Embed Size (px)
Citation preview
1
1
7. Deductive Arguments:
Propositional Logic
2
Background
In contrast to categorical logic (the logic of class membership), propositional logic (PL) deals with logical relationships between simple, discrete propositions and their logical operators (‘not’, ‘and’, ‘or’, ‘if/then’).
Arguments in propositional form are also said to be deductively valid or invalid, but, in contrast to categorical syllogisms, their validity or invalidity does not depend on the relation between subject and predicate.
2
3
Govier’s Example
1. If global warming continues, parts of the polar ice cap will melt.
2. Global warming will continueTherefore,3. Parts of the polar ice cap will melt.
We can see intuitively that this argument is valid, but notice that neither 1 or 2 is in subject/predicate form (and that there is no middle term)
4
The Example Formalized
G = “global warming will continue”M = “parts of the polar ice cap will melt”
1. G > M2. G
Therefore,
3. M
The symbol “ > ” (horseshoe) is used to represent the logical relation “if/then”
3
5
Some Ground Rules in PL
If a letter is used to represent one statement in an argument, it cannot be used to represent a different statement in the same argument. (One variable per letter)
Provided that an argument can be accurately put into propositional form, its deductive validity can then be tested using truth tables(and by other means)
6
Truth tables can also be used to define basic logical symbols (logical connectives):
‘not’, ‘and’, ‘or’, ‘if/then’
~, . , v , >
As Govier notes, the meaning of these logical operators does not always correspond exactly to our everyday understanding of those words.
(I suggest: Don’t think of them as representing English words at all, except sometimes when translating.)
4
7
Negation (“Not” ~)
Suppose K was used to symbolize the simple statement “Kerry won the 2004 US Presidential election”
The negation (or denial), of that statement can be written “~K”. “K” and “~K” are contradictories.
TF
FT
~PP
8
Conjunction (“and” .)
“Kerry lost the election and Bush was very happy.”(K . B)
The conjunction of two statements is true if and only if (‘iff’) both conjuncts are true.
FFT
FTF
F
T
Q
FF
TT
P . QP
5
9
Something to Notice About Truth Tables …
In the case of negation, only one proposition is involved and that proposition has two possible truth values. So only two rows in the truth table are required to present all of the possible truth values:
TF
FT
~PP
10
In the case of conjunction, two propositions are involved and each of those propositions has (again) two possible truth values. So four rows are required to represent all of the possibilities:
6
11
Disjunction (“or” v)
TFT
TTF
F
T
Q
FF
TT
P v QP
A disjunction asserts that either one or the other or both of two statements (called “disjuncts”) is true.
12
Inclusive “Or”
The operator “v” represents an inclusive “or”(A or B or both A and B).
To see this, consider the first line of its truth table.
The English word “or,” by contrast, is often used exclusively, as on a menu:
“Your choice of soup or salad (but not both)”
7
13
Conditionals (“if/then” >)
The basic form a conditional statement is “If such-and-such, then so-and-so”.
“If it is nice out, then we’ll go for a walk”
“If it nice out” is the antecedent
“we’ll go for a walk” is the consequent
14
Conditionals (“if/then” >)
FFT
TTF
F
T
Q
TF
TT
P > QP
“If Martin runs for PM, then he will be elected”(R > E)
A conditional statement does not assert the truth of either its antecedent or its consequent. Instead, it asserts that if the antecedent is true, then the consequent is true.
8
15
Some Advice
As the truth table shows, the logical operator > does not capture all natural language uses of “if / then”
For present purposes, think of > simply as an abstract logical invention.
FFT
TTF
F
T
Q
TF
TT
P > QP
16
Translation I
The logical connectives (i.e., symbols) of propositional logic can be used to represent arguments in an elegant, compact way, such that the logical structure of the argument is made obvious …
9
17
English:
Either the university budget will continue to increase or the quality of its library holdings will be undermined. If the university budget increases, the opportunities for students willbe better. If the quality of its library holdings is undermined, the great care will be needed to protect its reputation. So, either the opportunities for students will be better or great care will be needed to protect the university’s reputation.
Formalization:
Let I = “the university budget will increase”; U = “the quality of the university’s library holdings will be undermined”; W = “the opportunities for students will be better; G = “great care will be needed to protect the universities reputation” …
18
1. I v W2. I > W3. U > G
Therefore,
4. W v G
10
19
Brackets
Brackets play an important role in arithmetic:
“(45 + 10)/5” is not equivalent to
“45 + (10/5)”
Similarly in logic brackets serve to indicate how things should be grouped together …
20
Exclusive “Or” Again
Suppose we needed to translate the typical menu phrase “your choice of soup or salad”
Since we know how restaurants operate, we understand that the “or” in this statement must be an exclusive “or” – soup or salad, but not both.
