More on Conditionals

1
More on Conditionals

• Section 1.4

2
Introduction

• Given a conditional of the form if p then q p ?
  q, we can then form 3 other forms of the
  conditional.
• if q then p, q ? p is called the converse.
• if not p then not q, p ? q, is called the
  inverse.
• if not q then not p, q ? p, is called the
  contrapositive.

3
Converse

• Example let us have 2 statements
• p You play poker.
• q You win money.
• The conditional p ? q is if you play poker, then
  you win money.
• The converse of the conditional q ? p is if you
  win money then you play poker.
• The converse is formed by interchanging the
  premise with the conclusion.

4
Inverse

• The inverse of the conditional p ? q is the
  negation of the premise and the negation of the
  conclusion and keeping the same order.
• The inverse is p ? q.
• Using the previous p and q. The inverse of the
  conditional is if you do not play poker then you
  do not win money.

5
Contrapositive

• When you negate the premise and conclusion of the
  conditional, and interchange the premise an
  conclusion, then you form the contrapositive of
  the conditional.
• The contrapositive of p ? q is q ? p.
• Using the previous p and q the contrapositive is
  if you do not win money, the you do not play
  poker.

6
Equivalent Conditionals

• The reason we look at these different forms of
  the conditional is that there are 2 pairs of
  equivalent statements.
• The conditional is equivalent to the
  contrapositive.( p ? q q ? p)
• The inverse is equivalent to the converse. ( q ?
  p p ? q)

7
Truth Tables
8
The only if connective

• You have a picnic only if today is a holiday.
• This is translated as if you have a picnic, the
  today is holiday.
• The statement that follows the only if is the
  conclusion of the conditional.
• p only if q is equivalent to p ? q or
  if p then q.

9
The if connective

• Be careful when reading conditionals that have
  the connective word if in the middle of the
  sentence.
• Example Lets look at a conventional
  conditional. If the Eagles beat the Falcons, then
  Mr. D will go to Jacksonville.
• This is of the form If p, then q. Where p is the
  simple statement The Eagles beat the Falcons. q
  is the simple statement Mr. D will go to
  Jacksonville. In symbols this is p ? q.
• Heres the tricky part, I can rewrite the
  original conditional with the word if in the
  middle of the sentence.
• Mr. D will go to Jacksonville, if the Eagles beat
  the Falcons.
• Notice that I reverse the p and q statements.
• In English the sentence is read q if p. This has
  the same meaning as If p, then q.

10
Biconditional p ?? q

• The biconditional is a two way conditional or two
  way implication.
• The symbol p ?? q is the same as saying p ? q and
  q ? p.
• We use the key words if and only if.
• A citizen is eligible to vote if and only if
  the citizen is at least 18 years of age.
• Which is the same as saying if the citizen is
  eligible to vote then the citizen is at least 18
  years of age and the citizen is at least 18
  years of age then the citizen is eligible to
  vote.