunctions (AND statements)
And that we will call a
conjunction TRUE if and only if
each of its conjuncts are true
Vagueness
• Truth and falsity are all-or-nothing, like the oddness and
evenness of numbers.
• Counterexamples?
– Vagueness, e.g. ‘Stealing is wrong’
– Response: This isn’t a complete thought. We clarify
and spell out details to eliminate vagueness where
possible…
– And ignore recalcitrant cases like the dread Sorites
Paradox.
The Sorites Paradox
We agree that 100,000 grains of sand are a heap…
And that one grain of sand is not a heap…
And…
Sorites Paradox
We agree that removing one grain of sand from a
heap won’t make it stop being a heap…
The Sorites Paradox
a.k.a the Paradox or the Heap or the Bald Man
1.
A 100,000 grain collection is a heap
2.
If a k-grain collection is a heap then a (k - 1)-grain collection
is a heap
3.
Therefore, a 9,999-grain collection is a heap [by 1, 2]
4.
Therefore, a 9,998-grain collection is a heap [by 2, 3]…
Uh-oh!
n.
Therefore, a one-grain collection is a heap [by 2, n - 1]
A hundred bottles of beer on the wall…
A Big Problem
• The Sorites argument, which leads to the ridiculous
conclusion that one grain of sand is a heap, is a proof by
mathematical induction.
• To say that the argument is no good would seem to commit
us to rejecting mathematical induction…
• And that would be
VERY BAD!
Mathematical Induction
Mathematical induction is a
method of mathematical proof
typically used to establish that a
given statement is true of all
natural numbers. It is done by
proving that the first statement
in the infinite sequence of
statements is true, and then
proving that if any one
statement in the infinite
sequence of statements is true,
then so is the next one.
Mathematical Induction
A proof by mathematical induction
consists of two steps:
The basis (base case): showing
that the statement holds for a
natural number, n, e.g. when n = 1
The induction step: showing that if
the statement holds for some n,
then the statement also holds
when n + 1 is substituted for n.
This proves that the statement
holds for all values of n.
Mathematical Induction
1. P holds for 1 [by base step]
2. If P holds for some natural number n then it
holds for n + 1 [by induction step]
3. So P holds for 2 [by 1, 2]
4. So P holds for 3 [by 2, 3]
5. So P holds for 4 [by 2, 4] …
So the dominos all fall!
Example of Math Induction
• We want to show that for any natural number n, the sum of
numbers 1 + … + n = n(n + 1)
2
n(n + 1)
2 ‘P’
• Call the proposition that 1 + … + n =
•
•
•
1(1+ 1) 2
= =1
2
2
P is true for n = 1 since
2(2 + 1) 6
= =3
2
2
P is true for n = 2 since 1 + 2 = 3 and
3(3 + 1) 12
= =6
2
2
P is true of n = 3 since 1 + 2 + 3 = 6 and
• And so on . . .
• But ‘and so on’ is not a proof!
This is how you prove it
• We want to prove P: 1 + … + n = n(n + 1)
2
• Base Step: we show that P holds where n = 1: 1(1+ 1) 2
= =1
2
2
• Induction Step: we show that if P holds for a number n then it
holds for n + 1
(n)(n + 1)
– Suppose P holds for n, i.e. 1 + … + n =
2
– We do some algebra to show that P holds for n + 1, i.e. that
1 + … + n + (n + 1) = (n + 1)((n + 1) + 1)
2
• We’re done! This shows that P holds for all n’s!
• See how it’s done here:
https://www.khanacademy.org/math/precalculus/seq_induction/p
roof_by_induction/v/proof-by-induction
Sorites is a Math Induction Argument!
Basis: A 100,000 grain collection is
a heap.
Induction step: If an k-grain
collection is a heap then an (k - 1)grain collection is a heap.
So all the dominoes fall…and there
seems no way to avoid the
conclusion that a one-grain
collection is a heap!
What should we do???
We run away fast!
We’ll ignore the Sorites in this class...So now for some easier problems.
(For further discussion see http://plato.stanford.edu/entries/sorites-paradox/)
Sorites
Sorites seeking to impale a wet philosopher on the Horns of a Dilemma
An easier problem
• We claim that truth value is not relative to persons, times,
places, etc.
• Counterexamples?
• ‘True-for’ sentences
– ‘For the ancient Greeks, the earth was at the center of
the universe.’
