Sentential Logic

1
Sentential Logic

• Symbolization and Syntax

2
Review

• Last day we looked at 3 truth-functional
  sentential connectives
• Conjunction
• Disjunction
• Negation

3
Conjunction

• P Q P Q
• T T T
• T F F
• F T F
• F F F
• John and Mary were late for class.
• J R

4
Disjunction

• P Q P v Q
• T T T
• T F T
• F T T
• F F F
• The keys are in my office or in my car.
• O v C

5
Negation

• P P
• T F
• F T
• Cook is never late for class.
• C

6
Combinations of Sentential Connectives

• Some sentences contain more than one
  truth-functional sentential connective.
• E.g.
• John and Mary were late for class, but Cook
  wasnt.
• There are 3 connectives in this sentence.

7
Main Sentential Connective

• For each molecular sentence, there is one
  connective which is the main sentential
  connective.
• The main sentential connective (defn) is the
  connective which takes as its scope the whole
  sentence.
• The scope of a connective (defn) are the
  sentences that the connective ranges over.

8
Scope

• For conjunction and negation (as well as the
  material conditional and biconditional) the scope
  of the connective are the shortest sentences to
  the left and right of the connective.
• For negation, its scope is the shortest sentence
  to its right.

9
Scope

• Compare to mathematics
• E.g., compare
• (405)3 The main mathematical operator here is
  
• 40(53) The main mathematical operator here is
  

10
Main Sentential Connective

• The main sentential of a connective in SL is
  determined, as in mathematics, by punctuation
  (i.e., parentheses).
• (A B) v C
• A (B v C)
• (A (B v C))

11
Main Sentential Connective

• Unlike SL or mathematics, however, the main
  connective in English is determined by several
  factors punctuation (parentheses, commas, colons
  and semi-colons, etc.) tone, pauses, etc.

12
Combinations of Sentential Connectives

• E.g., try this
• John and Mary were late for class, but Cook
  wasnt.
• (J M) C

13
Neither nor

• Neither John nor Mary were late for class.
• (J vM)
• J M

14
Not both and

• John and Mary will not both be late for class.
• (J M)
• J v M

15
Exclusive or

• Either Manchester or Liverpool will win the
  league this year.
• M v L (no!this is inclusive)
• (M v L) (M L)

16
Material Conditional

• The material conditional is symbolized in SL as
  É (hook, or horseshoe).
• A molecular sentence is made with the material
  conditional by placing sentences on either side
  of the É.
• E.g.
• J É R
• The sentence to the left of the hook is referred
  to as the antecedent, and the sentence to the
  right is referred to as the consequent.

17
Material Conditional

• P Q P É Q
• T T T
• T F F
• F T T
• F F T

18
Material Conditional

• The material conditional, e.g., P É Q is
  equivalent to
• P v Q
• It is best captured in English by sentences that
  are in the if then form.
• E.g.,If Mary is in Paris, then she is in France.
• Equivalent to Either Mary is not is Paris, or
  she is in France.

19
Material Conditional

• Not all conditional sentences (if then
  sentences) are material conditionals.
• E.g., causal statements
• If this rod is made of metal, then it will expand
  when heated.
• E.g., subjunctive conditionals (counterfactuals)
• If Oswald hadnt killed Kennedy, someone else
  would have.

20
Material Conditional

• Other English words that can be symbolized by P É
  Q
• If P then Q
• Q if P
• P only if Q

21
Material Conditional

• These are equivalent
• If Manchester beats Chelsea, then theyll win the
  league.
• Manchester will win the league if they beat
  Chelsea.
• Manchester beats Chelsea only if they win the
  league.

22
Unless

• The word unless in English can be translated as
  v or É.
• E.g.,
• Manchester will win the league unless Rooney gets
  injured.
• M v I
• I.e., Manchester will win the league or Rooney
  will get injured.

23
Unless

• Manchester will win the league unless Rooney gets
  injured.
• Can be reread as
• If Rooney doesnt get injured, then Manchester
  will win the league.
• or
• If Manchester wins the league, then Rooney didnt
  get injured.

24
Unless

• P unless Q
• P Q P v Q P É Q P É Q
• T T T T T
• T F T T T
• F T T T T
• F F F F F

25
Material Biconditional

• The material biconditional is symbolized in SL as
  (triple bar).
• A molecular sentence is made with the material
  biconditional by placing sentences on either side
  of the .
• E.g.
• J R

26
Material Biconditional

• P Q P Q
• T T T
• T F F
• F T F
• F F T

27
Material Biconditional

• In English, the material biconditional is usually
  captured by the phrase if and only if.
• Its best to think of the material biconditional
  and the conjunction of two conditionals, where
  the antecedent and consequent are switched.
• E.g.
• P Q is equivalent to (P É Q) (Q É P)

28
Material Biconditional

• P Q P Q (P É Q) (Q É P)
• T T T T
• T F F F
• F T F F
• F F T T

29
Material Biconditional

• E.g.,
• You will pass this course if and only if you can
  differentiate a subjunctive conditional from a
  material conditional.