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• 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

• 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.
    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?
with reality
               Truth Value
• There are just two truth values: true and false

• 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 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

    – 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
Defending our idealized account
        of truth value
• 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

    – Where truth value seems to be a matter of degree

    – Where truth value seems to be relative

• And respond to them.

• But first let’s consider the idea of counterexample…
• We’ve already considered a special case of counterexample:
  showing that an argument was invalid by showing that another
  argument of the same form was invalid.

• The idea was that in claiming an argument to be valid we were
  claiming that it was an instance of a valid argument form, i.e. that
  all arguments of that form were valid.

• A counterexample argument is a counterexample to that general

• Now let’s consider more generally how counterexamples work…
• A case that shows a general claim to be

• 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

• And there are lots more.
Rebutting apparent
• 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.

  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
Apparent Counterexample

 – For Sale: 1996, 4-door Nissan Sentra. New
   clutch, low mileage [um, it’s almost true--
   everything except the low mileage]

 – Response: we treat this as a conjunction and

   stipulate that a conjunction is true only if all its

   conjuncts are true.
Conjunction: “and” statement
       • My car is a 1996 and it’s got four
         doors and it’s a Nissan Sentra and
         it’s got a new clutch and it’s got low

       • False! It’s got 209,173 miles on it.

       • If we want to get more specific, we
         can ask: is it a 1996? Does it have 4
         doors, etc.
• 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

 • 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]…


 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

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.
    Example of Math Induction
• We want to show that for any natural number n, the sum of
  numbers 1 + … + n = n(n + 1)
                                        n(n + 1)
• Call the proposition that 1 + … + n =          “P”

• P is true for n = 1 since 1(1+ 1) = 2 = 1
                               2      2
                                             2(2 + 1) 6
• P is true for n = 2 since 1 + 2 = 3 and             = =3
                                                2       2
                                                3(3 + 1) 12
•   P is true of n = 3 since 1 + 2 + 3 = 6 and           = =6
                                                    2     2

• 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)
                                                     1(1+ 1) 2
• Base Step: we show that P holds where n = 1:              = =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 =
    – We do some algebra to show that P holds for n + 1, i.e. that
      1 + … + n + (n + 1) = (n + 1)((n + 1) + 1)
• We’re done! This shows that P holds for all n’s!

   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!

    However the same form of argument
    gives us the Sorites Paradox.
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 run away fast!
  We’ll ignore the Sorites in this class...So now for some easier problems.
(For further discussion see


     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
        Context Dependence
    A                                         B

            I like chocolate
                               I don’t like

Not a counterexample! the truth value of these
context-independent 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.”
             2. Alice likes chocolate

       •   We’ll say that truth value belongs to
           propositions expressed by context-
           independent 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


                        Truth Value

    (“the World,” the way things are)
         Our working definition:
Truth is correspondence with reality

              Roses are red.

A propositional attitude
 Propositional Attitudes

• Ways in which people are related to


• Propositions are expressed by that clauses

• X _____ that p [hopes, is afraid, believes]
• 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
Belief: a propositional attitude

                  Propositional Attitude
 Proposition                               Person
    Truth Value

                     Believing doesn’t make it so!
                     The relation between propositions
  Reality            and reality is completely separate
                     from the relation between persons
                     and propositions!
  Controversial Beliefs

   God exists.
                           God doesn’t

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…


 …but it’s alright to be wrong!
Having good reasons for
   what you believe
    “Reasons” for belief
• Causal: what causes a person to hold a


• Pragmatic: the beneficial effects of

  holding a belief

• Evidential: evidence for the truth of a


• 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
        x knows that p:

        1. x believes that p

        2. x’s belief that p is


        3. p is true
Sources of knowledge

 • Sense perception

 • Introspection
 • Memory             but not
 • Reason

 • Expert testimony
Knowledge doesn’t require certainty!

 Now what?

                       I think,
                   therefore I am
      Truth and Justification

               True     False

      Truth and Justification

                  True   False

Not         e.g. lucky
Justified   guesses
      Truth and Justification

                  True         False

Not         e.g. lucky   e.g. unlucky
Justified   guesses      guesses
  Truth and Justification

                  True         False
Justified                e.g. “Smoking
             KNOWLEDGE   gun” example

Not         e.g. lucky   e.g. unlucky
Justified   guesses      guesses
           The Ethics of Belief

W. K. Clifford
                The Ethics of
                Is it ever rational for a person
                to believe believe anything
                for which he has no
                compelling evidential

                To be continued…
William James

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