Reason and Argument

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Reason and Argument

• Chapter 6 (1/4)

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Common misperceptions about logic

• The science of Deduction and Analysis is one
  which can only be acquired by long and patient
  study, nor is life long enough to allow any
  mortal to attain the highest possible perfection
  in it. From A Study in Scarlet

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A joke

• Sherlock Holmes and Dr. Watson go on a camping
  trip. After a good dinner and a bottle of wine,
  they retire for the night, and go to sleep. Some
  hours later, Holmes wakes up and nudges his
  faithful friend. "Watson, look up at the sky and
  tell me what you see.""I see millions and
  millions of stars, Holmes" replies Watson."And
  what do you deduce from that?"Watson ponders for
  a minute."Well, astronomically, it tells me that
  there are millions of galaxies and potentially
  billions of planets. Astrologically, I observe
  that Saturn is in Leo. Horologically, I deduce
  that the time is approximately a quarter past
  three. Meteorologically, I suspect that we will
  have a beautiful day tomorrow. Theologically, I
  can see that God is all powerful, and that we are
  a small and insignificant part of the universe.
  What does it tell you, Holmes?" Holmes is silent
  for a moment.

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The punchline

• "Watson, you idiot!" he says. "Someone has stolen
  our tent!"

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Common misperceptions about logic

• Typically, Spock said nothing about logic, per
  se.
• Whenever Spock would claim that something was or
  was not logical, he generally meant rational.

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A note about the rules of logic

• The rules of logic are not made up or stipulated,
  or even proved (though they can be derived from
  one another like the axioms of Euclidian
  geometry, and as with any axiomatic system, at
  least one axiom will always remain unproven)
• The rules of logic are discovered, and are lent
  force by the very fact that theyre obvious,
  otherwise we wouldnt be able to call them rules
  of logic.
• Well start with the simplest rules the rules
  for and

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But first, Propositions

• The textbook authors are (understandably)
  imprecise about talk of propositions. To wit
  John is Tall is not a proposition, just like
  3 is not a number.

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Propositions

• John is tall is a sentence that expresses the
  proposition that John is tall.
• 3 is a numeral that expresses what we mean by
  the number 3.
• This is important because one should not get the
  idea that any sentence can express a proposition.
  To express a proposition, a sentence must be an
  example of a linguistic act.

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Propositional Form and Substitution Instances

• p q represents any two joined propositions.
• Though the rule in the book (p. 144) allows that
  different variables be replaced by the same
  proposition, for practical purposes we will never
  do this.
• Also contrary to the text, Roses are red and
  violets are blue does not represent a single
  proposition, but instead a conjunction of two of
  them.
• The 8th edition of the book inserts a
  justification for this based on an analogy to
  mathematics. My reply logic, whatever it looks
  like, is not precisely math. We have good reasons
  to limit one proposition to one propositional
  variable and vice versa.

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Something else about propositions

• Propositions are bearers of truth-values.
• That means that any given proposition can have
  the property of being true or the property of
  being false, and all propositions have one or the
  other (which is why we insist on the linguistic
  act constraint)
• Does that mean that we must view truth as a
  black-and-white kind of thing? Well, yes, it
  does.

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Got a problem with black and white? Why dont you
tell me that to my face! Thought so
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Bearers of truth-value

• Since propositions are expressed by sentences
  that are meaningful, they reflect states of
  affairs. In other words, they reflect the way
  things are or are not.
• Take the proposition expressed by the sentence
  John is tall.
• The proposition is true if it is considered in a
  state of affairs in which John is tall and it is
  false if it is considered in a state of affairs
  in which John is not tall.
• Notice that whether we agree about the state of
  affairs is a different question.

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How truth tables work

• The leftmost columns are called reference columns
  and contain each individual propositional
  variable (or sentence), usually in alphabetical
  order.
• There is one remaining column for each connective
  (, v, , ?) used.
• Each row of a truth table corresponds to one
  possible state of affairs.
• Every possible state of affairs is represented on
  a truth table. The number of rows is 2n where n
  is the number of reference columns.

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The truth table for
p q p q
T T T
T F F
F T F
F F F
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Propositional versus nonpropositional conjunction

• Exercise IV
• 1. nonpropositional
• 2. nonpropositional
• 3. nonpropositional
• 4. propositional
• 5. propositional
• 6. nonpropositional
• 7. nonpropositional
• 8. either (ambiguous)
• 9. nonpropositional

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Validity for
conclusion premise
p q p q
T T T
T F F
F T F
F F F
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How about this one?
conclusion premise
p q p q
T T T
T F F
F T F
F F F
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What about this?
premise conclusion
p q p q
T T T
T F F
F T F
F F F
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Notice
premise conclusion
p q p q
T T T
T F F
F T F
F F F
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How about
premise premise conclusion
p q p q
T T T
T F F
F T F
F F F
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So any substitution instances of the following
will ALWAYS be valid.

