A Quick Lesson on Negation Several of the argument forms we have looked at (modus tollens and disjunctive syllogism, for valid forms; denying the antecedent for invalid) involve a type of statement which logicians call a negation. A negation is a statement that uses the word not (or sometimes no ) to deny another statement. Consider the following statements: A. February is one of the summer months. B. It seems likely that there will be a tornado tomorrow. C. There are more than thirty students in this class. By adding not to each of these statements, we negate (deny, declare to be false, declare the logical opposite of) them, and the statements we generate are negations of A-C: Not-A. February is not one of the summer months. Not-B. It does not seem likely that there will be a tornado tomorrow. Not-C. There aren t more than thirty students in this class. Note, though, that you can negate a statement that already has a not (or no ) in it. Take D-F, each of which is itself a negation of another statement: D. It is not possible to attend two classes at one time. E. Not a single person in this room has climbed Mount Everest. F. She didn t mean to offend someone. If we negate D-F, we get what we call double negations: Not-D. It is not not possible to attend two classes at one time. Not-E. Not not a single person in this room has climbed Mount Everest. Not-F. She didn t not mean to offend someone. Just as in ordinary English language, two not s cancel each other out, so we can read these statements as saying: Not-D. It is possible to attend two classes at one time. Not-E. A single person in this room has climbed Mount Everest. Not-F. She did mean to offend someone. Let s look at how double negations work in real arguments. Here s a premise: 1. If you arrived on time, you weren t late. We can substitute the propositions in this argument with letters to look at its form more easily. Let P = you arrived on time (the antecedent of premise 1), and Q = you weren t late (the consequent of premise 1).
Now, let s make a modus tollens (which, recall, has the form If P then Q. Not-Q. Therefore Not-P.). To do so we must deny the consequent. In other words, we must make our second premise the negation of the consequent in premise 1. But in this argument, our consequent happens to already be a negation. That means that when we deny the consequent, we get a double negation: 2. You weren t not late. which is logically equivalent to the same statement with both not s cancelled out: 2. You were late. A modus tollens allows us to infer Not-P: the negation of the antecedent. So our full argument looks like: 1. If you arrived on time, you weren t late. 2. You were late. 3. Therefore you did not arrive on time. For each of these arguments, determine a) whether it is valid or invalid, b) whether it is sound or unsound, and c) what argument form is used, if it is one of the forms we covered in class.. When in doubt, swap out the propositions (phrases that can stand alone as sentences) within the propositions for letters to look at the argument form. The letters in the argument are the names of the following shapes: A B C D E (square) (rhombus) (triangle) (circle) (oval)
I. II. 1. If a shape is a square, then it has four sides 2. A is a square. 3. Therefore A has four sides. 1. If a shape has four sides, then it is a square. 2. B is not a square. 3. Therefore B does not have four sides. III. 1. If a shape is an equilateral triangle, then it has three sides. 2. C has three sides. 3. Therefore C is an equilateral triangle. IV. 1. If a shape does not have three sides, it is not a triangle. 2. D is not a triangle. 3. Therefore D does not have three sides. V. 1. If a shape has sides, then it is not a circle. 2. E doesn t have sides 3. Therefore E is a circle. VI. 1. If a shape has sides, then it is not a circle. 2. D does not have sides. 3. Therefore D is a circle. VII. 1. If a shape has three sides, it is a triangle. 2. If a shape is a triangle, then it has three equal-length sides. 3. If C has three sides, then C has three equal-length sides. VIII. 1. If a shape is not oblong, then it is not an oval. 2. E is an oval. 3. Therefore E is oblong. IX. 1. Either a shape with four sides is a square or it is a rhombus. 2. A is not a rhombus. 3. Therefore A is a square. X. 1. D is a circle or D is an oval. 2. D is an oval. 3. Therefore D is a circle.
