Sentences with and without truth values

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Sentences with and without truth values

• Sentences without truth values
• Orders Have a bannana.
• Questions May I have a bannana?
• Exclamations Shit! A bannana.
• Sentences with truth values
• Assertions Here is a bannana.

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Controversial Cases

• Taste
• Bannanas are delicious
• Carrot Top is funny
• Barak is pretty
• Morality
• Murder is wrong

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Truth

• We will only be concerned with sentences that
  have truth values
• So what is it for a sentence to be true?
• Pragmatists say usefulness.
• Coherentists say coherence.
• We will assume the correspondence theory of
  truth
• A sentence is true iff it corresponds with the
  world.
• This assumes i) there is a real world out there
  and ii) it is represented by sentences.

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The Correspondence Theory of Truth
True
False
The giraffe is at the window
The giraffe is at the window
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Realism

• Tempting thought The Earth is flat was true
  for the Ancient Greeks.
• But
• Whats the point of doing science if everything
  we all believe is true?
• We do science to discover which of our beliefs
  are false.

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Compound Statements
Negation (not)
Conjunction (and) Also a.b ab
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Disjunction
Disjunction (or)
Or in English is usually exclusive You will
either graduate or live under the bridge selling
your body for crack implies only one will
happen. Or in logic is inclusive. If both
disjuncts are true, the whole sentence is true.
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Material Conditional
(Material) Conditional If a then b
Consequent
Antecedent
Conditionals are at the centre of all
arguments. There are different views about what
we should say when the antecedent is false.
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Biconditional
Biconditional a if and only if b a iff b
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Types of statement Possibility

• Contradictions Statements that cannot possibly
  be true. e.g. p -p
• Tautologies Statements that are always true.
  e.g. p or -p.
• Contingent statements Statements that may or may
  not be true depending on how the world actually
  is. E.g. p, p -- q,.

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Types of possibility
Logical Possibility
I run faster than light
I run faster than 20mph
_at_
Nomological (law-like / physical) Possibility
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Knowledge and belief

• Think of a belief as a sentence in the head.
• If the sentence is true, the belief is true.
• If the sentence is false, the belief is false.
• What is it to have knowlegde?
• It is to have
• i) a belief that is
• ii) justified and
• iii) true.

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Knowledge and Certainty

• We know lots of things were not certain of.
• In fact there is hardly anything we can (should!)
  be certain of.
• Scientific hypotheses are never proved the aim
  is that they be strongly justified.
• Atheists and theists dont need proof, just
  evidence.

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Arguments

• But if they seriously believe that I am going to
  give up eating meat - in the hope of reducing the
  temperature of the planet - then they must be
  totally barmy. Boris Johnson
• Bernard Connolly, strategist at Banque AIG,
  warned "The virtual wiping-out of Fannie and
  Freddie shareholders was unavoidable. But it may
  make investors more reluctant to provide
  additional capital elsewhere."
• Words to watch for if..then, only if, but,
  because

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Stream-lining arguments

• An argument is a list of statements, one of which
  is designated the conclusion, and the rest of
  which are designated as premises
• Premise 1
• Premise 2
• ..
• ..
• Conclusion
• The conclusion states the point being argued for
  and the premises state the reasons being advanced
  in support of the conclusion.

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How to win / lose an argument

• Only two questions matter for evaluating an
  argument
• 1. Are the premises true?
• 2. Do the premises support the conclusion?
• The philosophical action is about 2.
• Support comes in degrees.
• Maximal support makes the argument deductive.
• Weaker support makes the argument inductive.

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Deductive Arguments

• An argument is deductively valid if and only if
  it is (logically) impossible that its conclusions
  are false while its premises are true.
• Not deductively valid
• George is 103 years old
• Therefore, George will not go clubbing this
  weekend.
• Deductively valid
• Bob is a Martian
• Martians like Ricky Martin
• Therefore, Bob likes Ricky Martin

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Deductive Arguments

• Deductive arguments make no claim that is not
  already made by its premises (perhaps implicitly).

a
b
b
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Venn Diagrams
Logically Possible worlds
Bob is on the team
Bob in on defence
Bob is on the team but not on defence
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• Bob is on the defence
• Bob is on the team
• Bob is on the team
• Bob is on the defence
• Deductive arguments make no claim that is not
  already made by its premises (perhaps
  implicitly).

