Chapter One
Sentential Logic with ‘if’ and ‘not’
1 SYMBOLIC NOTATION
In this chapter we begin the study of sentential logic. We start by formulating the basic part of the symbolic notation mentioned in the Introduction. For purposes of this chapter and the next, our symbolic sentences will consist entirely of simple sentences, called atomic sentences, together with molecular sentences made by combining simpler ones with connectives. The simple sentences are capital letters, which can be thought of as abbreviating sentences of English, as in the Introduction. In this chapter, the connectives are the negation sign, ‘~’, and the conditional sign, ‘→’.
Copyright By PowCoder代写 加微信 powcoder
The negation sign, ‘~’, is used much as the word ‘not’ is used in English, to state the opposite of what a given sentence says. For example, if ‘P’ abbreviates the sentence ‘Polk was a president’, then ‘~P’ abbreviates the sentence ‘Polk was not a president’.
The conditional sign, ‘→’, is used much as ‘if . . , then . . .’ is used in English. If ‘P’ abbreviates the sentence ‘Polk was a president’ and ‘W’ abbreviates ‘Whitney was a president’ then ‘(P→W)’ abbreviates the sentence ‘If Polk was a president, then Whitney was a president’.
We need to be precise about exactly what the symbolic sentences of Chapter 1 are:
Some terminology:
• A symbolic sentence containing no connectives at all is an atomic sentence. In this chapter and
the next, only sentence letters are atomic.
• Any symbolic sentence that contains one or more connectives is called a molecular sentence.
• We call ‘~□’ the negation of ‘□’.
• We call any symbolic sentence of the form ‘(□ → ○)’ a conditional sentence; we call ‘□’ the
antecedent of the conditional, and ‘○’ the consequent of the conditional.
Examples of symbolic sentences with minimal complexity are: U
The first is an atomic sentence. The second is the negation of that atomic sentence. The last is a conditional whose antecedent is the atomic sentence ‘U’ and whose consequent is the atomic sentence ‘V’.
Once a molecular sentence is constructed, it can itself be combined with others to make more complex molecular sentences:
CHAPTER 1 SECTION 1
• Any capital letter between ‘P’ and ‘Z’ is a symbolic sentence. (Numerical subscripts may also be used, as in ‘P3’, ‘Q24’.)
• If ‘□’ is a symbolic sentence, so is ‘~□’
• If ‘□’ and ‘○’ are symbolic sentences, so is ‘(□→○)’.
Nothing is a symbolic sentence of Chapter 1 unless it can be constructed by means of these provisions.
Copyrighted material
Chapter One — 1 Version of Aug 2013
(~V → (U→V)) ~~(V → (U→V))
it is not the case that if U then V
if it is not the case that V then if U then V
it is not the case that it is not the case that if V then if U then V
The formation rules determine when parentheses occur in a symbolic sentence. When adding a negation sign to a sentence you do not add any parentheses. These are not symbolic sentences because they contain extra (prohibited) parentheses :
~(U), ~(~U), ~((U→V))
Although ‘~(U→V)’ has a parenthesis immediately following the negation sign, that parenthesis got into the
sentence when constructing ‘(U→V)’, and not because of the later addition of the negation sign.
When combining sentences with the conditional sign, parentheses are required. For example, this is not a
sentence: U→V→W
There is one exception to the need for parentheses. If a sentence appears all by itself, not as part of a larger sentence, then its outer parentheses may be omitted. So these sentences are taken informally to be conditional symbolic sentences:
~U → V (U→V) → ~U
CHAPTER 1 SECTION 1
For purposes of Chapter 1:
A sentence is in official notation if it can be constructed by using the processes given in the box above.
It is in informal notation if it can be put into official notation by enclosing it in a single pair of parentheses.
Anything that is not in either official notation or informal notation is not a sentence at all.
Any well-formed sentence can be “parsed” into its constituents. You begin with the sentence itself, and you indicate below it how it is constructed out of its constituents. First you locate the main connective, which is the last connective introduced when constructing the sentence. If the sentence is a negation, the main connective is the negation sign; you draw a vertical line under it and write the part of the sentence to which the negation sign is applied. If it is a conditional, the main connective is the conditional sign; you draw branching lines below the main conditional sign and write the antecedent and consequent:
If the parts are themselves complex, the parsing may be continued:
~~(P → Q) P → ~(Q→R) (P→Q) → (~R→S) |22
~(P→ Q) P ~(Q→R) P→Q ~R→S ||22
P→Q Q→R P Q~R S 22|
Copyrighted material Chapter One — 2 Version of Aug 2013
CHAPTER 1 SECTION 1
1. For each of the following state whether it is a sentence in official notation, or a sentence in informal notation, or not a sentence at all. If it is a sentence, parse it as indicated above.
