Reflections: In this chapter, we go over valid argument forms--called the Rules of Inference--and then look at the two formal fallacies. The combination of the two (Rules of Replacement from last chapter and Rules of Inference from this chapter) allows us to make any number of legitimate moves and, therefore, greatly expand our reasoning capacity. After you have worked through this chapter, you will be able to analyze and evaluate that reasoning. Having a facility with logic gives us the techniques to examine and evaluate the many kinds of arguments we confront. This can be both useful and empowering.
Goals for this chapter: My goal here is to give you a strong basis in one area of logic--the valid argument forms and expressing arguments in . Being able to recognize the different valid (deductive) arguments has a value beyond enhancing your analytical skills. It also helps us organize our thought so we think in a more systematic way. I provide a number of exercises so you can get the forms down pat and go from the form to an example and vice versa. It's not always easy working with a complex argument--and being able to streamline it by symbolizing the propositions can be extremely useful.
Patterns of Deductive Reasoning:Rules of Inference
RULES OF INFERENCE
Rules of inference are valid argument forms. This means if we assume the premises are true, then the conclusion could not be false--it would have to be true as well. That does not mean the argument is sound, as you know. We also need to know the premises really are true. Remember: a sound argument is a valid argument that has true premises.
1. Modus Ponens
If A then B
A
Therefore, B
Example:
If you don't like vegetables, you won't like carrots.
Ivan does not like vegetables.
So, Ivan will not like carrots.
2. Modus Tollens
If A then B
Not B
So, not A
Example:
If Diana comes for dinner, Leo will make lasagna
Leo did not make lasagna.
So, Diana did not come for dinner.
3. Hypothetical Syllogism
If A then B
If B then C
thus, If A then C
Example:
If we fly to Detroit we can see the Pistons play.
If we see the Pistons play, we'll go to Dearborn for shishkabob.
So, if we fly to Detroit, we'll go to Dearborn for shishkabob.
4. Disjunctive Syllogism
Either A or B Not A (or not B) So, B (or A)
Example
Either there's a cyborg in the living room, or that guy is just weird.
There is not a cybog in the living room.
So, that guy is just weird.
5. Conjunction
A
B
Therefore, A and B
Example
Nick likes anime.
Nick does not like reality TV.
Therefore, Nick likes anime and does not like reality TV.
6. Simplification
A and B
Therefore, A. (or B)
Example.
Marie was afraid of small birds and squirrels.
Therefore, Marie was afraid of small birds. (or: Therefore, Marie was afraid of squirrels.)
7. Logical Addition
A
Therefore, A or B (where B is any proposition whatsoever)
Example:
James is not fond of melted cheese sandwiches.
Therefore either James is not fond of melted cheese sandwiches or he (add any proposition here----e.g., or he likes burnt toast).
8. Constructive Dilemma
(If A then B) and (If C then D). Either A or C So, Either B or D.
Example
If Anita mows the lawn, Gary will trim the hedges, but if Anita scrapes the paint, then Gary can sand the windows today.
Either Anita will mow the lawn or she'll scrape the paint.
Therefore, either Gay will trim the hedges or he'll sand the windows today.
9. Destructive Dilemma
(If A then B) and (If C then D). Not B or Not D So, Not A or Not C.
Example:
If Anita spreads mulch, Gary will go after the gophers, but if Anita cuts off dead branches, Gary will run turn over sod.
Either Gay did not go after the gophers or he did not turn over sod.
So, either Anita did not spread mulch or Anita will not cut off dead branches.
10. Absorption
If A then B.
Therefore, if A then (A and B).
Example:
If Trent catches the kittens, he'll give them to Gloria.
Therefore, if Trent catches the kittens, then he caught the kittens and he'll give them to Gloria.
FORMAL FALLACIES
These two fallacies are, basically, mutations of Modus Ponens and Modus Tollens. But where both Modus Ponens and Modus Tollens are valid forms of argument, the two Formal Fallacies are both invalid and unsound.
1. Fallacy of Denying the Antecedent:
If A then B.
A does not happen.
Therefore, B does not happen.
Example:
If the thunder gets louder, the puppy will hide under the bed.
The thunder did not get louder.
So, the puppy did not hide under the bed.
2. Fallacy of Affirming the Consequent
If A then B.
B does happen.
Therefore, A happens
Example:
If Eric has a flat tire, he won't make the concert.
Eric did not make the concert.
So, he must have had a flat tire.
List: Rules of Inference, Rules of Replacement, and Formal Fallacies
Rules of Inference -- Valid Argument Forms
- Modus Ponens: If A then B. A, therefore B.
- Modus Tollens: If A then B. Not B, therefore not A.
- Hypothetical Syllogism: If A then B, If B then C, therefore, if A then C.
- Disjunctive Syllogism: Either A or B. Not A. Therefore, B.
- Constructive Dilemma:
If A then B, and, if C then D. Either A or C. Therefore, Either B or D. - Destructive Dilemma:
- If A then B, and, if C then D. Either not B or not D.
- Therefore, Either not A or not C.
- Simplification: A and B. Therefore, A (or Therefore, B).
- Logical Addition: A. Therefore, Either A or B.
- Conjunction: A. B. Therefore, A and B.
- Absorption: If A then B. Therefore, If A, then (A and B).
Formal Rules of Replacement
DeMorgan's Laws:Not Both: Not (A and B) = Not A or not B. Neither/Nor: Neither A nor B = Not A and not B.Material Implication: "If A then B" is equivalent to "Either not A or B."
Transposition:"If A then B" is equivalent to "If not B then not A." Exportation: "If (A and B) then C" is equivalent to "If A then, (if B then C)."
Equivalence: "A if and only if B" is equivalent to "If A then B and if B then A."
Formal Fallacies
Fallacy of Affirming the Consequent: If A then B. B. Therefore, A.Fallacy of Denying the Antecedent: If A then B. Not A. Therefore not B.