• Context-dependent sentences
– I like chocolate
Response to ‘True-for’
• ‘True-for’ is an idiom: it means ‘believed by’
• Example: “For the ancient Greeks, the earth was the center
of the universe.
• Translation: ‘The ancient Greeks believed that the earth
was the center of the universe’
• Compare to the ‘historical present’ e.g. ‘Socrates is in the
Athens Jail awaiting execution’.
Context Dependence
A
B
I like chocolate
I don’t like
chocolate
Not a counterexample! the truth value of these contextindependent sentences isn’t relative:
1. Alice likes chocolate
2. Bertie doesn’t like chocolate
Response to context-dependence
For any utterance of a context-dependent
sentence, there’s a context-independent
sentence that makes the same statement.
1. [uttered by Alice] ‘I like chocolate’.
A
2. Alice likes chocolate
•
We’ll say that truth value belongs to
propositions expressed by contextindependent sentences.
•
Given this restriction, truth value is not
relative to persons, places, times, etc.
What’s the point?
• In doing formal logic we will make some idealizing assumptions
about truth value that seem crazy.
• The point of considering and responding to apparent
counterexamples is to argue that these assumptions aren’t so
crazy.
• We argue for the legitimacy of this idealization
What is truth?
But we still haven’t answered the Big Question
Correspondence Theory of Truth
Proposition
Truth Value
Reality
(“the World,” the way things are)
Our working definition:
Truth is correspondence with reality
Roses are red.
True!
Does this tell us anything?
• Not really.
• Because we haven’t made sense the idea of ‘correspondence’
• So, as with sorites, we’ll leave this sit for further philosophy
classes…
Belief
A propositional attitude
Propositional Attitudes
• Ways in which people are related to propositions
• Propositions are expressed by that clauses
• X _____ that p [hopes, is afraid, believes]
Belief
• We call beliefs ‘true’ or ‘false’ in virtue of the truth
value of the propositions believed.
• By ‘belief’ we don’t mean ‘mere belief’
• Believing doesn’t make it so - denial doesn’t make it
not so.
• We may believe with different degrees of conviction.
Belief: a propositional attitude
Person
Truth Value
Proposition
Propositional Attitude
Reality
Believing doesn’t make it so!
The relation between propositions
and reality is completely separate
from the relation between persons
and propositions!
Controversial Beliefs
God exists.
God doesn’t
exist.
People disagree. Who’s to say? No one knows.
Who’s to say??!!?
• That’s a different question from
the true or false question!
• A proposition is either true or false--even if we
don’t (or can’t) know which.
– Example: No one now knows, or can know,
whether Lucy, an early hominid who lived
3.18 million years ago had exactly 4 children
or not. But “Lucy had exactly 4 children” is
either true or false.
So when there’s a genuine
disagreement, someone is wrong…
Atheists
Welcome
…but it’s alright to be wrong!
Justification
Having good reasons for
what you believe
‘Reasons’ for belief
• Causal: what causes a person to hold a
belief
• Pragmatic: the beneficial effects of
holding a belief
• Evidential (Epistemic): evidence for
the truth of a belief
Justification
• X is justified in believing that p if x has
good enough evidential reasons for
believing that p
• Knowledge doesn’t require certainty
• Justification is relative to persons
The JTB Account of
Knowledge
x knows that p:
1. x believes that p
2. x’s belief that p is
justified
3. p is true
Sources of knowledge
• Sense perception
• Introspection
• Memory
• Reason
• Expert testimony
Reliable…
but not
infallible!
Knowledge doesn’t require certainty!
Now what?
I think,
therefore I am
Truth and Justification
True
Justified
KNOWLEDGE
Not
Justified
False
Truth and Justification
True
Justified
KNOWLEDGE
Not
Justified
e.g. lucky
guesses
False
Truth and Justification
True
False
Justified
KNOWLEDGE
Not
Justified
e.g. lucky
guesses
e.g. unlucky
guesses
Truth and Justification
True
Justified
KNOWLEDGE
Not
Justified
e.g. lucky
guesses
False
e.g. “Smoking
gun” example
e.g. unlucky
guesses
The Ethics of Belief
W. K. Clifford
The Ethics of
Belief
Is it ever rational for a person
to believe believe anything
for which he has no
compelling evidential
reasons?
To be continued…
William James