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.
69
Valid Argument Forms Each of the following argument forms can be shown to be valid (e.g., using truth tables). Once we have established that these argument forms are valid (and, possibly, committed them to memory), we can use them to derive the conclusion of an argument from its premises. If we can do this, we will have shown that the argument is valid. 70
1
Valid Argument Forms P ∴ ~(~P) ~(~P) ∴ P
or
Double negation
PvP∴P
Tautology
P; Q; ∴ P . Q
Conjunction
P.Q∴P P.Q∴Q
Simplification
P > Q; P ∴ Q
Modus Ponens
P > Q; ~ Q ∴ ~P
Modus Tollens
P > Q ∴ ~Q > ~P or ~Q > ~P ∴ P > Q
Transposition
71
P > Q; Q > R ∴ P > R
Hypothetical syllogism
P v Q; ~P ∴ Q P v Q; ~Q ∴ P
Disjunctive Syllogism
P∴PvQ P∴QvP
or
Addition
or
~(P . Q) ∴ ~P v ~Q ~P v ~Q ∴ ~(P . Q) P > Q ∴ ~P v Q ~P v Q ∴ P > Q
or
or
De Morgan’s Rule Implication
P > Q; R > S; P v R ∴Q v S Constructive dilemma P > Q; R > S; ~Q v ~S ∴ ~P v ~R
Destructive dilemma
~(P . ~P)
Noncontradiction 72
2
Simple Proof: An Example Let’s say we had an argument that, once formalized, looked like this: 1. (A v B) > ~ C 2. ~C > D 3. A Therefore, 4. D Once again: If we can derive the conclusion from the premises using only known to be valid argument forms, we will have shown that the argument is valid…
73
1. 2. 3. 4. 5. 6.
(A v B) > ~ C ~C > D A AvB ~C D
(premise) (premise) (premise) (from 3 by (from 1 by (from 2 by
to be proven: D
addition) modus ponens) modus ponens)
Valid (D was the conclusion of the original argument, and we have derived it from the premises of that argument using only known-to-be-valid argument forms) 74
3
Another example 1. A . B 2. A > ~C 3. B > ~D Therefore, 4. ~C . ~D
75
1. A . B 2. A > ~C 3. B > ~D 4. 5. 6. 7. 8.
A B ~C ~D ~C . ~D
(premise) to be proven: ~C . ~D (premise) (premise) (from (from (from (from (from
1, 1, 2, 3, 6,
simplification) simplification) modus ponens) modus ponens) 7, conjunction)
Valid
76
4
Example from Exam #2 1. C > D 2. D > E 3. C
(premise) to be proven: E (premise) (premise)
4. D (from 1 and 3, modus ponens) 5. E (from 2 and 4, modus ponens)
77
Something to Notice About Simple Proof If can derive the conclusion from an argument using only valid argument forms, you have shown that that argument is valid. However, if you cannot derive the conclusion this way, you have not shown that the argument is invalid. I.e., the simple proof technique requires a certain amount of ingenuity and you may have failed to hit upon a workable proof strategy. 78
5
So, if you cannot construct a proof for an argument, you cannot say for certain that the argument is invalid: It may be invalid or you may have failed to find the right proof strategy. (By contrast the truth table method, you’ll recall, always provides an answer.)
79
Conditional Proof In conditional proof, a statement is assumed “for the sake of argument.” In particular, we assume the antecedent of a conditional and then see if that conditional follows from the other premises using valid argument forms. Notice that a conditional proof is conditional in two senses: i) the proof of the conclusion is conditional upon the assumption and ii) the conclusion proven is itself a conditional statement. 80
6
Conditional Proof: Simple Example 1. A > B (premise) 2. B > C (premise) 3. 4. 5. 6.
