18 Rules of Deduction
The rules of deduction provide proof in a stronger and more elegant sense than do truth tables. The reader will recognize the process as very similar to the style of proof found in geometry. The justification for each proof will be based upon the rules employed and the steps given in the premises or deduced already.
First, we will introduce a number of rules (18) that will function as the theorems of this system. These are divided into two groups. The rules of inference work from left to right or from the top down (depending upon how they are structured). The rules of replacement work in both directions. The symbols on the left side of the equivalence symbol could just as well be on the right side.
Further the rules of inference apply only to full lines in the proof. If we are going to use the rule simplification, for example, the “and” must be the symbol with the greatest scope. The rules of replacement, however, may be used on whole lines or part of the line in the proof, as long as we are using the symbol that is appropriate for the rule.
Finally, it is the rules of inference that more likely function to get us somewhere in the proof. Typically, they function to discard variables, so have the effect of allowing us to progress from more complicated statements to those that are less so.
8 Rules of Inference
Among the 8 rules of inference are a number of “housekeeping “ rules. Their role is not so much to provide logical insight as to “tidy up” formulas in order to employ the other rules. I have placed the two “tidying” rules first (Simplification and Conjunction).
Rules of InferenceRule #1. Simplification (S): From pq can be inferred p, and from pq can be inferred q. From any conjunction of statements assumed true can be inferred the truth of any one of those statements taken separately.
Rule #2. Conjunction (CJ): From p and q can be inferred pq. From any statements assumed true separately can be inferred the truth of the conjunction of those statements.
Rule #3. Modus Ponens (MP): From pq and p can be inferred q. From a conditional statement and the affirmation of its antecedent can be inferred the affirmation of its consequent.
Rule #4. Modus Tollens (MT): From pq and ~q can be inferred ~p. From a conditional statement and the negation of its consequent can be inferred the negation of its antecedent.
Rule #5. Disjunctive Syllogism (DS): From pvq and ~p can be inferred q, and from pvq and ~q can be inferred p. From a disjunction and the negation of a disjunct can be inferred the affirmation of the other disjunct. This rule functions as a process of elimination for various alternatives and works for both meanings of “either...or.”
Rule #6. Addition (AD): From p can be inferred pvq. From any statement assumed true can be inferred a disjunction composed of the original statement and any other statement whether true or false.
Rule #7. Hypothetical Syllogism (HS): From pq and qr can be inferred pr. From any two conditional statements, where the consequent of one is identical to the antecedent of the other, can be inferred a conditional statement composed of the antecedent of the first and the consequent of the second.
Rule #8. Constructive Dilemma (CD): From pq and rs and pvr can be inferred qvs. From any two conditional statements and a disjunctive statement which affirms the antecedents of the two conditionals can be inferred a disjunction of the consequents of the two conditionals. This rule also works for the exclusive sense of “either...or.”
10 Rules of Replacement
Among the 10 rules of replacement are a number of “housekeeping” rules. Their role is not so much to provide logical insight as to “tidy up” formulas in order to employ the other rules. I have placed the four “tidying” rules first (Double Negation, Commutation, Association and Tautology).
Rules of ReplacementRule #9. Double Negation (DN): p~~p. Any statement is equivalent to its double negation.
Rule #10. Commutation (CM): (pq)(qp) and (pvq)(qvp). The order of statements in conjunctions and disjunctions in no way affects the truth value of the conjunction or disjunction (both exclusive and inclusive).
Rule #11. Association (AS): ((pq)r)(p(qr)) and ((pvq)vr)(pv(qvr)). The grouping of conjuncts or disjuncts in no way affects the truth value of conjunctions or disjunctions (both exclusive and inclusive).
Rule #12. Tautology (T): (pp)p and (pvp)p. The repetition of statements in conjunctions or disjunctions is of no consequence logically.
Rule #13. Implication (I): (pq)(~pvq). A conditional statement is equivalent to a disjunction where the antecedent has been negated.
Rule #14. Contraposition (CP): (pq)(~q~p). A conditional statement is equivalent to another conditional statement whose antecedent and consequent are reversed and negated.
Rule #15. Exportation (EX): ((pq)r)(p(qr)). A conjunct, making up part of the antecedent of a conditional statement, can be exported to the consequent and connected to it as its antecedent.
Rule #16. Distribution (D): (p(qvr))((pq)v(pr)) and (pv(qr))((pvq)(pvr)). A conjunct connected to another composed of disjuncts, can be connected to those disjuncts separately. A disjunct, connected to another composed of conjuncts, can be connected to those conjuncts separately.
Rule #17. Equivalence (EQ): (pq)((pq)(qp)) and (pq)((pq)v(~p~q)). Two statements are materially equivalent if each can be inferred from the other. Also, two statements are materially equivalent if either both are true or both are false.
