PHIL 120: Third meeting

I. What to know for Test 1 (in general terms).II. Symbolizing compound sentences (cont’d)

a. Paying attention to English punctuation when symbolizing into SL and when and how to use SL punctuation: () and []

b. Special phrases and terms

III. Basic notions: their inter-relationships and implications continued

IV. Open: review questions

Part 1

Preparing for Test 1

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

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

IVitamin C cures all colds.Vitamin C never cures

colds.-------------------------------The moon is made of

green cheese.

IIThere are 385 days in a

yearFebruary has 31 days-------------------------------A rose is a roseIIII’m here and nobody is

here--------------------------2 + 2 = 5

Part 2 (a)

Symbolizing compound sentencescontinued

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.

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 regularlyJ: 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

‘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)

Truth tables of the two forms of the consequent demonstrate that the sentences are logically equivalent

J T ~J v ~T ~(J & T)~(J & T)

TT TT FF FF

T F T T

F T T T

F FF T T

Paying attention to English punctuation ‘Alison works hard although Mark doesn’t; 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 hardS: Mark is a success D: Alison is a success

So we can use:(A & ~M) & (S D)

Paying attention to English punctuation ‘Either Michael, or Roxanne, or Shirley works

hard; but if Michael works hard, then either Roxanne doesn’t or Shirley doesn’t’

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 hardS: Shirley works hardThe left conjunct can be symbolized as

(M v R) v S OR M v (R v S)

Using the punctuation of SL

‘Either Michael, or Roxanne, or Shirley works hard; but if Michael works hard, then either Roxanne doesn’t work hard or Shirley doesn’t’

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 &

Using the punctuation of SL

If we don’t 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.

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)]

What is and what is not a sentence of SL

These are sentences of SL:~~AA22

~(A B)[A (B v A)] M

These are not sentences of SL (why not?)A ~ B

~(A BA (B v A)] M

Part 2 (b)

“Special” phrases and terms

‘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.

‘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.Let’s use:O: the operation is a success P: the patient survives(a) can be symbolized as ‘O S’(b) can be symbolized as ‘S O’

Truth table for (a) and truth table for (b):Note differences in rows 2 and 3

OO SS O O S S S S O O

TT TT TT TT

TT FF FF TT

FF TT TT FF

FF FF TT TT

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

conditionalsTake the sentence

A BIt 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.

‘If and only if’

AA BB A A B B (A (A B) & (B B) & (B A) A)

TT TT TT TT

TT FF FF FF

FF TT FF FF

FF FF TT TT

‘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 MOR ‘If Mary does not jog, then Mary is sick’

~M S

Truth table for ‘unless’

MM SS M v SM v S ~S ~S M M ~M ~M S S

TT TT TT TT TT

TT FF TT TT TT

FF TT TT TT TT

FF FF FF FF FF

‘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 JRight 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)

‘Either/or and not both’(or at most one)

SS JJ (S v J) & ~(S & J)(S v J) & ~(S & J) (S v J) & (~S v ~J)(S v J) & (~S v ~J)

TT TT FF FF

TT FF TT TT

FF TT TT TT

FF FF FF FF

‘Neither/nor'

Consider the sentence:‘Neither Alice nor Bruce plays poker’This is logically equivalent to:‘Alice doesn’t play poker and Bruce doesn’t 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’.

‘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)

‘Neither/nor’

AA BB ~A & ~B~A & ~B ~(A v B)~(A v B)

TT TT FF FF

TT FF FF FF

FF TT FF FF

FF FF TT TT

Connectives that are not truth-functionalA connective is truth functional if and only if it

determines the truth value of a sentence given the truth values of the sentence’s 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’

Implications of logical notions

Main connectives matter because they Main connectives matter because they determine the truth value of a given sentence determine the truth value of a given sentence (T or F) based on their characteristic truth (T or F) based on their characteristic truth tables and the truth values (T or F) of a tables and the truth values (T or F) of a sentence’s immediate components on each sentence’s immediate components on each possible truth value assignmentpossible truth value assignment

Compare:Compare:(A v B) (A v B) ~A ~A

with:A v (B ~A)

The sentences’ truth tables demonstrate that they are not logically equivalent (rows 1 and 2)

AA BB (A v B) (A v B) ~A ~A A v (B A v (B ~A) ~A)

TT TT FF TT

TT FF FF TT

FF TT TT TT

FF FF TT TT

Part 4

Open review