1. Proofs by demonstration using inference rules: use at least one of these patterns to reconstruct rational reasoning
a. modus ponens (method of affirming)
if P then Q
P
therefore, Q
1. If this green, growing thing is a plant, then it undergoes photosynthesis. 4. If wolves are a species, then they can't interbreed with foxes. 2. It is a plant. 5. Wolves are a species. 3. Therefore, this undergoes photosynthesis. 6. Therefore, wolves cannot interbreed with foxes.
b. modus tollens (method of denying)
if P then Q
not Q
therefore, not P
1. If Spike is a racist, then he discriminates on the basis of race. 4. If you respect my authority, then you do as I say. 2. Spike does not discriminate on the basis of race. 5. You do not do as I say. 3. Thus, Spike is not a racist. 6. You do not respect my authority.
c. disjunctive syllogism (an either-or argument)
either P or Q
not P
therefore, Q
1. Either God created humans or humans evolved. 4. You are either with us or with the terrorists. 2. Humans did not evolve. 5. You are not with us. 3. Therefore, God created humans. 6. Therefore, you are with the terrorists.
d. hypothetical syllogism (a conditional argument)
if P then Q
if Q then R
therefore, if P then R
1. If we develop nuclear fusion power, then power will become cheap and plentiful. 4. If an action is not legal, then it is not moral. 2. If power becomes cheap and plentiful, then the economy will flourish. 5. If it is not moral, then one should not do it. 3. Therefore, if we develop nuclear fusion power, then the economy will flourish. 6. Thus, if an action is not legal, then one should not do it.
2. Proof by contradiction
In an indirect proof, instead of showing that the conclusion to be proved is true, you show that all of its alternatives are false. One form is called a reductio ad absurdum because you derive what is absurd or impossible from what is assumed, which permits you to reject the original assumption. To do this, you must assume the negation of the proposition to be proved. Then, use deductive reasoning to derive a contradiction: two statements that cannot both be true. A contradiction shows that the assumption made earlier is impossible, and therefore false. Thus, the statement to be proved must be true, because its negation is false.
Suppose you are trying to establish the truth of proposition p by means of a reductio ad absurdum. You must do the following:
- Assume (assert) the denial of p, namely, not-p.
- Using only valid rules of inference,
- Deduce a known falsehood from not-p together with
- Other known-to-be-true propositions.
If you can do this, you will have shown that the assumption (not-p) is false; and if not-p is false, then p is true. This is what you set out to prove. Below are two examples of indirect proof.
a. Some people say that every claim is relatively true, that is, any claim is true from some perspectives but false from others. I disagree, I think the opposite is absolutely true, namely, some claims are not relatively true. Suppose I want to prove that every claim is relatively true isn't itself true, so what I do is just assume it for the sake of argument and see if I can generate a contradiction. If I can, then I show it is false because it leads to an absurdity. Whenever an assumption entails a contradiction, a falsehood or an implausibility, it is disproven.
Suppose I want to disprove the claim, "No proposition is absolutely true or absolutely false." That is, I want to prove its opposite is true: "Some propositions are not relatively true."
- Assume: No proposition is absolutely true or absolutely false, that is, any proposition is true from some perspectives, false from others. (Call this "REL".)
- Let S stand for any specific proposition.
- Therefore: S is neither absolutely true nor absolutely false; it is true from some perspectives and false from others. Thus we may:
- Let P be any specific perspective from which S is true. Then:
- The proposition that S is true from perspective P is true. But then:
- At least one proposition is just-plain true, that is, some propositions are not relatively true.
- Therefore: REL is contradictory (since line 6 contradicts line 2).
b. Most people think that the world is so complex that it had to be designed by a supremely intelligent creator. Suppose one argues that the world is like a beautiful, orderly, well-built house and that since the world is similarly well-designed and amazingly complex but on a far grander scale, that the world too must have an intelligent designer. Suppose I find this comparison dubious, and I want to show this.
Imagine I want to disprove the claim, "The world has a creator in the way a house does." That is, I want to prove its opposite is true: "The world does not have a creator in the way a house does."
