Logical Laws and Valid Inferences

Question 1

Law of Non-Contradiction

Answer 1

No sentence is both true and false, or for no A, A and ¬A. In symbols: |= ¬(A ∧ ¬A).
The Law of Non-Contradiction specifies what is wrong with contradictions: they cannot be true. The Law of Non-Contradiction entails that no sentence is both, true and false.

Question 2

Law of Excluded Middle.

Answer 2

(tertium non datur) For every proposition A, either A or ¬A. In
symbols: |= A ∨ ¬A.
There is no third option.

Question 3

Law of Double Negation

Answer 3

The double negation of A, ¬¬A, is logically equivalent to A: ¬¬A≡A.

Question 4

Law of Identity

Answer 4

Every sentence is logically equivalent to itself, or for any sentence A, A is logically equivalent to A, or |= A ≡ A.

Question 5

De Morgan’s Laws

Answer 5

(i) A negated disjunction is logically equivalent to the conjunction of its negated disjuncts. In symbols: |= ¬(A ∨ B) ≡ (¬A ∧ ¬B).
(ii) A negated conjunction is logically equivalent to the disjunction of its negated conjuncts. In symbols: |= ¬(A ∧ B) ≡ (¬A ∨ ¬B).

Question 6

Modus (ponendo) ponens

+example

Answer 6

A is true and so is if A then B. Therefore: B is true. In symbols: A,A → B |= B

Example. Tormenting the cat is wrong. If tormenting the cat is wrong, then getting your little brother to torment the cat is wrong. Therefore: getting your little brother to torment the cat is wrong.

Question 7

Modus (tollendo) tollens. (+ example)

Answer 7

If A then B is true, but B is false. Therefore: A is false. In symbols: A → B, ¬B|= ¬A

Example. If Mary did it, then she was in Berlin at the time of the murder. Mary was not in Berlin at the time of the murder. Therefore: Mary didn’t do it.

Question 8

Rejecting a conditional. (+ example)

Answer 8

A is true, but B is false. Therefore: if A then B is false. In symbols: A, ¬B |= ¬(A → B).

Example. The sample contained traces of arsenic. Fred did not die. Therefore: it is not true that if the sample contained traces of arsenic, then Fred died.

Question 9

Conditional proof. (+example)

Answer 9

If there is a valid argument of the conclusion B from A (together with premises Γ), then there is a valid argument for the conclusion A → B (from premises Γ).

Example: If John wins the lottery, then Mary is happy. If Mary is happy, then John is happy. John wins the lottery. Therefore (by two applications of modus ponens): John is happy. Therefore the following argument is valid: If John wins the lottery, then Mary is happy. If Mary is happy, then John is happy. Therefore: If John wins the lottery, then John is happy.

Question 10

Transitivity of Implication. (+example)

Answer 10

If A then B, and if B then C. Therefore: if A thenC. Insymbols: A→B,B→C|= A→C.
Example. If John wins the lottery, then Mary is happy. If Mary if happy, then John is happy. Therefore: if John wins the lottery, then John is happy.

Question 11

Modus tollendo ponens or Disjunctive Syllogism. (+example)

Answer 11

A or B is true, but A is false. Therefore: B is true. In symbols: A ∨ B, ¬A|= B.

Example. Alice drinks the potion or Alice eats the cake. Alice does not eat the cake. Therefore: Alice drinks the potion.

Question 12

Alternation. (+example)

Answer 12

A is true. Therefore: A or B is true. B is true. Therefore: A
or B is true In symbols: A|= A ∨ B and B|= A ∨ B

Example. I’m having tea. Therefore: I’m having either tea or I’m having
coffee.

Question 13

Proof by Cases. (+example)

Answer 13

Whatever follows from each disjunct, follows from their disjunction: A or B is true, and so are if A then C and if B then C: therefore C is true. More generally, if there is a valid argument of C from A (together with premises Γ) and a valid argument for C from B (together with premises Σ), then there is a valid argument for C from A ∨ B (together with premises Γ and Σ). In symbols: If Γ,A|=C and Σ,B |=C, then Γ, Σ, A ∨ B|= C.

Example. We lost the map. We have no idea where we are and where to go to. We can either search for the map or we continue without it. If we continue without the map, we’ll never find our way back. If we look for the map, we’ll lose time and resources, so we won’t concentrate. If we don’t concentrate, we’ll never find our way back. So either way, we’ll never find our way back.

Question 14

Simplification (+example)

Answer 14

A and B is true. Therefore A is true (and so is B). In symbols: A∧B |=A and A∧B|= B.

Example. John Lee had one Bourbon, one Scotch and one beer. There- fore: John Lee had one beer.

Question 15

Adjunction

Answer 15

A and B are both true. Therefore A and B is true. In symbols: A, B|= A ∧ B.

Question 16

Modus ponendo tollens (+example)

Answer 16

A is true, but A and B is false. Therefore B is false. B is true, but A and B is false. Therefore A is false. In symbols: A, ¬(A ∧ B)|= ¬B and B, ¬(A ∧ B)|= ¬A .

Example. I ate the cake. But I didn’t eat the cake and the ice cream. Therefore: I didn’t eat the ice cream.

Question 17

Reductio ad absurdum. (+example)

Answer 17

If A entails a contradiction, then A is false. If A entails its own falsehood, then A is false. More generally, if there is a valid argument of a contradictory conclusion from premise A (together with premises Γ), then there is a valid argument for the conclusion ¬A (from premisesΓ). Insymbols: If Γ,A|= B∧¬B, thenΓ|= ¬A.

Example. Assume an arrow moves. Then it must move from one place to another during a period of time, let’s say from p1 to pn during times t1 to tn. Consider a moment of time in that interval, say ti. At ti the arrow is in place pi. While it is there, the arrow is at rest. Similarly for any other moment during the interval. But what is at rest at each moment during an interval of time is at rest during the whole interval. What is at rest does not move. So the arrow does not move. So the arrow moves and the arrow does not move (from the assumption and adjunction). We reached a contradiction on the assumption that the arrow moves. Therefore the arrow does not move. (After Zeno of Elea)

Question 18

Consequentia mirabilis.(+example)

Answer 18

If the falsehood of A entails a contradiction, then A is true. If the falsehood of A entails that A is true, then A is true. More generally, if there is a valid argument of a contradictory conclusion from the premise ¬A (together with premises Γ), then there is a valid argument for the conclusion A (from premises Γ). In symbols: If Γ, ¬A |= B ∧ ¬B, then Γ|= A.

Example. If we should not do philosophy, we are must investigate why we should not do philosophy. But investigation is the root of philosophy. So if we must investigate, we should do philosophy. So we should and should not do philosophy (by assumption and adjunction). We’ve reached a con- tradiction on the assumption that we should not do philosophy. Therefore we should do philosophy. (After Aristotle)