# Natural Deduction 1

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### Natural Deduction:

Natural Deduction 7.1

### Rules of Inference I:

Rules of Inference I Modus Ponens Modus Tollens Disjunctive Syllogism Hypothetical Syllogism

### Modus Ponens:

Modus Ponens p É q p _________ q

### Modus Tollens:

Modus Tollens p É q ~q _________ ~p

### Disjunctive Syllogism:

Disjunctive Syllogism p v q ~p _________ q

### Hypothetical Syllogism:

Hypothetical Syllogism p É q q É r _________ p É r

### Solving proofs in natural deduction is based on pattern recognition. Think of it as using the rules to crack a code! :

Solving proofs in natural deduction is based on pattern recognition. Think of it as using the rules to crack a code!

### Example One:

Example One 1. F v (D É T) 2. ~F 3. D /T

### Example One:

Example One 1. F v (D É T) 2. ~F 3. D /T Using one of the rules find a pattern in these premises. The pattern here is DS.

### Example One:

Example One 1. F v (D É T) 2. ~F D /T D É T 1,2 DS So we draw the conclusion from DS on line 4.

### Example One:

Example One 1. F v (D É T) 2. ~F D /T D É T 1,2 DS Now use the rules to find another pattern with what we’ve just derived. This time the rule we use to “crack the code” is MP.

### Example One:

Example One 1. F v (D É T) 2. ~F D /T D É T 1,2 DS T 3,4 MP Once we reach the final conclusion we’ve solved the proof!

### A more involved example:

A more involved example ~M v (B v ~T) B É W M ~W /~T Here there are several patterns to identify. We need all of them to crack this code! Here’s one: MT Here’s another: DS

### A more involved example:

A more involved example ~M v (B v ~T) B É W M ~W /~T ~B 2,4 MT B v ~T 1,3 DS So we derive lines 5 and 6 using those rules. The order doesn’t matter. We could have reversed 5 & 6.

### A more involved example:

A more involved example ~M v (B v ~T) B É W M ~W /~T ~B 2,4 MT B v ~T 1,3 DS There’s one last code to crack to solve this proof. For this one we’ll use DS again.

### A more involved example:

A more involved example ~M v (B v ~T) B É W M ~W /~T ~B 2,4 MT B v ~T 1,3 DS ~T 5,6 DS There’s one last code to crack to solve this proof. For this one we’ll use DS again. Notice, to solve this proof we needed 5 & 6 in order to deduce line 7.

### Slide 17:

Practice some more proofs from 7.1 using the rules. Try to have fun with it! Crack the code! 