Lessons 6–7 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
6
Laws of correct reasoning; sophisms and paradoxes
Textbook, Part 1: pp. 33–38
GoalWrite the logical form of an argument and check its correctness through a tautology; tell a sophism from a paradox.
Every argument has several statements, the premises, and a conclusion drawn from them, joined by “therefore”. If the premises are S and the conclusion is R, the argument has the form S ⇒ R and is correct if this expression is a tautology. Four basic correct forms are ((p ⇒ q) ∧ p) ⇒ q; ((p ⇒ q) ∧ ¬q) ⇒ ¬p; ((p ∨ q) ∧ ¬p) ⇒ q; ((p ⇒ q) ∧ (q ⇒ r)) ⇒ (p ⇒ r). But ((p ⇒ q) ∧ q) ⇒ p is not a tautology (it is false when p is false and q is true), so such a conclusion is wrong. A sophism is a false “proof” whose mistake is cleverly hidden (often a hidden division by zero). A paradox is an unexpected statement that contradicts an accepted view; for example “This sentence is false” leads to a contradiction whether we call it true or false.
Worked examples
“If a number ends in 0, it is divisible by 10. The number 340 ends in 0. Therefore 340 is divisible by 10.” The form is ((p ⇒ q) ∧ p) ⇒ q — a correct conclusion.
“If it rains, the ground is wet. The ground is wet. So it rained.” The form ((p ⇒ q) ∧ q) ⇒ p is wrong: with p false and q true (the ground was watered) the premises are true and the conclusion false. A sophism: let a = b; then a² = ab, a² − b² = ab − b², (a − b)(a + b) = b(a − b). Cancelling (a − b) gives a + b = b, i.e. 2b = b, 2 = 1; the mistake is division by a − b = 0.
Class activity
“Find the mistake” contest: each group gets a simple sophism prepared by the teacher (for example the “proof” of 2 = 1); the group finds the step with a division by zero or a wrong conclusion and proves it.
Practice
1
Write the form of reasoning by the contrapositive (modus tollens).
((p ⇒ q) ∧ ¬q) ⇒ ¬p
2
“Karim is either at school or in the library. He is not at school. So he is in the library.” Is the conclusion correct? Name the form.
Correct; the form is ((p ∨ q) ∧ ¬p) ⇒ q.
3
In the sophism with a = b = 4, what is a − b? Why does it make cancelling impossible?
a − b = 4 − 4 = 0; division by zero is impossible, so this factor cannot be cancelled.
4
Why is ((p ⇒ q) ∧ q) ⇒ p not a tautology?
When p is false and q is true, p ⇒ q is true and q is true — the premises hold — but the conclusion p is false, so the whole expression is false.