- Prove: The world does not have a creator in the way a house does.
- Assume the opposite: The world does have a creator in the way a house does.
- Argue that from the previous assumption we would have to conclude: The creator is imperfect (because the world, like any house, is imperfect).
- But: God (the creator) cannot be imperfect.
- Conclude: The world does not have a creator in the way a house does.
Erroneous because their conclusions do not follow from their premises. Even if their premises are true, these forms are not truth-preserving...
Fallacy of affirming the consequent:
"When you have a cold, your sinuses become congested, your eyes itch, and you have headaches. You are congested, your eyes itch and you have a headache. So you have a cold."
Fallacy of denying the antecedent:
"If abortion is murder, then it is wrong. But abortion is not murder. So abortion is not wrong."
Fallacy of affirming a disjunct:
"Jesus was the son of God or Jesus was a liar. Since Jesus was the son of God, Jesus was not a liar."
Fallacy of undistributed middle:
"All reptiles lay eggs, and all birds lay eggs. Therefore, all birds are reptiles."
1. If Spike is a racist, then he discriminates on the basis of race. Spike discriminates on the basis of race, therefore he is a racist.
2. If you study, you will pass the test. You do not study, so you will not pass the test.
3. If you don't let him buy a Hummer, then you don't love him. But you let him buy a Hummer, so you love him.
4. Unless she has a fever, she doesn't have the flu. She doesn't have the flu, so she doesn't have a fever.
5. If it is raining my car will get wet. But it is not raining. So my car will not get wet.
6. Every person should avoid keeping loaded guns around the house. People who have the capacity to kill should avoid keeping loaded guns around the house. Every person has the capacity to kill.
7. Liars mislead and deceive; Ollie is a liar because he gave misleading and deceiving testimony.
8. I should not diet, so I should jog. I want to get into shape. If I want to get into shape, either I should jog or I should diet.
9. Mice fed saccharin develop bladder cancer. It follows that humans who consume saccharin also develop bladder cancer, because substances that cause cancer in mice cause cancer in humans.
10. Nobody should be forced to risk their health against their will unless there is some greater benefit. Allowing cigarette smoking in public places provides no greater benefit. Cigarette smoking in public places should not be allowed because doing so forces the nearby non-smoker to risk her health against her will.
11. Capital punishment is an acceptable social policy only if it either deters murder or is justifiable revenge. Since capital punishment does not deter murders and is not justifiable revenge, capital punishment is not an acceptable social policy.
12. Time has neither a beginning nor an end, that is, time is eternal. If time had a beginning, then there would have been a time before time. If time had an end, then there would be a time after time. The idea of there being a time before time or a time after time is absurd since before and after mean before and after in time.
1. Induction by enumeration
"All ravens we have ever observed are black, so (we may conclude) that all ravens are black."
Presumes: If all observed X are Y, then (probably) all X are Y.
2. Induction by analogy
1. Person A has properties p, q, r, and s.
2. Person B has properties p, q, and r.
3. Therefore, (probably) person B has property s also.
[p: has a backpack; q: has a class schedule; r: has this text; s: is a student]
Presumes: If X and Y are very similar, then (probably) X and Y are similar in another respect.
3. Statistical induction
"On standard intelligence tests, asians consistently outscore whites and whites outscore blacks. Thus, whites have higher IQs than blacks and asians have higher IQs than both whites and blacks."
Presumes: If the sample accurately represents the population from which it is drawn, then (probably) whatever is a property of the sample is also a property of the population.
4. Causal induction
"Many smokers are afflicted by chronic bronchitis, asthma, emphysema, heart disease, mouth and lung cancer. Heavy smokers suffer these problems even more so than do light smokers. Further, non-smokers living with smokers suffer these problems more than non-smokers who do not. Obviously smoking causes these problems."
Presumes: If there is a strong correlation between X and Y, where X and Y do NOT accidentally coincide, X and Y do NOT have a common cause, and Y does NOT cause X, then (probably) X is a cause of Y.