Let S = “you may have soup”A = “you may have salad”
11
21
What we need to capture in logical terms is the idea that “you may have soup or you may have salad and it is not the case that you can have both soup and salad”
Symbolized correctly:
(S v A) . ~(S . A)
This, by contrast, would be ambiguous and a mistake:
(S v A) . ~S . A
22
Another Example
English: “If you have multimedia skills or have worked on video productions, you can apply for the job”
Formalization:
Let M = “You have multimedia skills”V = “You have worked on video productions”A = “You can apply for the job”
(M v V) > A
12
23
Testing for Validity Using Truth Tables
Consider the following (dubious) argument:
If Doug’s investment pays off, he will have enough money for his trip to Sweden. Doug’s investment does not pay off. Therefore, it is not the case that Doug will have enough money to go to Sweden.
D > S~D
Therefore,
~ S
24
We can show that this argument is invalid using truth tables.
Recall that, since there are two propositions (two letters), there will be four rows. The number of columns will depend on the number of distinct statements and compound statements that occur in the premises and in the conclusion.
In this case …
13
25
F
F
T
T
D
F
T
F
T
S
FTF
TFT
T
F
~S
TT
TF
D > S~D
D > S~DTherefore, ~ S
Conclusion Premise 1Premise 2
Basic Propositions
26
F
F
T
T
D
F
T
F
T
S
FTF
TFT
T
F
~S
TT
TF
D > S~D
Recall what we mean by validity: An argument form is valid iff the truth of the conclusion is guaranteed by the truth of the premises. So, if we can find any row in the truth table where all the premises are represented as true and the conclusion false, we will have shown that the argument is invalid.
D > S~DTherefore, ~ S
Conclusion false; All premises true, Invalid
14
27
Constructing Truth Tables
1. Rows: For an argument with n distinct propositions (letters) you will need 2n rows.
I.e., for two letters you will 2 x 2 (4) rows; for three letters you will need 2 x 2 x 2 (8) rows, etc.
2. Represent all possible combinations of truth values: Start on the left and fill half with Ts half with Fs. In the next column, fill one quarter with Ts, one quarter with Fs and repeat. In a third column, one eighth Ts, one eighth Fs and repeat twice, and so on.
28
3. Columns: You will need one column for each distinct statement (each letter), and at least one for each premise and one for the conclusion.
It may sometimes be useful also to include a separate column for a significant component of an argument even if it is not a premise or a conclusion.
E.g., if you have a premise of the form (P.Q) > R, it will be useful to have a column for “P.Q” since it is the antecedent of a conditional and seeing the relationship of the antecedent to the conditional will be helpful in avoiding mistakes.
15
29
An Example
English: If John does not practice his singing, he will hinder the work of the choir director. If John hinders the work of the choir director, he should not be allowed to continue as a member of the choir. John does not practice his singing. We can only conclude that John should not be allowed to continue as a member of the choir.
Formalization: Let P = “John practices his singing”; H = “John will hinder the work of the choir director”; B = “John should be allowed to continue as a member of the choir”
1. ~P > H2. H > ~B
3. ~P
Therefore,
4. ~ B
30
F
F
F
F
T
T
T
T
P
T
T
T
F
T
T
T
F
H > ~B
TTFFF
TFTTT
TTTFT
FFTTF
F
F
T
T
H
F
T
F
T
B
TTF
TFF
T
F
~B
FT
TF
~P > H~P
1. ~P > H2. H > ~B3. ~PTherefore,4. ~ B
Are there any rows in which all the premises are T and conclusion F? If so, the argument is invalid.
16
31
Quick Review: Truth Table Technique
Let’s say we needed to assess the validity of the following argument using truth tables:
If the Liberals or the Conservatives win the next federal election, the NDP will be in trouble. But the Conservatives will win only if the Liberals don’t. So, whatever happens, the NDP will be in trouble.
32
A Step by Step Process …
1. Identify the premises and the conclusion:
If the Liberals or Conservatives the win the next federal election, the NDP will be in trouble. But the Conservatives will win only if the Liberals don’t. So, whatever happens, the NDP will be in trouble.
1. If the Liberals win the next federal election or if the Conservatives win the next federal election, then the NDP will be in trouble.
2. The Conservatives will win the next federal election only if it is not the case that the Liberals win the next federal election.
Therefore,3. The NDP will be in trouble.
17
33
2. Identify all propositions (all basic statements, i.e., all letters).
C = the Conservatives win the next federal election
L = the Liberals win the next federal election
T = the NDP is in trouble
34
3. Identify all the logical connectives (i.e., the symbols)
1. If the Conservatives win the next federal election or if the Liberals win the next federal election, then the NDP will be in trouble. (v and >)
2. The Conservatives will win the next federal election only if it is not the case that the Liberals win the next federal election. (> and ~)
Therefore,3. The NDP will be in trouble.
