Conditional Propositions and Logical Equivalence

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Conditional Propositions and Logical Equivalence
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Johnsonbaugh Chapter 1.1
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The purpose of studying logic is to determine if
our reasoning is correct. It is not concerned
with determining if a statement is true or false.

• There is a big difference between mathematical
  argument and rhetorical argument the former is
  concerned primarily with valid, incontrovertible
  reasoning, while the latter is more concerned
  with persuasion.
• Persuasion depends heavily on the content or
  meaning of the statements used. Logic depends on
  the relationship between statements, and is
  otherwise unconcerned with content.

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For this reason, logic lends itself to a
mathematical treatment, leading to a kind of
algebra of symbolic manipulation.

• We will therefore concentrate on three steps
• Translation of common English statements into
  symbolic notation. Why do we do this? Because
  English statements can often be misinterpreted or
  taken out of context. Consider for the example
  the following actually headlines
  http//www.antion.com/humor/speakerhumor/headline
  s.htm or http//freespace.virgin.net/mark.fryer/
  headlines2.html Add to this the line from a
  police blotter Officers sent to kick teenagers
  off roof of
• Symbolic manipulation of this notation.
• Translation of symbolic notation back into common
  English statements.

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First we define our "playing pieces"

• Definition A proposition is a statement that is
  either true or false, but not both.
• We need to worry about this definition a little,
  and fine-tune our understanding of what this
  means. English has so many shades of meaning and
  is subject to so many different interpretations
  that it can be rather ambiguous at times.

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First of all, the statement needs to be
explicitit cannot contain any "variables" that
could take on different values and change whether
the statement is true or false, and must be
objective.

• "October 26, 2001 is a Friday." is a proposition
  (it is true).
• "Charles Dickens wrote Moby Dick." is also a
  proposition (it is false).
• "Alfred Hitchcock was a brilliant director." is
  NOT a proposition, since it is subjective, and
  not everyone would agree on the truth or falsity
  of the claim.
• "Go home!" is NOT a proposition. It is neither
  true nor false.
• "JFK was shot by a lone gunman." is a
  proposition. It is either true or false, though
  we may not know which.
• "x 22" is NOT a proposition. Its truth value
  depends on what x is.
• A little more subtle would be something like
• "Today is Wednesday."
• "I danced with your grandfather."

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Sentences like these contain variables today,
I, you, your grandfather. The truth or falsity of
these sentences depends on the value of these
variables. However, these statements are
generally made in context, and the context
determines the truth of the sentence.

• If I said "Today is Wednesday" today, and said it
  again tomorrow, the sentence might be true one
  day, and false the next.
• If your grandmother said "I danced with your
  grandfather," the sentence would likely be true.
  If I said the same thing, it would certainly be
  false unless you have a square dancing
  grandfather, in which case it could be true. So
  this sentence depends on who says it, to whom
  it was said, and which of your grandfathers was
  referred to!
• We will often accept these context-dependent
  sentences as propositions, as long as we are
  aware of their context-dependency.

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Fortunately, in mathematics, we can avoid most of
the context dependency, and when we cannot, we
make the context explicit.

• Notation We will denote propositions with
  lower-case letters p, q, r, s, ...
• Connectives In ordinary speech, we often
  combine small statements into larger, more
  complex statements. This happens routinely in
  mathematics, too.
• "If
• is invertible (has an inverse), then it is
  row-equivalent to I (eros can transform it into
  I), and its determinant is nonzero."

1 2 3 7
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We concern ourselves with a handful of ways to
build such compound propositions from simple
propositions. The main four are

• AND, OR, NOT, IMPLIES.
• There are others (XOR, NAND, NOR, ...) but they
  can all be constructed from the basic four.
• Definition If p and q are propositions, we
  define the compound proposition p and q,
  denoted p?q to be true whenever both p and q are
  true, and false whenever at least one of p and q
  is false.

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Notation To compactly and precisely represent
this type of compound statement, we use a truth
table.

• A truth table lists all possible combinations of
  the truth values of the component propositions
  (one combination per line ), and has a column for
  each connective to be executed. This column
  gives the truth value of the compound proposition
  at the top of the column for each possible
  combination of truth values of the component
  propositions.
• As usual, this is easier to do than to explain
• The truth table for AND

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Example

• Let p be Alfred Hitchcock directed Rear Window
• Let q be Rear Window starred Cary Grant
• Then p?q is the proposition Alfred Hitchcock
  directed Rear Window and Rear Window starred Cary
  Grant.
• This might be shortened to Alfred Hitchcock
  directed Rear Window which starred Cary Grant,
• Or even
• Alfred Hitchcock directed Rear Window, starring
  Cary Grant.
• If you know your movie trivia, you know this is a
  false statement. Although Hitchcock did direct
  Rear Window, the male lead was Jimmy Stewart, not
  Cary Grant. So even though half of the
  statement was true, the second half forced the
  combination to be false. (This statement falls
  into row 2 of our truth table.)

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Definition If p and q are propositions, we
define the compound proposition p or q, denoted
p?q to be true whenever at least one of p and q
are true, and false whenever both p and q are
false.

• The truth table for OR
• Example
• Let p be Rear Window starred Jimmy Stewart
• Let q be Rear Window starred Cary Grant
• Then p?q is the proposition Rear Window
  starred Jimmy Stewart, or Rear Window starred
  Cary Grant, which would probably be rewritten
  as Rear Window starred Jimmy Stewart or Cary
  Grant.
• This is a true statement, because Rear Window did
  indeed star Jimmy Stewart, and the truth of this
  part of the compound statement was enough to make
  the entire statement true. (row 2 of the OR table)

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Example

• Let p be Rear Window starred Jimmy Stewart
• Let q be Rear Window starred Grace Kelly
• Then p?q is the proposition Rear Window
  starred Jimmy Stewart, or Rear Window starred
  Grace Kelly, which would probably be rewritten
  as Rear Window starred Jimmy Stewart or Grace
  Kelly.
• This is a true statement, because Rear Window did
  not only star Jimmy Stewart, but also starred
  Grace Kelly. This was a kind of overkill,
  because the truth of just one part of the
  compound statement would be enough to make the
  entire statement true. Note, however, that OR
  does not require one of its components to be
  false!
• This is called an inclusive orthe statement is
  true even when both components are true.
• If you want to use OR in the sense of one or the
  otherbut you cant have both, you need an
  exclusive or. This is a different connective
  (XOR).
• Dont confuse the two!

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Definition If p is a proposition, we define
the compound proposition not p, denoted ?p, p,
or p to be true whenever p is false and false
whenever p and q is true.

• The truth table for NOT
• Note NOT is often referred to as negation.

p p
T F F T
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This will be a tricky one to translate, since we
negate statements in a variety of waysalmost
none of them have the word not in front of an
expression....
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Example

• Let p be the proposition Alfred Hitchcock won
  an academy award for directing.
• Then is the proposition It is not the case
  that Alfred Hitchcock won an academy award for
  directing,
• which would more commonly be phrased Alfred
  Hitchcock did not win an academy award for
  directing.

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Exercises 9-12, 13-23 (odd), 31-57 (odd). There
will be a quiz on this material.