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Form and validity

• Some arguments are valid purely in virtue of
  form
• All flurgs are turvy
• This is a flurg
• This is turvy
• Some are not
• Shaq is tall
• Shaq is more than 5ft tall

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Validity and Truth

• Whether or not the premises are actually true
  makes no difference to the validity of the
  argument.
• What if the premises are contradictory?
• Vancouver is a big city
• Vancouver is not a big city
• Sarah Palin is a red-neck

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Validity and Truth

• You can never determine that an argument is valid
  by being told the truth values of its sentences.
• You can only determine that an argument is
  invalid by being told the truth values of its
  sentences.
• And you can do that only if the premises are true
  and the conclusion false.

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Why deduction isnt enough

• Scientific argument
• Evidence
• Hypothesis
• The hypothesis usually says more than the
  evidence.

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Inductive Arguments

• An argument is inductively strong if and only if
  it is improbable that its conclusions are false
  given that its premises are true, and it is not
  deductively valid.
• Snow White is doing mushrooms
• Dopey is doing mushrooms
• Doc is doing mushrooms
• Sleepy is doing mushrooms
• Sneezy is doing mushrooms
• Bashful is doing mushrooms
• Happy is doing mushrooms

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Given that

• A good inductive argument
• There is a 2000 year old man in Cleveland
• There is a 2000 year old man in Cleveland with 1
  head.
• But it is improbable that the conclusion and
  premises are true.
• What is probable is the conclusion given the
  premises.

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Specific and General

• There is no necessary connection between the
  deductive vs. inductive distinction and the
  general vs. specific distinction.
• Paris is smart
• Anyone who knows Paris knows someone smart
• Car A is a Delorian
• Car B is a Delorian
• Car A is in mint condition
• Car B is in mint condition
• Car A has a top speed of 88 mph
• Car B has a top speed of 88mph.

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Key Features of Inductive Arguments

• 1. They are knowledge expanding
• 2. The premises never confer certainty on the
  conclusion
• Now well go back to thinking about conditionals
  and deductive arguments.

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Sufficient Conditions

• P is sufficient for Q
• If P then Q
• Two mgs of cyanide is sufficient for death.
• If you ingest two mgs of cyanide, you will die.
• (But it may not be necessary.)

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Necessary Conditions

• Q is necessary for P
• If P then Q
• Being 19 (Q) is necessary for drinking alcohol
  (P)
• If you are drinking alcohol (P), then you are
  19 (Q)
• There is no causation from antecedent to
  consequent.
• Note that being 19 isnt sufficient for drinking
  alcohol no-ones forcing you (though it is
  probably sufficient for being allowed).

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If, and only if

• P only if Q
• Q is necessary for P
• You are drinking alcohol only if you are 19
• Being 19 is necessary for drinking alcohol

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If, and only if

• Potenitally confusing grammatical point
• P if Q If Q then P
• Important philosophical point
• P if Q is totally different from P only if Q
• P if Q If Q then P, which gives a sufficent
  condition for P
• But P only if Q gives a necessary condition for
  P
• So, P if Q is completely different from P only
  if Q.
• You are drinking alcohol only if youre 19 --
  True
• You are drinking alcohol if youre 19 False,
  Chris is drinking coke.

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Contrapositive

• If P then Q If not Q then not P
• If you eat the poison cookie you will go blind.
• If you dont go blind, then you didnt eat the
  cookie.
• The antecedent need not be the cause of the
  consequent. The antecedent may even be later.

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Valid form 1 Affirming the antecedent

• If P then Q
• P
• Q
• If Socrates is a man then he is mortal
• Socrates is a man
• So Socrates is mortal

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Valid form 2 Denying the consequent

• If P then Q
• Not Q
• Not P
• If you drink from the Holy Grail you are immortal
• You are not immortal
• So you didnt drink fro the Holy Grail

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Invalid form 1 Affirming the consequent

• If P then Q
• Q
• Therefore P
• If Socrates is a man then he is mortal
• He is mortal
• Therefore he is a man
• Nope. Socrates is the name of my dog.

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Invalid form 2 Denying the antecedent

• If P then Q
• Not P
• Not Q
• If youre my boss I quit
• Im not your boss
• So youre not quitting
• Nope. I won the lottery and Im quitting anyway.

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Denying a conjunction

• Not (P and Q)
• Therefore either not P or not Q
• Its not the case that Madonna and Beyonce will
  win an Oscar
• Therefore either Madonna wont win one or Beyonce
  wont win one.

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Disjunctive Syllogism

• P or Q
• Not P
• Therefore Q
• Bob is either human or Martian
• Bob is not human
• Bob is Martian