• p q
• p
• p q
• q
• p
• q___
• p q

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Exercise V

• 1. valid
• 2. not valid
• 3. valid (trivially)
• 4. valid
• 5. valid (though conversationally, a different
  meaning is implied)
• 6. nonpropositional, but valid (would require
  predicate logic to demonstrate)

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Exercise VI

• 1. True
• 2. True
• 3. True

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Exclusive vs. Inclusive or

• Sometimes when a person says something of the
  form p or q they mean p or q or both and
  sometimes they mean p or q and not both. The
  former is an inclusive or and the latter is
  exclusive.
• Most logicians default to the inclusive or.
  Some even claim that all uses of or are
  inclusive, and it is conversational implication
  that makes some of them exclusive.
• In any case, it is important to examine cases
  where or is used to determine which is which,
  because it will affect the validity of any
  argument that or is used in.

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Disjunction
p q p v q
T T T
T F T
F T T
F F F
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Negation

• It is tempting to say that Smurfs are blue and
  Smurfs are not blue are sentences that express
  two propositions.
• That is not the case. What is going on is that
  the same proposition is involved, and in one case
  the proposition is negated.
• If s stands for Smurfs are blue and is
  our symbol for negation, then Smurfs are not
  blue is formalized as s.

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Be careful with Negation

• Sometimes not is syntactically ambiguous.
  Translating as it is not the case that can
  help to disentangle ambiguity.
• Be careful with opposites.
• nobody owns Mars is the negation of somebody
  owns Mars because it is not the case that
  somebody owns Mars means the same thing as
  nobody owns Mars
• However, some opposites are not binary. Consider
  Cheering for the Yankees is moral. The
  negation of this should just be It is not the
  case that cheering for the Yankees is moral.
  Resist the temptation to translate the negation
  as Cheering for the Yankees is immoral. This
  is because actions that are not moral could be
  either amoral or immoral (but not both).
• The point is, just be strict in translating
  as it is not the case that

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Further ambiguity in negation

• Consider (Everyone loves running)
• Not everyone loves running
• Everyone does not love running
• Everyone loves not running
• No one loves running
• Everyone hates running
• Everyone loves walking
• For the sake of Pete, just say It is not the
  case that everyone loves running

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Disjunctive Syllogism

• Consider the argument
• p v q
• p
• q

C P2 P1
p q p p v q
T T
T F
F T
F F
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Disjunctive Syllogism

• Consider the argument
• p v q
• p
• q
• VALID

C P2 P1
p q p p v q
T T F T
T F F T
F T T T
F F T F
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Consider the Argument

• p v q
• p___
• q

P2 C P1
p q q p v q
T T
T F
F T
F F
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Consider the Argument

• p v q
• p___
• q
• INVALID

P2 C P1
p q q p v q
T T F T
T F T T
F T F T
F F T F
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Pay attention to parentheses

• Notice that a g means something different than
  (a g).
• Substitute Annie is rich for a and Gina is
  happy for g.
• The first phrase translates to It is not the
  case that Annie is rich and it is the case that
  Gina is happy.
• The second phrase translates to It is not the
  case that both Annie is rich and Gina is happy.
• How about a g?

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Logic and Math

• I know that logic LOOKS for all the world like
  math, but resist the temptation to treat
  mathematical symbols and logical symbols as
  interchangeable.
• For example, math has parentheses, and also has a
  negative symbol, - that looks a bit like
  logics negation symbol , so since changing
  (2 3) to -2 -3 is a mathematically valid
  procedure, changing (p q) to p q should be
  logically valid, right?

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Equivalence of a g, (a g), a g
a g a g a g a g (a g) a g
T T F F T F F F
T F F T F F T F
F T T F F T T F
F F T T F F T T
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Exercise XII

• 15. A v ((B C) v (B v (Z v B)))
• T v ((T T) v (T v (F v T)))
• T v ((T T) v (T v T))
• T v ((T T) v (T v T))
• T v ((F T) v (F v F))
• T v ((F T) v (F v F))
• T v (F v F)
• T v (F v F)
• T v (F v T)
• T v (F v T)
• T v T
• T v T
• T