PHIL 120: Third meeting

1
PHIL 120 Third meeting

• What to know for Test 1 (in general terms).
• Symbolizing compound sentences (contd)
• Paying attention to English punctuation when
  symbolizing into SL and when and how to use SL
  punctuation () and
• Special phrases and terms
• Basic notions their inter-relationships and
  implications continued
• Open review questions

2
Part 1

• Preparing for Test 1

3
Test 1

• You need to know
• The definitions of basic notions of logic
  introduced in lecture on Chapter 1 (see glossary
  p. 27 you may ignore inductive strength) and
  what they imply
• The recursive definition of SL and what it
  implies about what is (and is not) a sentence of
  SL, that sentences with connectives have a main
  connective, etc.
• The characteristic truth table for each of the 5
  connectives of SL
• How to symbolize simple and compound sentences
  into SL

4
For example, you need to know that and why the
following arguments are deductively valid

• I
• Vitamin C cures all colds.
• Vitamin C never cures colds.
• -------------------------------
• The moon is made of green cheese.

• II
• There are 385 days in a year
• February has 31 days
• -------------------------------
• A rose is a rose
• III
• Im here and nobody is here
• --------------------------
• 2 2 5

5
Part 2 (a)

• Symbolizing compound sentences
• continued

6
Terminology

• Sentences of the form P ? Q are called material
  conditionals. The sentence that follows the
  logical operator if and is symbolized to the
  left of the horseshoe is called the antecedent.
  The sentence that follows the logical operator
  then and is symbolized to the right of the
  horseshoe is called the consequent.
• The sentences connected by the in a conjunction
  are called conjuncts.
• The sentences connected by the v in a disjunction
  are called disjuncts.

7
Paying attention to English punctuation

• If Sarah skis regularly and Otis does too, then
  it is not the case that Sarah jogs regularly.
• Choose atomic sentences to symbolize the simple
  declarative sentences, e.g.,
• S Sarah skis regularly O Otis skis regularly
• J Sarah jogs regularly
• The coma after does too suggests that all that
  comes before it is a compound sentence (S O)
  that is the antecedent of a conditional, the
  consequent of which is J
• So we can use (S O) ? J

8

• If Sarah skis regularly and Otis does too, then
  either it is not the case that Sarah jogs
  regularly or it is not the case that Otis does
  OR
• If Sarah skis regularly and Otis does too, then
  it is not the case that both Sarah jogs regularly
  and Otis does
• Use the coma again to identify the main
  connective as if, then (?) and to identify the
  antecedent and consequent of this conditional
• Using T for Otis jogs regularly
• We can use
• (S O) ? (J v T)
• OR
• (S O) ? (J T)

9
Truth tables of the two forms of the consequent
demonstrate that the sentences are logically
equivalent
J T J v T (J T)
T T F F
T F T T
F T T T
F F T T
10
Paying attention to English punctuation

• Alison works hard although Mark doesnt but if
  Mark is a success, then Alison is too
• The semi-colon indicates that there is a sentence
  before and after it.
• But indicates that we use an to connect the
  sentences before and after the semi-colon.
• A Alison works hard M Mark works hard
• S Mark is a success D Alison is a success
• So we can use
• (A M) (S ? D)

11
Paying attention to English punctuation

• Either Michael, or Roxanne, or Shirley works
  hard but if Michael works hard, then either
  Roxanne doesnt or Shirley doesnt
• Use the semi-colon to identify the sentence as a
  conjunction whose left conjunct is a disjunction
  and right conjunct is a material conditional
• M Michael works hard R Roxanne works hard
• S Shirley works hard
• The left conjunct can be symbolized as
• (M v R) v S
• OR M v (R v S)

12
Using the punctuation of SL

• Either Michael, or Roxanne, or Shirley works
  hard but if Michael works hard, then either
  Roxanne doesnt work hard or Shirley doesnt
• Left conjunct can be symbolized as
• M v (R v S) OR (M v R) v S
• The right conjunct can be symbolized as
• M ? (R v S) OR M ? (R S)
• As each conjunct includes a binary connective, we
  need to use brackets so that it is clear which
  sentences are combined using

13
Using the punctuation of SL

• If we dont use brackets we have the following
• M v (R v S) M ? (R v S)
• (or one of the other versions)
• But this is not a sentence of SL because it has
  no main connective. We have no idea what
  conditions would make it true or false.
• We want one of the versions that includes
  brackets
• M v (R v S) M ? (R v S)
• Here we have a conjunction, with a disjunction as
  the left conjunct and a material conditional as
  the right conjunct.

