1. fitch rules for negation
Lecture 13
fitch rules for negation
Reminder: Start with previous class’s i>clicker question

2. Disjunction elimination and fitch
Example: Disjunction 1

3. I>clicker question
Is this disjunction elimination? A

4. Negation elimination
If a double negation ¬¬P is true, then P must be true too.
¬Elim:
¬¬P . P

5. Prelude to Negation Introduction: Proof by contradiction
AKA: Reductio ad absurdum, which is Latin for “reduction to the absurd”.
In this kind of proof, we presuppose something for the sake of argument in order to show that it will lead us into contradiction.
If a presupposition leads us into a contradiction, we can conclude that it’s false.

6. Reductio example 1
Some people say that all beliefs are equally true. Suppose that that were the case, and that at least one belief were true. Then all beliefs are true. But I believe that not all beliefs are true. So from our supposition, not all beliefs are true. Therefore, all beliefs are and all beliefs aren’t true. Since the supposition leads to contradiction, we should reject it.

7. Reductio example 2
There is no chess computer program that will win every game it plays. To see this, suppose that there were such a program. Then we could run it on two computers, A and B, and have them play each other. Since A is running a program that always wins, it will win the game and B will lose. But since B is running the program, it will win and A will lose. Thus the assumption that the unbeatable chess program is possible leads to contradiction and must be rejected.

8. Reductio example 3
The legal example on p. 139 of the text.

9. Reductio and contradiction
In a reductio proof, we may infer the negation of the supposition if it leads to contradiction.
Q: But what’s a contradiction?
A: A logical impossibility, i.e., a sentence that’s false in all possible worlds.
Examples:
P∧¬P
Tet(a)∧Cube(a)
a≠a

10. Negation introduction
‘⊥’ is the contradiction symbol; we will use it to indicate in Fitch that a logical impossibility has been reached.
¬Intro: P . ⊥ ¬P

11. Contradiction introduction
The logical impossibilities that allow us to introduce ‘⊥’ aren’t single sentences, but rather pairs of sentences of the form P and ¬P.
That is, instead of P∧¬P on one line, we may introduce ‘⊥’ only when we have P on one line of the proof and ¬P on another.
⊥Intro: P . ¬P ⊥

12. Example
DeMorgan’s Laws

13. Contradiction and ana con
Cube(b) and Tet(b) are mutually inconsistent.
However, since there is no way to get them into the form, P and ¬P, ⊥Intro does not recognize them as inconsistent.
Instead, if a proof has sentences that are inconsistent in virtue of the meanings of the Block Language predicates next to the same vertical line, we can infer ⊥ using Ana Con.
Example:
1. LeftOf(a,b)
2. RightOf(a,b)
3. ⊥ Ana Con: 1,2