Sentential Logic

1
Sentential Logic

• Symbolization and Syntax

2
Review and Examples

• Key
• A Anna will pass the course.
• B Bob will pass the course.
• C Chuck will pass the course.
• D Anna is always late for class.
• E Bob studies hard.
• F Chuck failed all the quizzes.

3
Review and Examples

• They wont all pass the course.
• (I.e., Its not the case that Anna and Bob and
  Chuck will pass the course)
• (A (B C))

4
Review and Examples

• At least two of them will pass the course.
• (Either Anna and Bob will, or Anna and Chuck
  will, or Bob and Chuck will)
• (A B) v ((A C) v (B C))

5
Review and Examples

• Exactly two of them will pass the course.
• Remember, this is the same as saying that at
  least two will pass and at most two will pass.
• At least (A B) v ((A C) v (B C))
• At most A v (B v C) (i.e., at least one
  wont pass).
• ((A B) v ((A C) v (B C))) (A v (B v
  C))

6
Review and Examples

• Anna will pass unless she is always late.
• A v D
• A É D
• D É A

7
Review and Examples

• Anna and Bob will pass the course only if she
  isnt always late for class and he studies hard.
• (A B) É (D E)

8
Review and Examples

• If any of them passes the course, so will the
  other two.
• (If Anna or Bob or Chuck pass the course, then
  the other two will)
• (A v (B v C)) É (A (B C))

9
Review and Examples

• If Anna is always late for class then she will
  pass the course only if neither Chuck nor Bob do.
• D É (A É (B v C))

10
Review and Examples

• Provided Chuck didnt fail all the quizzes and
  Bob studied hard, Anna will pass the course if
  either of the other two do.
• (F E) É ((B v C) É A)

11
The Syntax of SL

• We want to provide a rigourous definition of a
  sentence of SL.
• This is achieved by using a recursive definition.
• In the definition we will use the metavariables P
  and Q to represent sentences of SL.

12
The Syntax of SL

• A sentence of SL is defined as follows
• Every sentence letter is a sentence of SL.
• If P is a sentence of SL, the P is a sentence of
  SL.
• If P and Q are sentences of SL, then (P Q) is a
  sentence of SL.
• If P and Q are sentences of SL, then (P v Q) is a
  sentence of SL.
• If P and Q are sentences of SL, then (P É Q) is a
  sentence of SL.
• If P and Q are sentences of SL, then (P Q) is a
  sentence of SL.
• Nothing is a sentence unless it can be formed by
  repeated applications of clauses 1-6.

13
The Syntax of SL

• Using the recursive definition, we now have an
  effective way of determining for any string
  whether it is a sentence of SL.

14
The Syntax of SL

• Are the following sentences of SL?
• H
• F
• (A B v C)
• M N
• A B
• (A B) v ((C É D) v F

15
Sentential Logic

• Semantics

16
Semantics

• In semantics, we are concerned with the concepts
  of
• truth-functional truth
• truth-functional falsity
• truth-functional indeterminacy
• truth-functional consistency
• truth-functional validity
• truth-functional entailment
• truth-functional equivalence

17
Semantics

• We will be using truth-tables to test the various
  sentences, sets, arguments for these semantic
  properties.
• In the same way that every molecular sentence is
  constructed out of atomic sentences (according to
  our definition of sentence of SL), so to the
  truth value of every sentence can be determined
  solely by determining the truth-value of its
  atomic components.

18
How to construct a truth table

• Identify all the atomic components of the
  sentence.
• e.g. if you are constructing a t-table for
• (A B) É (C B)
• the atomic components are
• A B C

19
How to construct a truth table

• Construct a table by putting the atomic
  components (in alphabetical order) on the left
  and the molecular sentence on the right.
• e.g.
• A B C (A B) É (C B)

20
How to construct a truth table

• List all the possible truth-value assignments of
  the atomic components.
• If there are n atomic components, there are 2n
  rows in the table.
• In our example, since there are 3 atomic
  components, there will be 8 rows on our table.

