Tweedledum:%20

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• Tweedledum I know what youre thinking, but it
  isnt so. No how.
• Tweedledee Contrariwise, if it was so, it might
  be and if it were so, it would be but as it
  isnt, it aint. Thats logic.

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PHIL 120 Introduction to Logic

• Professor
• Lynn Hankinson-Nelson
• lynnhank_at_u.washington.edu
• 206.543.5094
• Instructors
• Lars Enden
• Cheryl Fitzgerald
• Mitch Kaufman
• Joe Ricci

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PHIL 120 Introduction to Logic

• Course website
• http//faculty.washington.edu/lynnhank/PHIL120.htm
  l
• Syllabus and course requirements
• Power point lectures
• Sample tests
• Office hours and locations e-mail addresses.
• Announcements
• Course Text
• The Logic Book, 5th edition. McGraw Hill.
• A solutions manual is available online.

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PHIL 120 Requirements

1. Attendance and participation in lectures and
   discussion sections, including assigned homework
   and pop quizzes (20)
2. 5 tests (16 each)
3. You may take 1 test over to raise the grade
4. Practice, practice, practice logic is not a
   spectator sport!
5. Ask questions!

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Logic

• The study of reasoning
• The drawing of inferences what follows from what
  and what doesnt follow
• We can talk in terms of ideas, beliefs, and
  the like, but its more concrete to talk about
  sentences those entities that we use to
  express ideas, beliefs, and claims.
• One focus arguments
• A set of at least two sentences, one of which is
  the conclusion and the other or others is/are
  reasons (premises) that support it.

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Good arguments vs. bad arguments

• All men are mortal.
• Socrates is a man.
• __________________
• Socrates is mortal.
• This argument is truth-preserving. It is
  deductively valid.
• If it is true that all men are mortal, and true
  that Socrates is a man, then it must be true that
  Socrates is mortal.

• All men are mortal.
• Socrates is mortal.
• ____________________
• Socrates is a man.
• This argument is not truth preserving. It is
  deductively invalid.
• Even if the premises are true, they do not
  guarantee the truth of the conclusion.
• Socrates could be the name of any living thing.

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Logic and humor
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Good arguments vs. bad arguments

• If you studied a lot, you did well in the logic
  course.
• You studied a lot.
• _________________________
• You did well in the logic course.
• This argument is truth-preserving. It is
  deductively valid.
• It is not possible for the premises to be true
  and the conclusion false.

• If you studied a lot, you did well in the logic
  course.
• You did well in the logic course.
• ____________________
• You studied a lot.
• This argument is not truth preserving. It is
  deductively invalid.
• It is possible for the premises to be true and
  the conclusion false.

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The language SL

• A symbolic language used to illustrate the
  logical structure of sentences, of sets of
  sentences, of arguments, and of other
  relationships between sentences
• In sentential logic, the most basic unit is the
  simple declarative sentence.
• Simple no logical connectives
• Declarative either true or false
• We assume bivalence

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The language SL

• The vocabulary of SL
• Roman capital letters, A through Z, with or
  without subscripts (e.g., S and S3) used to
  symbolize simple, declarative sentences
• 5 (sentential) connectives
• (tilde)
• (ampersand)
• v (wedge)
• ? (horseshoe)
• ? (triple bar)
• The first is a unary connective.
• The rest are binary connectives.
• Punctuation ( ) and

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The language SL

• Every sentence of SL is either simple/atomic or
  compound/molecular.
• Simple/atomic sentences have no connectives.
• Compound/molecular sentences have at least one
  connective.
• Meta variables
• Object language and meta language
• P, Q, R, and S are meta variables used to talk
  about sentences of SL.

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The recursive definition of SL

1. Every sentence letter is a sentence.
2. If P is a sentence of SL, P is a sentence of SL.
3. If P and Q are sentences, then (P Q) is a
   sentence.
4. If P and Q are sentences, then (P v Q) is a
   sentence.
5. If P and Q are sentences, then (P ? Q) is a
   sentence.
6. If P and Q are sentences, then (P ? Q) is a
   sentence.
7. Nothing else is a sentence.