c. ~(Q~→R)
d. ~(~P)→~R
e. (P→Q) → (R→~Q)
f. P→(Q→R)→Q
g. (P→(Q→R)→Q)
h. (~S→R) → ((~R→S) → ~(~S→R))
i. P → (Q→P)
Copyrighted material Chapter One — 3 Version of Aug 2013
2 MEANINGS OF THE SYMBOLIC NOTATION
The negation sign: The logical import of the negation sign is this: it makes a sentence that is false if the sentence to which it is prefixed is true, and true if the sentence to which it is prefixed is false. It is common to summarize this behavior of the negation sign by means of what is called a truth table:
□ ~□ TF FT
In this table, ‘T’ stands for ‘true’ and ‘F’ for ‘false’. Reading across the rows, the table says that when a sentence ‘□’ is true, its negation, ‘~□’ is false, and when a sentence ‘□’ is false, its negation ‘~□’ is true. When a connective can be defined by a truth table in this way, the connective is said to be “truth functional”. This means that the truth value of the whole sentence that is created by combining this connective with another sentence is completely determined by (“is a function of”) the truth value of the sentence with which it is combined. All connectives in our logical notation will be truth functional.
The negation sign corresponds naturally to either of two locutions in English. One is the locution ‘it is not the case that’ placed on the front of a sentence. The other is the word ‘not’ when used within a simple sentence. So if ‘S’ is taken to abbreviate ‘The salad was tasty’, these have the same logical content:
It is not the case that the salad was tasty The salad was not tasty
In the logical tradition, the locution ‘fail to’ is often taken to have the meaning of negation. This is limited to certain uses of English. For example, if you say Samantha failed to reach the summit you are probably reporting a situation in which she tried to reach the summit, but did not reach it. However, if you say that the number of buttons on a shirt fail to match the number of buttonholes, you are probably not saying that the buttons tried to match the buttonholes but did not match them, you may only be saying that the buttons and buttonholes do not match. In this usage, ‘fail to’ may report only negation. The sentence:
The salad failed to be poisonous
then reports the negation of the sentence ‘The salad was poisonous’, and it may be symbolized ‘~P’. In
the exercises we will assume that ‘fail to’ only amounts to negation.
The conditional sign: The conditional sign is meant to capture some part of the logical import of ‘if . . , then’ in English. But it is not completely clear under what circumstances an ‘if . . , then’ claim in English is true. It seems clear that any English sentence of the form ‘If P then Q’ is false when ‘P’ is true but ‘Q’ is false. If you say ‘If the Angels win there will be a thunderstorm’, then if the Angels do win and if there is no thunderstorm, what you said is false. In other cases things are not so clear. Consider these conditional sentences uttered in normal circumstances:
If it rains, the game will be called off.
If the cheerleaders are late, the game will be called off.
Now suppose that it rains, and the cheerleaders are late, and the game is called off. Are the sentences above true or false? Most people would be inclined to say that the first is true. But the second is less obvious. After all, the game was not called off because the cheerleaders were late. So there is something funny about the second sentence. If it is false, it will be impossible to capture the logical import of conditionals by means of any truth functional connective. For the truth of the first sentence above requires that some conditionals be true when both their parts are true, and the second would require that some conditionals be false when both their parts are true.
However, you might hold that the second sentence above is true. Granted, the game was not called off because the cheerleaders were late, but so what? The second sentence doesn’t say anything at all about why the game was called off. It only says that it will be called off if the cheerleaders are late; and they were late, and the game was called off, so it is true. If so, perhaps conditionals are truth functional. There is no universal agreement about how conditionals work in natural language. The position taken in
CHAPTER 1 SECTION 2
Copyrighted material Chapter One — 4 Version of Aug 2013
this text is that ‘if . . , then. . .’ is sometimes used to express what is called the “material conditional”. This is the use of ‘if . . , then . . .’ where a conditional sentence is false in case the antecedent is true and the consequent false, and it is true in every other case. This use is truth functional. It is described by means of this truth table:
□ ○ □→○ TTT TFF FTT FFT
The conditional is used in this way by mathematicians, and by others. We will assume in doing exercises and examples that the logical import of ‘if . . , then’ is intended to coincide with our symbolic ‘→’. There may be other uses of ‘if . . , then’ that convey more than ‘→’, but we will not address them in this text.
