PHIL 120: Jan 8

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PHIL 120 Jan 8

• Basic notions of logic
• More implications of the recursive definition of
  SL
• Kinds of sentences
• Main connectives and sentential components
• Symbolizing sentences with more than one
  connective

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Argument

• An argument is a set of 2 or more sentences, one
  of which is the conclusion and the others
  premises.
• Standardized form
• Premise
• Another premise (if there is one)
• And another (if there is one)
• -------------------------------------------------
  ---
• Conclusion

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Arguments

• Premise indicator terms because, since, for
  the reason that
• Conclusion indicator terms therefore, thus,
  it follows that, so
• One compound sentence can contain an argument
• Sarah will do well in logic because she studies
  and anyone who studies does well in logic

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Arguments

• Sarah will do well in logic because she Sarah
  studies and anyone who studies does well in
  logic
• Sarah studies.
• Anyone who studies does well in logic.
• --------------------------------------------------
  -
• Sarah will do well in logic.

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Arguments

• Famous arguments
• Descartes cogito
• I think, therefore I am
• Cogito, ergo sum
• John Locke (and Samuel Johnson)
• Men (humans) are free by nature.
• As they are born free, no other man has the
  right to claim complete and total power over
  another man.
• --------------------------------------------------
  -------------
• Governments can only legitimately rule just as
  long as men, who are free by nature, consent to
  it.

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Arguments

• Arguments and philosophy
• 2008 UW Philosophy Club tee shirt
• On the front
• Sketches of Plato, Socrates, Aristotle, Newton,
  Descartes, etc.
• Great minds dont just think
• On the back
• They argue

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Basic notions of logic validity

• An argument is deductively valid if and only if
  it is not possible for the premises to be true
  and the conclusion false.
• An argument is deductively invalid if and only if
  it is possible for the premises to be true and
  the conclusion false.
• Every argument is either deductively valid or
  deductively invalid.
• Only arguments are valid or invalid.

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Validity A matter of form rather than content

• All men are mortal.
• Socrates is a man.
• -------------------------
• Socrates is mortal.
• If it is true that all men are mortal and that
  Socrates is a man, then it must be true that
  Socrates is a mortal.

• All men are green.
• Socrates is a man.
• -------------------------
• Socrates is green.
• If it true that all men are green

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Venn Diagram of the two categories and of their
potential overlap
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The first premiseThere are no men who arent
mortal, hence the part of men that doesnt
overlap with mortals is empty
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The second premise using X for SocratesAs he is
a man, he must be in that portion of the category
men that is not empty and that overlaps with
mortal. So, once we diagram the two premises,
the conclusion is already diagrammed.
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Basic notions of logic that apply to sentences

• Logical truth a sentence is logically true if
  and only if it is not possible for the sentence
  to be false.
• 2 2 4
• A rose is a rose
• If it rains, then it rains
• Logical falsity a sentence is logically false
  if and only if it is not possible for the
  sentence to be true.
• If it rains, then it doesnt rain.
• A rose is not a rose.

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Basic notions of logic that apply to sentences

• Logical indeterminacy a sentence is logically
  indeterminate if and only if it is neither
  logically true nor logically false.
• Most sentences are logically indeterminate though
  all sentences are either true or false.
• The truth or falsity of logically indeterminate
  sentences is determined by the way the world is.
• The earth is a sphere
• Copper conducts electricity
• All men are mortal

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Basic notions of logic that apply to sentences

• Logical equivalence the members of a pair of
  sentences are logically equivalent iff it is not
  possible for one of the members of the pair to be
  true while the other is false.
• Neither Mary nor John went to the store
• Both Mary didnt go to the store and John didnt
  go to the store
• It follows that
• The sentences of any pair of logical truths
  (logically true sentences) are logically
  equivalent
• The sentences of any pair of logical falsehoods
  (logically false sentences)

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Basic notions of logic that apply to sentences

• Strange sentences
• This sentence is false.
• (The liars paradox)
• Strange pairs of sentences
• On the front of a tee shirt
• The sentence on the back of this shirt is false.
• On its back
• The sentence on the front of this shirt is true.

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Basic notions of logic that apply to sets of
sentences

• Logical consistency A set of sentences is
  logically consistent iff it is possible for all
  the members of that set to be true.
• A set of sentences is logically inconsistent iff
  it is not possible for all the members of that
  set to be true.
• Put another way logically consistent sets of
  sentences do not contain any contradictions
  logically inconsistent sets of sentences do.

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Basic notions of logic that apply to sets of
sentences

• Questions
• If a set of sentences includes only logically
  indeterminate sentences could it be logically
  consistent?
• If a set of sentences includes one logically
  false sentence could it be logically consistent?
• Could a set of sentences each one of which is
  logically true be logically inconsistent?
• Could a set of sentences, none of which includes
  the , be logically inconsistent?

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Implications of the definition of validity

• An argument is deductively valid iff it is not
  possible for the premises to be true and the
  conclusion false.
• From which it follows that valid arguments can
  have
• True premises and a true conclusion
• False premises and a false conclusion
• False premises and a true conclusion
• Only arguments with true premises and a false
  conclusion cannot be valid.

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Implications of the definition of validity

• A valid argument with a false premise and false
  conclusion
• All men are green.
• Socrates is a man.
• -------------------------
• Socrates is green.
• A valid argument with false premises and a true
  conclusion
• If the sun didnt rise 01/06/2009, PHIL 120 met
  at 1130 that day as scheduled.
• The sun didnt rise 01/06/2009.
• ------------------------------------------------
• Phil 120 met at 1130 that day as scheduled.