21
How to construct a truth table

• A B C (A B) É (C B)
• T T T
• T T F
• T F T
• T F F
• F T T
• F T F
• F F T
• F F F

22
How to construct a truth table

• Determine the truth-values for each connective
  for every possible truth-value assignment.
• Start with the inner-most connectives and work
  your way out to the main sentential connective.
• That is, you cant determine the truth value of
  the main sentential connective until the
  truth-values for all the other connectives has
  been determined.

23
How to construct a truth table

• A B C (A B) É (C B)
• T T T
• T T F
• T F T
• T F F What is the main sentential
• F T T connective?
• F T F
• F F T
• F F F

24
How to construct a truth table

• A B C (A B) É (C B)
• T T T
• T T F
• T F T
• T F F To determine the t-values of
• F T T É we first need to determine
• F T F the values of (A B) and
• F F T (C B)
• F F F

25
How to construct a truth table

• A B C (A B) É (C B)
• T T T
• T T F
• T F T
• T F F Remember, conjunctions are true
• F T T only when both conjuncts are true.
• F T F
• F F T
• F F F

26
How to construct a truth table

• A B C (A B) É (C B)
• T T T T
• T T F T
• T F T F
• T F F F
• F T T F
• F T F F
• F F T F
• F F F F

27
How to construct a truth table

• A B C (A B) É (C B)
• T T T T
• T T F T
• T F T F To get the value of
• T F F F (C B) we need to
• F T T F determine the values of
• F T F F C and B
• F F T F
• F F F F

28
How to construct a truth table

• A B C (A B) É (C B)
• T T T T
• T T F T
• T F T F Negations take the
• T F F F opposite value of the
• F T T F sentence it negates
• F T F F
• F F T F
• F F F F

29
How to construct a truth table

• A B C (A B) É (C B)
• T T T T F
• T T F T T
• T F T F F
• T F F F T
• F T T F F
• F T F F T
• F F T F F
• F F F F T

30
How to construct a truth table

• A B C (A B) É (C B)
• T T T T F F
• T T F T T F
• T F T F F T
• T F F F T T
• F T T F F F
• F T F F T F
• F F T F F T
• F F F F T T

31
How to construct a truth table

• A B C (A B) É (C B)
• T T T T F F
• T T F T T F
• T F T F F
  T Biconditionals
• T F F F T T are
  truth when
• F T T F F F both
  sides have
• F T F F T F the
  same value.
• F F T F F T
• F F F F T T

32
How to construct a truth table

• A B C (A B) É (C B)
• T T T T F T F
• T T F T T F F
• T F T F F F
  T Biconditionals
• T F F F T T T are
  truth when
• F T T F F T F both
  sides have
• F T F F T F F the same
  value.
• F F T F F F T
• F F F F T T T

33
How to construct a truth table

• A B C (A B) É (C B)
• T T T T F T F
• T T F T T F F
• T F T F F F T Now for
  the
• T F F F T T
  T conditional.
• F T T F F T F Look at
  the
• F T F F T F F values
  of
• F F T F F F T the
  sentences
• F F F F T T T on the L
  and R

34
How to construct a truth table

• A B C (A B) É (C B)
• T T T T F T F
• T T F T T F
  F Conditionals
• T F T F F F T are only
  false
• T F F F T T T when
  antec. is
• F T T F F T F true and
  consq.
• F T F F T F F is
  false.
• F F T F F F T
• F F F F T T T

35
How to construct a truth table

• A B C (A B) É (C B)
• T T T T T F T F
• T T F T F T F F
• T F T F T F F T
• T F F F T T T T
• F T T F T F T F
• F T F F T T F F
• F F T F T F F T
• F F F F T T T T

36
How to construct a truth table

• What does this table tell us?
• In general, it tells the truth-value of the
  molecular sentence given every possible
  truth-value assignment of the atomic sentences.
• In particular, this t-table tells us that (A B)
  É (C B) is true on every possible truth-value
  assignment of the atomic components, except when
  A is T, B is T and C is F, in which case, (A B)
  É (C B) is false.

37
How to construct a truth table

• A B C (A B) É (C B)
• T T T T T F T F
• T T F T F T F F
• T F T F T F F T
• T F F F T T T T
• F T T F T F T F
• F T F F T T F F
• F F T F T F F T
• F F F F T T T T

38
Examples

• (A B)
• A v (B É C)