Answer Key I. valid, sound, modus ponens 1. If a shape is a square, then it has four sides. TRUE. If P, then Q. 2. A is a square. TRUE. P. 3. Therefore A has four sides. TRUE. Q. I is valid because it has the form of modus ponens, and sound because premises 1 and 2 are both true. II. valid, not sound, modus tollens 1. If a shape has four sides, then it is a square. FALSE. (If s shape has four sides, it could be a rectangle, a rhombus, a trapezoid, and parallelogram ). If P, then Q. 2. B is not a square. TRUE. Not-Q. 3. Therefore B does not have four sides. FALSE. Not-P. II is valid because it has the form of modus tollens, but not sound because premise 1 is false. III. not valid, not sound, affirming the consequent 1. If a shape is an equilateral triangle, then it has three sides. TRUE. If P, then Q. 2. C has three sides. TRUE. Q. 3. Therefore C is an equilateral triangle. FALSE. P. III is not valid because it uses an invalid form. Though both premises are true, it cannot be sound, because to be sound is to be valid and have all true premises; since it is not valid, it cannot be sound. It yields a false conclusion. IV. not valid, not sound, affirming the consequent 1. If a shape does not have three sides, it is not a triangle. TRUE. If P, then Q. 2. D is not a triangle. TRUE. Q. 3. Therefore D does not have three sides. TRUE. P. This is a tricky one. It gives us a true conclusion, but only by chance not by using an argument form that guarantees the truth of the conclusion, as a valid argument form would. Like in III above, both premises are true, but IV cannot be sound because it is not valid. V. Not valid, not sound, denying the antecedent 1. If a shape has sides, then it is not a circle. TRUE. If P then Q. 2. E doesn t have sides. TRUE. Not-P. 3. Therefore E is a circle. FALSE. Not-Q. V is not valid because it uses an invalid form. We can see that it is invalid because the truth of its premises cannot guarantee the truth of the conclusion.
VI. Not valid, not sound, denying the antecedent. 1. If a shape has sides, then it is not a circle. TRUE. If P then Q. 2. D does not have sides. TRUE. Not-P. 3. Therefore D is a circle. TRUE. Not-Q. VI is not valid and not sound, even though it has a true conclusion. Much like in argument IV, the conclusion is true only by chance not because it uses a form of argument that guarantees a true conclusion to follow from true premises. VII. Valid, not sound, hypothetical syllogism 1. If a shape has three sides, it is a triangle. TRUE. If P, then Q. 2. If a shape is a triangle, then it has three equal-length sides. FALSE. (A triangle can also have two equal-length sides, or no equal-length sides at all.) If Q then R. 3. If C has three sides, then C has three equal-length sides. FALSE. If P, then R. VII is valid, but not sound, because its second premise is false. VIII. Valid, sound, modus tollens. 1. If a shape is not oblong, then it is not an oval. TRUE. If P, then Q. 2. E is an oval. TRUE. Not-Q. 3. Therefore E is oblong. TRUE. Not-P. VIII is valid and sound, since it uses a valid form and has all true premises. IX. Valid, not sound, disjunctive syllogism 1. Either a shape with four sides is a square or it is a rhombus. FALSE. (A shape with four sides could also be a rectangle, a trapezoid, etc.) P or Q. 2. A is not a rhombus. - TRUE. Not-Q. 3. Therefore A is a square. - TRUE. P. IX is valid but not sound, because premise 1 is false. X. Not valid, not sound, dysfunctional syllogism 1. D is a circle or D is an oval. TRUE (or at least, there s no reason to think it false). P or Q. 2. D is an oval. FALSE. Q. 3. Therefore D is a circle. TRUE. P. X has a true conclusion, but it doesn t use a valid pattern of reasoning to infer that conclusion. It starts with a disjunction like a disjunctive syllogism, but it does not rule out one option and infer the other by process of elimination. Instead, it makes it seem like D could be both an oval and a circle which contradicts the implication made by premise 1, that D could be either a circle or an oval, but not both.