Rule #18. DeMorgan’s Theorem (DM): (pq)~(~pv~q) and (pvq)~(~p~q). Any conjunction is equivalent to a disjunction where the statement variables (taken separately) and the entire formula (taken as a whole) are negated. Any disjunction is equivalent to a conjunction where the statement variables and the entire formula have been negated.
Recognizing Valid Patterns
We are now in a position to recognize rudimentary arguments that follow the rules (and are therefore valid), as compared to those that do not.
The pattern symbolized below is alternately called affirming the antecedent (since in this presentation the second premise affirms the antecedent of the conditional statement) or modus ponens (Rule 3).
Example 27:
(P1)PQ
(P2)P
(C1)Therefore, Q
This pattern can be set up in a variety of different ways in English. All have the same logical form. The order in which the three statements are presented makes no difference to the pattern presented, or to the validity of the argument. All are instances of affirming the antecedent—modus ponens. All are valid.
Example 28:
(P1) If you run in the marathon in Niagara Falls next weekend, you will experience leg cramps.
(P2) You ran the race there.
(C1) Consequently, you had the leg cramps.
(C1) You had leg cramps, (P1) because if you run the 26 miles in the race at Niagara Falls, you will get leg cramps, (P2) and you went ahead and ran the race anyway.
(P2) You ran the race last weekend at Niagara Falls.
(P1) If you run that far, then you get leg cramps.
(C1) So you got the leg cramps. No surprise!
Some patterns are used much more often than others. These include: MP (modus ponens), MT (modustollens), DS (disjunctive syllogism), and HS (hypothetical syllogism). However, no rule is any more necessary than any other in a given argument. Each of the rules has an equal status since each is one of the subset of the minimum 18 needed to show that all valid arguments are really so. Even the lowliest rule among the 18, a rule like simplification or association, has no substitute among the rules. This list of 18 is complete, and what each rule allows us to accomplish is not repeated by any other rule.
An explanation of the formal fallacies associated with this section can be found in Section C of this chapter. Those fallacies include affirming the consequent, denying the antecedent, and affirming the disjunct.
EXERCISE 6K.
Identify the premises and conclusions in the following arguments. What rule or rule violation (formal fallacy) does each exhibit?
1.If you smoke, then you will eventually get lung disease because if you smoke you will irritate the lining of your lungs, and if you irritate the lining of your lungs, you will get lung disease.
2.Eight is greater than six because eight is greater than seven and seven is greater than six.
3.If you have enough money, you can go to the Lil Kim concert. But you don’t have enough money. Therefore, you can’t go.
4.If you have enough money, you can go to the Dave Matthews concert. But you can’t go. Therefore, you did not have enough money.
5.If you have enough money, you can go to the Who concert. You went to the Who concert. Therefore, you must have had enough money.
6.Either the battery is bad, or something is wrong with the starter. Yes, the battery will not hold a charge. Therefore, nothing is wrong with the starter.
7.If you are a Democrat, then you voted for Clinton. You voted for Clinton in '96. Therefore, you must be a Democrat.
8.At the end of this course: If you do well in this course, then you get a good grade. You did not do well in the course; therefore, you will not get a good grade.
9.God must not have wanted us to wear clothes all the time, because if God wanted us to wear clothes all the time, then He would not have created such hot summers, and He did create any very hot summers.
10.If she loved you, she would not have sent you that letter. She loves you. So, the letter cannot be from her.
11.She was born either in Indiana or in Ohio. We know for sure that she was not born in Indiana. Therefore, she must have been born in Ohio.
12.Notre Dame's football team went to a bowl game last year. Therefore, either all Bona students are valedictorians of their high school classes or Notre Dame's football team went to a bowl game last year.
13.If Jesus says to pay the tax, he loses the respect of the Jews for agreeing with Roman tyranny. If Jesus advises the Jews not to pay the tax, then those who are out to get him will have reason for asking that he be jailed, that is, for sedition. Jesus must respond in one way or the other. Therefore, he loses either way.
14.If the U.S. assassinates Osama bin Laden, it will be thought a murderer. If it does not, it will be responsible for all the deaths that come about because of this Saudi terrorist. The U.S. must do one or the other. Therefore, it loses either way.
For the arguments in Exercise 6J, can you recognize any of the patterns you now know as the rules of logic?
EXERCISE 6L.
Some additional uses of deduction.
1.You have played Clue before. Can you name some of the logical rules that are employed in the deductions you typically make to try to win the game? Can you symbolize some of these?
2.Toward the end of every NFL season, deductive calculations can be made concerning certain teams’ chances. Select the team of your choice and express what would need to be done in order to make the playoffs, given that the team has just completed its 14th week of play.
SELECTED ANSWERS TO EXERCISE 6K.
1.Hypothetical syllogism, valid.
3.Denying the antecedent—violation of modus tollens.
4.Modus tollens, valid.
5.Affirming the consequent—violation of modus ponens.
6.Affirming the disjunct—violation of disjunctive syllogism.
12.Addition, valid.
14.Constructive dilemma, valid.
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