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What the recursive definition of SL does

• It tells us what will count as a sentence of SL
• It also tells us what will not.
• For example, these are not sentences of SL
• A
• clause 3 is a binary connective
• B C ?
• SL does not include ?
• must be used before a sentence (clause 2)
  
• and must connect 2 sentences (clause 3)

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What the recursive definition of SL does

• It tells us what will count as a sentence of SL
• These are sentences of SL
• A B (clause 3)
• (B B) B (clause 3)
• B (clause 2)
• A v B (clause 4)
• A ? B (clause 5)
• A ? B (clause 6)

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Using SL to symbolize sentences

• Roman capital letters A through Z (with or
  without subscripts) to symbolize simple
  declarative sentences
• Mary went to the store (we could symbolize as
  M).
• John went to the store (we could symbolize as
  J).
• These are NOT simple declarative sentences
• Either Mary went to the store or John did.
• Mary did not (or didnt) go to the store.

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Using SL to symbolize sentences

• Mary did not (or didnt) go to the store
• The logic of this sentence is
• It is not the case that Mary went to the store
• symbolizes it is not the case that
• So, using M for Mary went to the store, we use
  M to symbolize It is not the case that Mary
  went to the store
• Sentences whose main connective is the tilde are
  called negations.

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The characteristic truth table for the
P P
T F
F T
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Using SL to symbolize sentences

• Mary and John went to the store
• This is a compound/molecular sentence to be
  translated as
• Mary went to the store and John went to the
  store
• We can use M to symbolize Mary went to the
  store, and J to symbolize John went to the
  store
• We use the for and
• So we have
• M J

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The characteristic truth table for the
P Q P Q
T T T
T F F
F T F
F F F
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Using SL to symbolize sentences

• Sentences whose main connective is the are
  called conjunctions and each of the sentences
  connected by the is called a conjunct. We can
  refer to them as the right or the left conjunct.
• We use as a connective in all cases in which
  the compound sentence is only true if both of its
  component sentences are true.
• This is the case for
• Mary went to the store but John did too
• the logical structure of which is
• Mary went to the store and John went to the
  store

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Using SL to symbolize sentences

• Mary went to the store but John did not
• Is paraphrased as
• Mary went to the store and it is not the case
  that John went to the store
• M can symbolize Mary went to the store
• And is used for but. So far we have
• M
• J can symbolize it is not the case that John
  went to the store
• So we have
• M J

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Using SL to symbolize sentences

• Either Mary went to the store or John did
• Is translated as
• Either Mary went to the store or John went to
  the store
• We use the v to symbolize either/or
• If we use M to symbolize Mary went to the store
  and J to symbolize John went to the store we
  symbolize the whole sentence as
• M v J

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The characteristic truth table for v
P Q P v Q
T T T
T F T
F T T
F F F
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Using SL to symbolize sentences

• Because either/or and the v assume the
  inclusive sense of or (at least one is true),
  we will need to do more if we believe a sentence
  makes use of the exclusive sense of or (at most
  one of the two) and should be symbolized to
  reflect this.
• On a restaurant menu, for example, the phrase
  either soup or salad is included reflects the
  exclusive sense of or.
• Context may tell us that a claim comes to Either
  Mary went to the store or John did, but not both

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Using SL to symbolize sentences

• If Mary went to the store, John did
• If Mary went to the store, then John went to the
  store
• M Mary went to the store
• J John went to the store
• We use ? to symbolize if, then
• So we have
• M ? J

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The characteristic truth table for the ?
P Q P ? Q
T T T
T F F
F T T
F F T
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Using SL to symbolize sentences

• The reasoning behind the ?
• Consider the following claim
• If the operation is a success, the patient
  survives.
• The condition, the operation is a success, is a
  sufficient but not a necessary condition for the
  patients survival.
• The docs decided not to operate and the patient
  survived they discovered they were wrong about
  the need for an operation
• If the claim was Only if the operation is a
  success, the patient survives, then the
  operations success would require the patients
  survival for the claim to be true but that is
  not what If by itself entails.

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Using SL to symbolize sentences

• Mary went to the store if and only if John did
• Mary went to the store if and only if John went
  to the store
• We use ? for if and only if
• So we have
• M ? J
• P if and only if Q is equivalent to
• (If P then Q) and (If Q then P)

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The characteristic truth table for the ?
P Q P ? Q
T T T
T F F
F T F
F F T