The word ‘if’ in English has many synonyms. In at least some contexts these are all interchangeable:
CHAPTER 1 SECTION 2
provided that assuming that
given that
on the condition that
If Maria sings, Xavier will leave
Provided that Maria sings, Xavier will leave Assuming that Maria sings, Xavier will leave
Given that Maria sings, Xavier will leave
In case Maria sings, Xavier will leave
On the condition that Maria sings, Xavier will leave
Using ‘S’ for ‘Maria sings’ and ‘X’ for ‘Xavier will leave’, these can all be symbolized as S→X
‘If’ clauses in English may also occur at the end of a sentence instead of at the beginning. So these also may be symbolized as ‘S→X’:
Xavier will leave if Maria sings
Xavier will leave provided that Maria sings
Xavier will leave assuming that Maria sings
Xavier will leave given that Maria sings
Xavier will leave in case Maria sings
Xavier will leave on the condition that Maria sings
In either use, the word ‘If’ immediately precedes the antecedent of the conditional.
Many of these “synonyms” of ‘if’ can be used to say more than what is said with a simple use of the word ‘if’. For example, a person who says ‘assuming that’ may want to convey that s/he is indeed making a certain assumption, and not just saying ‘if’. But in other contexts no assuming is indicated. A physicist who says ‘Assuming that there are planets with orbits outside the orbit of Pluto, we will need to send space probes to investigate them’ may simply be responding to the question ‘What if there are planets beyond Pluto?’, and not doing any assuming at all. In doing the exercises we will take for granted that the locutions identified above are being used in the most minimal sense of ‘if’, which we take to be that of the connective ‘→’.
Only if: The word ‘only’ can be added to the word ‘if’, to make ‘only if’. The ‘only’ has the effect of reversing antecedent and consequent. As a result, whereas ‘if’, when used alone, immediately precedes the antecedent of a conditional, ‘only if’ immediately precedes the consequent. So we have these equivalences:
If P, Q P→Q Only if P, Q Q→P
P if Q Q→P P only if Q P→Q
Some will find it more natural to represent ‘P only if Q’ by ‘If not Q then not P’, or ‘~Q → ~P’. It will turn out that this is logically equivalent to ‘P → Q’. We will generally use the latter form because it’s simpler.
Copyrighted material Chapter One — 5 Version of Aug 2013
When ‘only if’ comes first, there are grammatical changes in the last clause, which converts into its interrogative word order:
The game will be called off only if it rains = Only if it rains will the game be called off ‘Only’ may also precede any of the synonyms of ‘if’, so these all may be symbolized as ‘X → S’:
Xavier will leave only if Maria sings
Xavier will leave only provided that Maria sings
Xavier will leave only assuming that Maria sings
Xavier will leave only given that Maria sings
Xavier will leave only in case Maria sings
Xavier will leave only on the condition that Maria sings
Only if Maria sings will Xavier leave
Only provided that Maria sings will Xavier leave
Only assuming that Maria sings will Xavier leave
Only given that Maria sings will Xavier leave
Only in case Maria sings will Xavier leave
Only on the condition that Maria sings Xavier will leave
CHAPTER 1 SECTION 2
For these exercises assume that ‘S’ abbreviates ‘Susan will be late’ and ‘R’ abbreviates ‘It will rain’.
a. Susan will be late only provided that it rains
b. Only on condition that it rains will Susan be late c. Susan will be late only in case it rains
d. Susan will be late only if it rains
e. It is not the case that Susan will be late
1. For each of the following sentences say which symbolic sentence is equivalent to it.
a. Only if it rains will Susan be late S→R
b. Susan will be late provided that it rains S→R
c. Susan won’t be late ~S
d. Susan will be late only if it rains S→R
e. Given that it rains, Susan will be late S→R
2. Symbolize each of the following:
Copyrighted material Chapter One — 6 Version of Aug 2013
3 SYMBOLIZATION: TRANSLATING COMPLEX SENTENCES INTO SYMBOLIC NOTATION
In representing simple sentences of English by sentential letters you need to say which letter abbreviates which sentence. So far this has been done informally, by choosing sentential letters that are already used in a prominent place in the English sentence, as in using ‘S’ for ‘Sally will be late’. But if different English sentences share their prominent letters, we can soon run out of natural sentential letters to choose. So instead we give a scheme of abbreviation, which pairs off sentence letters with the English sentences that they abbreviate. An example is:
X Susan will be late
Y It will rain
With this scheme we would represent ‘Given that it rains, Susan will be late’ by: Y→X
Any way of “symbolizing” English sentences in logical notation, or of “translating” English into logical notation, is done relative to a scheme of abbreviation. A translation or symbolization that is correct on one scheme may be incorrect on others.