14
Using the punctuation of SL

• So we can use any of the following (logically
  equivalent) symbolizations
• M v (R v S) M ? (R v S)
• M v (R v S) M ? (R S)
• (M v R) v S M ? (R v S)
• (M v R) v S M ? (R S)

15
What is and what is not a sentence of SL

• These are sentences of SL
• A
• A22
• (A ? B)
• A ? (B v A) ? M
• These are not sentences of SL (why not?)
• A B
• (A ? B
• A ? (B v A) ? M

16
Part 2 (b)

• Special phrases and terms

17
Only if

• Compare
• (a) If the operation is a success, then the
  patient survives
• (or Provided that the operation is a success,
  then the patient survives)
• with
• (b) Only if the operation is a success, the
  patient survives
• The only time (a) will be false is when the
  operation is a success but the patient does not
  survive.
• If the operation is cancelled or a failure, the
  conditional is true.

18
Only if

• (a) If the operation is a success, the patient
  survives
• (b) Only if the operation is a success, the
  patient survives
• The only time (b) will be false is if the patient
  survives and the operation was not a success.
• Lets use
• O the operation is a success P the patient
  survives
• can be symbolized as O ? S
• can be symbolized as S ? O

19
Truth table for (a) and truth table for (b)Note
differences in rows 2 and 3
O S O ? S S ? O
T T T T
T F F T
F T T F
F F T T
20
Why if and only if works as it does

• We said that a sentence of the form P ? Q
• is logically equivalent to the conjunction of 2
  material conditionals
• Take the sentence
• A ? B
• It is logically equivalent to the sentence
• (A ? B) (B ? A)
• If A then B (A ?B), and if B then A
• OR
• If A then B, and A only if B
• B ? A symbolizes both paraphrases of the right
  conjunct.

21
If and only if
A B A ? B (A ? B) (B ? A)
T T T T
T F F F
F T F F
F F T T
22
Unless'

• Mary jogs unless she is sick
• M Mary jogs. S Mary is sick.
• Can be paraphrased and symbolized as EITHER
• Either Mary jogs or Mary is sick
• M v S
• OR If Mary is not sick, then Mary jogs
• S ? M
• OR If Mary does not jog, then Mary is sick
• M ? S

23
Truth table for unless
M S M v S S ? M M ? S
T T T T T
T F T T T
F T T T T
F F F F F
24
Either/or but not both'

• The v and either/or reflect the inclusive sense
  of or.
• So consider the sentence
• Either Sarah plays poker well or Jack does, but
  not both
• S Sarah plays poker well J Jack plays
  poker well.
• Left conjunct can be symbolized as S v J
• Right conjunct can be symbolized as S v J OR
  (S J)
• The whole sentence can be symbolized as
• (S v J) (S v J)
• OR
• (S v J) (S J)

25
Either/or and not both(or at most one)
S J (S v J) (S J) (S v J) (S v J)
T T F F
T F T T
F T T T
F F F F
26
Neither/nor'

• Consider the sentence
• Neither Alice nor Bruce plays poker
• This is logically equivalent to
• Alice doesnt play poker and Bruce doesnt play
  poker which we paraphrase as
• It is not the case that Alice plays poker and it
  is not the case that Bruce plays poker.
• It is also logically equivalent to
• It is not the case that either Alice or Bruce
  plays poker.

27
Neither/nor'

• Neither Alice nor Bruce plays poker
• A Alice plays poker. B Bruce plays poker.
• It is not the case that Alice plays poker and it
  is not the case that Bruce plays poker
• A B
• It is not the case that either Alice plays poker
  or Bruce plays poker
• (A v B)

28
Neither/nor
A B A B (A v B)
T T F F
T F F F
F T F F
F F T T
29
Connectives that are not truth-functional

• A connective is truth functional if and only if
  it determines the truth value of a sentence given
  the truth values of the sentences immediate
  components.
• Because is not a truth functional connective.
• It can connect 2 sentences that are each true to
  form a true sentence Jan. 19 celebrates MLK
  because he was a great American
• Or connect 2 true sentences that are each true to
  form a false sentence Jan. 19 celebrates MLK
  because 2 2 4

30
Implications of logical notions

• Main connectives matter because they determine
  the truth value of a given sentence (T or F)
  based on their characteristic truth tables and
  the truth values (T or F) of a sentences
  immediate components on each possible truth value
  assignment
• Compare
• (A v B) ? A
• with
• A v (B ? A)

31
The sentences truth tables demonstrate that they
are not logically equivalent (rows 1 and 2)
A B (A v B) ? A A v (B ? A)
T T F T
T F F T
F T T T
F F T T
32
Part 4

• Open review