18
35
4. Formalize the argument
1. (L v C) > T
2. C > ~L
Therefore,
3. T
36
5. Design a truth table
2n rows for an argument with n letters
One column for each proposition (each letter) and at least one for each premise. (It may sometimes be useful to add an additional column, e.g. to represent separately the antecedent of a conditional.)
Populate all possible truth values for the basic propositions (the letters) using the 1/2, 1/4, 1/8 rule.
So …
19
37
~LC (L v C) > TL T L v C C > ~L
Phase 1
Basic Propositions
Premise 1 Premise 2Conclusion
Antecedent of Premise 1
38
~L
F
F
F
F
T
T
T
T
C (L v C) > T
FF
TT
FT
TF
F
F
T
T
L
F
T
F
T
T L v C C > ~L
Phase 2
20
39
T
T
F
F
T
T
F
F
~L
F
F
F
F
T
T
T
T
C
T
T
F
T
F
T
F
T
(L v C) > T
TTFF
TTTT
TTFT
TFTF
F
F
T
T
L
F
T
F
T
T
FT
TT
F
T
L v C
T
F
C > ~L
Phase 3
Test for Validity: Any row where conclusion is false and all premises true?
40
T
T
F
F
T
T
F
F
~L
F
F
F
F
T
T
T
T
C
T
T
F
T
F
T
F
T
(L v C) > T
TTFF
TTTT
TTFT
TFTF
F
F
T
T
L
F
T
F
T
T
FT
TT
F
T
L v C
T
F
C > ~L
Phase 3
Invalid
21
41
Shorter Truth Table Technique
The truth table method for assessing validity is rather lengthy and cumbersome. So it would be useful to have a shorter version.
It turns out that this is possible.
Recall: An argument was shown to be invalid if we could find a row in the truth table in which all of the premises were false and the conclusion true …
42
… so, in order to use the shorter technique, we
1) set that values of the component statements so as to guarantee that the conclusion will be false and then …
2) see whether we can consistently set the values of the basic propositions in the premises so that the premises turn out to be true.
If we can do this, the argument is invalid. If we cannot, it is valid.
22
43
An example
1. A v B2. B > C3. C
Therefore,
4. ~A
1) For ~A to be false, A must be true
2) If A is true, then A v B is true, regardless of the truth value of B. If we assume that C is true, B > C is true regardless of the truth value of B.These assumptions are consistent with each other (i.e., nothing else in the premises rules them out).
So, if we assume that A is true (the denial of the conclusion); C is true and B is either true or false, then the premises are all true and the conclusion false. The argument is invalid.
44
Another
1. B > D2. D > E3. B . A
Therefore,
4. D . E
1) For D . E be false, at least one of (D, E) will have to be false
2) Suppose that D is false. If D is false, then for the first premise to true, B must be false. The second premise will be true if D is false and E is true. The third premise, however, cannot be true if B is false (which was our assumption).
Likewise, if we assume E is false, then second premise must be false. If we assume both D and E are false, then B must again be false for the first premise to be true, but then B cannot be true for the third premise to be true.
23
45
More on Translating
Sometimes (as we’ve seen) translating English statements into propositional form is relatively easy (!).
Alas, this is not always the case, however.
On pp. 259-76 Govier offers extensive advice about translating from English into propositional form.
Students should read and take account of allof this. Here, for the moment, are some highlights …
46
Translating ‘Not’
If you represent some statement in argument with the letter P and another statement as ~P, these statements must truly be logically contradictory (not merely contraries).
Consider:
“Saskatoon is beautiful”
If we symbolize this as B, we cannotaccurately symbolize “Saskatoon is ugly”as ~B
24
47
Translating ‘And’
In ordinary speech, “and” often carries with it the implication that there is some connectionbetween the statements it conjoins.
Not so with “.”
G = “George W. Bush is President of the US”D = “I like doughnuts”
“G . D” is just fine as a propositional statement.
48
Similarly, the order of statements conjoined by an “and” often makes a difference in English. They have the sense of “and … then”
“George W. Bush was elected and the U.S. Supreme Court became more conservative”
In logic, by contrast, “B . S” and “S . B” have the same truth table.
25
49
Moreover, the English “and” can sometimes conjoin not entire statements, but subjects or predicates:
“Saddam Hussein and Osama bin Laden are in hiding”
This can be symbolized correctly as “S . O” : Saddam is in hiding and bin Laden is in hiding.
50
But consider: “Dick and Jane got married.”
(or, in light of recent events, “Scott and Darren got married” or “Lenore and Margaret got married” )
It would be a mistake to symbolize these as two separate statements pertaining to two different subjects, since the usual implication of “got married” includes “to each other”
26
51
Translating ‘Or’
We have already seen that “v” represents the logical relation of inclusive disjunction: A v B means either A is true, or B is true, or bothare true.