Complex sentences of English generally translate into complex sentences of the logical notation. Here it is important to be clear about the grouping of clauses in the English sentence. Consider, the sentence:
If Roberta doesn’t call, Susan will be distraught
This is a conditional whose antecedent is a negation. Using ‘P’ for ‘Roberta calls’ and ‘Q’ for ‘Susan will be
distraught’, this may be symbolized: ~P → Q
It is not the negation of a conditional: ~(P→Q)
To make the negation of a conditional, you need to say something like:
It is not the case that if Roberta calls, Susan will be distraught
which is symbolized as: ~(P → Q)
There are a few fundamental principles that govern symbolizations of English sentences in the logical notation.
Notice that in the sentence ‘It is not the case that Willa will leave if Sam does’ there are two grammatical sentences that immediately follow ‘It is not the case that’, namely, ‘Willa will leave’ and ‘Willa will leave if Sam does’. So there are two ways to symbolize the sentence:
S → ~W ~(S→W)
The sentence is in fact ambiguous.
CHAPTER 1 SECTION 3
SOURCES OF ‘~’
The locution ‘fail to’ always yields a negation sign that applies to the symbolization of the
smallest sentence that ‘fail to’ is part of.
Likewise for the word ‘not’.
The expression ‘it is not the case that’ applies to a sentence immediately following it.
Copyrighted material Chapter One — 7 Version of Aug 2013
Illustration: These principles determine that the sentence: ‘Pat won’t call only if it is not the case that the quilt is dirty’ is symbolized:
The (contracted) ‘not’ in ‘Pat won’t call’ yields a negation that applies directly to ‘P’. The ‘it is not the case that’ yields a negation that applies directly to ‘Q’, since ‘the quilt is dirty’ is the only sentence immediately following ‘it is not the case that’. The only sentence immediately to the left of the ‘only if’ is ‘Pat won’t call’, so that is the antecedent of the conditional, and the only sentence immediately to the right of the ‘only if’ is ‘it is not the case that the quilt is dirty’, so that is the consequent.
Illustration: In the sentence ‘If Wilma leaves then Xavier stays if Yolanda sings’ the first ‘if . , then . . .’ exactly encloses ‘Wilma leaves’, so W is the antecedent of the first conditional. There are two sentences immediately following ‘then’; they are the whole ‘Xavier stays if Yolanda sings’ and just ‘Xavier stays’. So the sentence must have the form:
W → (Xavier stays if Yolanda sings)
(W → X) if Yolanda sings
The second ‘if’ comes between its consequent and antecedent. It must give rise to a conditional that has ‘Y’ as its antecedent, since the only sentence following the ‘if’ is ‘Yolanda sings’. The consequent of that conditional can be the symbolization of either just ‘Xavier stays’, or ‘If Wilma leaves then Xavier stays’, since each of these immediately precedes the ‘if’. The first of these fits with the first partial symbolization above, giving:
W → (Y→X) and the second fits with:
Y → (W→X).
Both of these symbolizations are possible, which agrees with the intuition that the original English sentence is ambiguous. (Some people find the first reading more natural than the second, but the second is a possible reading under some circumstances.)
Copyrighted material Chapter One — 8 Version of Aug 2013
CHAPTER 1 SECTION 3
SOURCES OF ‘→’
If: The word ‘if’ always gives rise to a conditional, ‘□→○’.
Wherever ‘if’ occurs (not as part of ‘only if’), the antecedent of the conditional is the symbolization of a sentence immediately following ‘if.
The consequent of the conditional is either the symbolization of a sentence immediately preceding ‘if’ (with no comma in between) — as in ‘○ if □’ — or it is the symbolization of a sentence immediately following the sentence that is symbolized as the antecedent — as in ‘if □ then ○’.
Then: If ‘then’ occurs it must be paired with a preceding ‘if’.
The antecedent of the conditional introduced by ‘if’ is the symbolization of the sentence exactly between ‘if’ and ‘then’.
Its consequent is the symbolization of a sentence immediately following ‘then’.
Only if: The expression ‘only if’ always gives rise to a conditional, ‘□→○’.
The consequent of ‘□→○’ is the symbolization of a sentence immediately following ‘only if’ — as in ‘□ only if ○’ — or as in ‘only if ○, □’.
The antecedent of ‘□→○’ is the symbolization of a sentence immediately preceding ‘only if’ (with no comma in between) — ‘□ only if ○’ — or of a sentence immediately following the consequent — as in ‘only if ○, □’. In the latter case, that sentence is grammatically changed (to its interrogative word order).
Illustration: In the sentence ‘If Wilma leaves then Xavier stays onl
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com