Inference

Inference means reasoning to draw conclusion from a given set of information. For instance,
Man is Mortal.
Socrates is human.
Therefore,
Socrates will die.

A few rules

Modus Ponens:

\(\begin{array}{rl} & p \\ & p \to q \\ \therefore & q \end{array}\)

• Test

• test 2

Modus Tollens

Hypothetical Syllogism

Simplification

Resolution

Conjunction

Disjunctive Syllogism

Example 1

Premises: It is not sunny this afternoon and it is colder than yesterday. We will go swimming only if it is sunny. If we do not go swimming, then we will take a canoe trip. If we take a canoe trip, then we will be home by sunset.

Conclusion: We will be home by sunset.

Let's apply rules of inference to see if the conclusion can be reached.

1. \(\neg p \wedge q \;\;\;(Premise\;1)\)

2. \( r \rightarrow p \;\;\;(Premise\;2)\)

3. \( \neg r \rightarrow s \;\;\;(Premise\;3)\)

4. \( s \rightarrow t \;\;\;(Premise\;4)\)

5. \( \neg p \;\;\; (Simplification\;rule\;on\;1)\)

6. \( \neg r \;\;\; (Moduls\;Tollens\;between\;1\;and\;5)\)

7. \( s \;\;\; (Modus\;Ponens\;between\;3\;and\;6)\)

8. \( t \;\;\; (Modus\;Ponens\;between\;4\;and\;7)\)
   

Example 2

Premises: A student in this class has not read the book. Everyone in this class passed the first exam. Therefore, someone who passed the first exam has not read the book.

Assuming All NSU students as the domain.

\(C(x)\): x is student in this class.
\(B(x)\): x has read the book.
\(P(x)\): x has passed the exam.

By applying rules of inference,

1. \(\exists x (C(x) \wedge \neg B(x))\)

2. \(\forall x (C(x) \rightarrow P(x))\)

3. \(C(a) \wedge \neg B(x)\;\;\;(Existential\;instantiation\; of\;1)\)

4. \(C(a) \;\;\;\;\; (Simplification\;of\;3)\)

5. \(C(a) \rightarrow P(a)\;\;\; (Universal\;instantiation\;of\;2)\)

6. \(P(a)\;\;\;(Modus\;Ponens\;of\;4\;and\;5)\)

7. \(\neg B(a)\;\;\;(Simplification\;of\;3)\)

8. \(P(a) \wedge \neg B(a) (Conjunction\;of\;6\;and\;7)\)

9. \(\exists x (P(a) \wedge \neg B(a)) (Existential\;generalization\;of\;8)\)