A B C A>C
to be proven: A > C
(assumption) (from 1 and 3, modus ponens) (from 2 and 4, modus ponens) (from 3-5, conditional proof)
81
Making assumptions in this way may seem arbitrary. Remember, however, that > asserts only that if the antecedent is true, then the consequent cannot be false. So, if from the stated premises and some assumption A we can derive X, then we can derive from the stated premises “if A, then X” P (assumption) … ∴ P > Q Conditional Proof
82
7
So, another rule (another valid argument form): P (assumption) … ∴ P > Q Conditional Proof Meaning: If Q can be derived from the assumption P through a series of valid intermediate steps, represented here by the ellipsis (…), then we can validly conclude P > Q.
83
Conditional proof is actually a fairly common technique in everyday reasoning. Imagine, e.g., that we believe that 1) we can solve the problems of ‘Third World’ nations and still maintain a reasonable standard of living by developing alternative forms of energy and that 2) there will be a greater chance of lasting peace if we solve problems of Third World nations…
84
8
This can be formalized as follows: E = we develop alternative forms of energy S = we solve problems of Third World nations R = we enjoy a reasonable standard of living P = there is greater chance of lasting peace 1. 2. 3. 4. 5. 6. 7.
E > (S . R) S>P E S.R S P E>P
(premise) to be proven: E > P (premise) (assumption) (from 1 and 3, modus ponens) (from 4, simplification) (from 2 and 6, modus ponens) (3-6, conditional proof)
85
Indirect Proof: Reductio ad Absurdum Reductio ad absurdum (< Latin “reduce to absurdity”) is a proof technique in which we prove a statement by showing how the premises can entail a contradiction. I.e., they entail a statement that cannot be true, a statement of the form “P . ~P” The technique: If we assume X and show by a series of valid intermediate steps that it leads to a statement of the form P . ~P, then we can validly conclude ~X. 86
9
Reductio ad Absurdum: An Example 1. C > ~R 2. C > A 3. A > R 4. C 5. ~R 6. A 7. R 8. R . ~R 9. C > (R . ~R) 10. ~(R . ~R) 11. ~C
(premise) to be proven: ~C (premise) (premise) (assumption) (from 1 and 4, modus ponens) (from 2 and 4, modus ponens) (from 3 and 6, modus ponens) (from 8 and 5, conjunction) (4-8, conditional proof) (noncontradiction) (from 9 and 10, modus tollens)
87
Two Fallacies Related to Conditionals modus ponens: modus tollens:
P > Q; P ∴ Q P > Q; ~Q ∴ ~P
Both of these argument forms are valid. They can be proven to be valid using truth tables and, moreover, they appear intuitively to be valid. They are “basic to human thinking.” Both of these argument forms are commonly misused, however …
88
10
Conditional Fallacies Affirming the consequent: P > Q; Q ∴ P * Consider: If you got the job, you must have impressed the interviewer. You impressed the interviewer. Therefore, you must have got the job (!?) Denying the antecedent: P > Q; ~P ∴~Q * Consider: If Derek is in Saskatoon, then he is in Saskatchewan. Derek is not in Saskatoon. Therefore, he is not in Saskatchewan (!?) 89
Propositional Logic and Cogency We have only touched on the basics of formal logic in this course: It can be pursued in much greater depth (e.g., U of R, PHIL 250, 350, 351, MATH 221, 301; U of S, PHIL 241, 243, 343, CMPT 260, 417, … etc.) For present purposes we should bear in mind the connection between formal logic and the topics in natural language (informal) logic that we encountered in Govier chs. 1-6 …
90
11
… we noted then that an argument is cogent iff it satisfies all of the ARG conditions. If an argument is deductively valid, then it satisfies the R and (if it premises are all true) G conditions. But just because an argument is valid does not mean that the A condition is satisfied. A special case of formally valid arguments failing the A test occurs in connection with dilemma arguments … 91
“Either employment levels will go up or there will be a revolt. Employment levels are not going up. Therefore, there will be a revolt.” E v R; ~E ∴ R (valid – disjunctive syllogism) But it the first premise really acceptable as it stands? Does anyone have good reason to believe that these are the only two alternatives? If we can specify a third alternative (a counterexample to premise 1) we can be said to have “escaped through the horns of the dilemma” 92
12