If we need to symbolize and exclusive “or” (as on a menu), we can simply specify “and not both”:
(S v A) . ~(S . A)
52
Translating ‘If Then’
As we’ve noted, > is defined in logic in way that does not correspond exactly to all possible uses of “if … then” in ordinary English.
Once again, however, this is in part because ordinary usage assumes that there is some connection (e.g., a causal connection) between the antecedent and the consequent of a conditional statement.
With the logical relation of >, however, no such connection (i.e., no non-logical connection) is assumed …
27
53
FFT
TTF
F
T
Q
TF
TT
P > QP
D = “I like doughnuts”W = “George W. Bush is President of the US”
D > W
That seems crazy and arbitrary, but note what the truth table indicates (and all that it indicates):
54
The “minimum truth value connection” of >
> asserts only that if the antecedent is true, the consequent must be true.
This assertion does not require that there is any (non-logical) relation between the antecedent and the consequent.
“premise + premise > conclusion” suggests why the minimal definition of > is basic to logic: > captures the basic idea of logical entailment.
28
55
Counterfactuals
“If George W. Bush had died in infancy, the American republic would be better off today”
“If George W. Bush had died in infancy, the US would still be a British colony”
Both of these are counterfactual statements (conditional statements the antecedent of which is known to be false).
In science and in ordinary life we might naturally say that one of these counterfactuals is much more plausible than the other. But propositional logic cannot capture this difference.
56
Translating ‘Both … and’
Expressions of the form “Both P and Q” are virtually always translated correctly using the symbol “.”
“Both the IMF and the ASEAN are concerned about the increasing value of the Chinese Yuan.”
I . A
29
57
Translating Neither … Nor
Expressions of the form “Neither P nor Q”usually assert “not either P or Q” – i.e., a disjunction of two statements neither of which is true.
“Neither Gary nor Arlene is able to attend the Grey Cup.”
~(G v A)
58
Translating “Implies that”
In logic “implies that” is a synonym for the logical relationship described by >. So too in most everyday speech:
“The increased value of the Chinese Yuan implies that wages are rising in the Chinese economy.”
I = the value of the Yuan has increasedW = wages in the Chinese economy are rising
I > W
30
59
Negating Conditionals
Either the antecedent or the consequent of a conditional can, of course, be a negated proposition. When a conditional statement as a whole is negated, however, we need to indicate this by means of brackets:
“The fact that he is smiling does not imply that he is happy”
~(S > H)
60
Translating “Provided that ...”
Statements of the form “Q provided that P”assert that Q will be the case if some condition P (some proviso) is met. Such statements can be represented by making the negation of the proviso the antecedent of a conditional.
“You may have a doughnut, provided that you finish all of your Brussels sprouts.”
~B > ~D
31
61
Sometimes the proviso may be negative:
“I can understand spoken French provided that it is not spoken too rapidly”
In which case, the consequent of the formalization is not negated:
~R > U
62
Translating “Only if”
“Ilene can apply to law school only if she has written the LSAT”
This asserts that it is necessary for Ilene to write the LSAT in order for her to be able to apply to law school. I.e., given that Ilene is eligible to apply, it must be true that she has written the LSAT.
A > L
32
63
“Ilene can apply to law school only if she has written the LSAT”
Notice the order of the propositions makes a real difference in translating this statement:
L > A * (wrong)
Asserts that if it is true that Ilene has written the LSAT, then it must be true that she can apply. Writing the LSAT would then be a sufficient condition for being able to apply to law school. But that is fairly clearly not what the statement means.
64
Necessary and Sufficient Conditions
Necessary condition: What is needed or required for something to be the case
Sufficient condition: What is sufficient, i.e., enough, for something to be the case.
E.g., having oxygen available is a necessary (but not a sufficient) condition for sustaining human life.
Consuming 3,000 calories/day is sufficient (but not necessary) for adequate human nutrition.
33
65
Necessity
“Availability of oxygen is necessary for human beings to remain alive.”
O = oxygen is available H = human beings remain alive
H > O
I.e., given that human beings remain alive, it must be true that oxygen is available.
66
Sufficiency
“The absence of oxygen is a sufficient condition for human death.”
A = oxygen is absentD = human beings die
A > D
I.e., given that oxygen is absent, it must be the case that human beings will die
34
67
Necessary and Sufficient
“That fact that Hank is an unmarried male is necessary and sufficient for his being a bachelor.”
A = Hank is an unmarried maleB = Hank is a bachelor
(B > A) . (A > B)
This sort of conjunction of two conditionals is called a biconditional. (iff – “if and only if”)
68
Simple Proofs in Propositional Logic
We do not need to use truth tables or the shorter truth table technique in order to asses the validity of arguments in propositional form.
Instead, we can show the validity of an argument by deriving its conclusion from its premises using argument forms that are known to be valid.
