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Chapter 24: Some Logical Equivalences

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Chapter 24:Some Logical Equivalences

- Two statements are logically equivalent if they are true or false under exactly the same circumstances.
- You can replace logically equivalent statements with each other wherever they occur in an argument.

- De Morgan’s Theorems
- The denial of a conjunction is logically equivalent to the disjunction of the denials of the original two conjuncts.
- The denial of a disjunction is logically equivalent to the conjunction of the denials of the original two disjuncts.
- Basically, when you move the denial “into” a statement, ‘and’ becomes ‘or’ and ‘or’ becomes ‘and’. When you move a denial “out of” a statement, ‘and’ becomes ‘or’ and ‘or’ becomes ‘and’. (The symbolic representation in the textbox might be clearer than the words.)

- Double Negation
- As the name suggests, two negatives make a positive.

- Material Equivalence
- Statements of material equivalence (“p if and only if q”) are complex. They claim both “If p then q” and “If q then p.”
- Typically, if you have a statement of material equivalence, you will convert it into its compound form and simplify: Typically only one of the conditionals will be useful in drawing out a conclusion.
- If you have a statement of material equivalence as a conclusion, the only way you can reach it , unless the statement itself was found in the original premises, is if you have the two appropriate conditionals, conjoin them, and use material equivalence.

- Transposition (Contraposition)
- When you flip the positions of the antecedent and the consequent in a conditional, negative statements become positive statements, and positive statements become negative statements.

- In symbolic form, where p and q are statements of any degree of complexity:
De Morgan’s Theorems (DeM)

~(p & q) (~p v ~q)

~(p v q) (~p & ~q)

Double Negation (DN)

p ~p

Material Equivalence (Equiv.)

(p q) [(p q) & (q p)

Transposition (Trans.)

(p q) (~q ~p)

- You use logical equivalences to put statements into the correct form to correspond with the argument forms in the last chapter.
- Sometimes more than one equivalent statement form will play a role in evaluating an argument.

- If Luis went to the dance, then Dora went fishing. If Frieda did not go to the movies, then Dora did not go fishing. So, if Luis went to the dance, then Frieda went to the movies.
- It’s a hypothetical syllogism. To recognize that, all you need to do is acknowledge that the second premise transposes to “If Dora went fishing, then Frieda went to the movies.”

- 1. L D
- 2. ~F ~D / L F
- 3. D F 2 Trans.

- If Luis went to the dance, the Marta made mincemeat; and if Gustav did not gather grapes, then Bianca did not make bratwurst. It is not the case that both Marta made mice meat and Gustav gathered grapes. So, either Luis did not go to the dance or Bianca did not make bratwurst.
- If you transpose the second conditional in the first premise and use De Morgan’s Theorem on the second premise, you will notice that this is a Destructive Dilemma.

1. (L M) & (~G ~B)

2. ~(M & G) / ~L v ~B

3. (L M) & (B G) 1 Trans.

4. ~M v ~G2 DeM.

- Matt makes mincemeat if and only if Briana doesn’t bat balls all day. Briana bats balls all day. So, Matt does not make mincemeat.
- The first premise is a biconditional, so you’ll need to restate it as a conjunction of conditionals: If Matt makes mincemeat, then Briana doesn’t bat balls all day; and if Briana doesn’t bat balls all day, then Matt makes mincemeat. The conclusion you are after is the denial of the antecedent of the first conditional, so begin by simplifying the conjunction to “If Matt makes mincemeat, then Briana doesn’t bat balls all day.” The second premise is logically equivalent “It is not the case that Briana does not bat balls all day.” So the conclusion follows by denying the consequent.

In symbolic form:

1. M ~B

2. B/ ~M

3. (M ~B) & (~B M) 1 Equiv.

4. M ~B 3 Simp.

5. ~~B 2 DN

6. ~M 4,5 DC

- The process is the same as in Example 1, though more involved, if you’re given an argument and asked to figure out what the final conclusion is. You treat a conclusion as the final conclusion if it is derived by using all the relevant premises. Typically, the final conclusion will either be one of the simple statements in the premises or the denial of such a premise or the denial of a compound component that cannot be broken down further.

- If Monica munches moo goo gai pan meditatively, then Boris builds boats boisterously. If Boris builds boats boisterously, then Lucretia licks lollypops lovingly. If Angelique angles ably for trout, then Tennis tumbles tenaciously through thorny thickets. Monica munches moo goo gai pan meditatively unless Angelique angles ably for trout. If Zelda zaps zits, then Lucretia does not lick lollypops lovingly; and Danielle drugs druids during dark days only if Tennis does not tumble tenaciously through thorny thickets. Danielle drugs druids during dark days. So, ...

- There are a number of ways you might approach the problem. On one approach, you’d do a hypothetical syllogism with the first two premises and conclude that “If Monica munches moo goo gai pan meditatively, then Lucretia licks lollypops lovingly.” Conjoin that statement with premise 3 and do a constructive dilemma with premise 4 to conclude: “Either Lucretia licks lollypops lovingly or Tennis tumbles tenaciously through dense thickets.” You can double negate both disjuncts, and use the resulting disjunctive statement with premise 5 to conclude that “Either Zelda does not zap zits or Danielle does not drug druids on dry days” by destructive dilemma. Premise 6 is logically equivalent to “It is not that case that Danielle does not drug druids on dry days,” so with that statement and the previous disjunctive conclusion, you can “Zelda does not zap zits” by disjunctive syllogism.

In symbolic form:

1. M B.

2. B L

3. A T

4. M v A

5. (Z ~L) & (D ~T)

6. D/ ???

7. M L1,2 HS

8. (M L) & (A T)7, 3 Conj.

9. L v T8, 4 CD

10. ~~L v T9 DN

11 ~~L v ~~T10 DN

12. ~Z v ~D5,11 DD

13. ~~D6 DN

14. ~Z12,13 DS