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Implications of the definition of validity

• Special cases of validity
• All arguments containing one logically-false
  premise are deductively valid.
• Why?
• All arguments whose premises form a logically
  inconsistent set are deductively valid.
• Why?
• All arguments with a logically true conclusion
  are deductively valid.
• Why?

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Logical soundess

• An argument is deductively sound if and only if
  it is deductively valid and its premises are
  true.
• So valid arguments with one or more false
  premises are not sound arguments.
• And valid arguments with one or more logically
  false premises are not sound arguments.
• And valid arguments whose premises form an
  inconsistent set are not sound arguments.

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Part 2

• Further implications of the recursive definition
  of SL

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Implications of the recursive definition

• All sentences of SL are one of the following
• An atomic or simple sentence (A, B Z)
• A negation (a sentence of the form P)
• A conjunction (a sentence of the form (P Q)
• A disjunction (a sentence of the form (P v Q)
• A material conditional (a sentence of the form
• (P ? Q)
• A biconditional (a sentence of the form (P ? Q)
• This means that no matter how many connectives a
  sentence may contain, it has one (and only one)
  main connective.

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Implications of the recursive definition

• Note in practice, we drop the outside premises
  of a whole sentence.
• Take the sentence
• (A v B) C
• Our question which kind of sentence is it? Put
  another way, what is the main connective of this
  entire sentence?
• One clue the is the only binary connective
  that is not inside parentheses
• So the entire sentence is a conjunction whose
  conjuncts are A v B and C

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Implications of the recursive definition

• The sentence (A v B) C
• is a conjunction whose conjuncts are
• A v B and C
• To determine whether the entire sentence is true
  or false, we need to determine if the left
  conjunct (A v B) is true or false and whether the
  right conjunct (C) is true or false.
• The parentheses around A v B indicate that it
  is a disjunction that forms one conjunct of the
  entire sentence.

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Implications of the recursive definition

• Compare the sentence
• (A v B) C
• To the sentence
• A v (B C)
• In the second sentence v is the only binary
  connective outside of parentheses indicating that
  the entire sentence is a disjunction whose
  disjuncts are A and B C
• The left disjunct is the atomic sentence A the
  right disjunct is the conjunction B C, the
  left conjunct is B and the right is C

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Implications of the recursive definition

• Compare the sentence
• (A v B) C
• To the sentence
• (A v B) ? C
• Here the only binary connective outside of
  parentheses is the ? indicating that the sentence
  is a material conditional whose antecedent is A
  v B and whose consequent is C.
• A v B is a disjunction.
• And C is a negation.

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Implications of the recursive definition

• Compare the sentence
• (A v B) C
• To the sentence
• (A v B) ? C
• Here the only connective outside of parentheses
  is the indicating that the sentence is a
  negation, a sentence of the form P, where P is
  the conditional (A v B) ? C
• The antecedent of this conditional is the
  disjunction A v B and the consequent is the
  atomic sentence C

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• So every sentence of SL that is not atomic has a
  main connective (and this determines what kind of
  sentence it is).
• For a sentence of the form P, P is its immediate
  component
• For sentences of the forms P Q, P v Q, P ? Q,
  or P ? Q, P and Q are the immediate components
• If an immediate component itself has a
  connective, it too has a component or 2
  components.
• And the atomic components of a sentence are all
  the components that are atomic.
• Finally, the sentential components of a sentence
  include the sentence itself.

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Main connectives and sentential components

• Take the sentence (A v B) ? (A B)
• Its sentential components include it
• and its immediate components
• (A v B) and (A B)
• That in turn have components, respectively
• A, B and A B
• A and B are atomic (so atomic components of the
  whole sentence)
• A B has the components A and B
• That, respectively, have the components A and B

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Part 3

• Symbolizing sentences
• with more than 1 connective

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Symbolizing

• Sentences with more than one connective
• Sentences with one binary connective and a tilde
• Sentences with more than one binary connective
  (with or without one or more tildes)
• Compare the sentences
• Bob jogs regularly but Carol doesnt.
• Although Carol doesnt jog regularly, Bob does.
• B C
• C B
• They are logically equivalent.

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Symbolizing

• Underline simple declarative sentences and choose
  sentence letters for them
• Identify the main connective (if there is one),
  and any other connectives
• Begin to build the sentence
• Bob jogs regularly if Carol doesnt
• B C
• Bob jogs regularly if it is not the case that
  Carol jogs regularly
• C ? B

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Symbolizing

• Alice and Bob jog regularly if and only if Carol
  or Dan do not jog regularly
• A Alice jogs regularly B Bob jogs regularly
• C Carol jogs regularly D Dan jogs regularly
• Main connective ?
• Immediate sentential components
• A B and C v D
• (A B) ? (C v D)
• OR
• (A B) ? (C D)

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Symbolizing

• Either Alice, Bob or Carol jogs regularly.
• A Alice jogs regularly B Bob jogs regularly
• C Carol jogs regularly
• Main connective v
• Either
• (A v B) v C
• OR
• A v (B v C)
• NOT
• A v B v C

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Symbolizing

• If Carol jogs regularly, then if Bob does too, so
  does Alice.
• C Carol jogs regularly
• B Bob jogs regularly
• A Alice jogs regularly
• Main connective ?
• But there are 2 sentences of the form P ? Q.
• Which contains the main connective?
• C ? (B ? A)