0.3 Discrete structures logic  (Page 17/23)

Thus altogether wff becomes ¬∀x E(x).

This given sentence can also be interpreted as "Some integers are not even". Then it can be restated as "For some object x in the universe, x is not integer". Then it becomes ∃x ¬E(x).

More examples: A few more sentences with corresponding wffs are given below. The universe is assumed to be the set of integers, E(x) represents x is even, and O(x), x is odd.

"Some integers are even and some are odd" can be translated as

∃x E(x) ⋀∃x O(x)

"No integer is even" can go to

∀x ¬E(x)

"If an integer is not even, then it is odd" becomes

∀x [ ¬E(x) →O(x)]

"2 is even" is

E(2)

More difficult translation: In these translations, properties and relationships are mentioned for certain type of elements in the universe such as relationships between integers in the universe of numbers rather than the universe of integers. In such a case the element type is specified as a precondition using if_then construct.

Examples: In the examples that follow the universe is the set of numbers including real numbers, and complex numbers. I(x), E(x) and O(x) representing "x is an integer", "x is even", and "x is odd", respectively.

"All integers are even" is transcribed as

∀x [ I(x) →E(x)]

It is first restated as "For every object in the universe (meaning for every numnber in this case) if it is integer, then it is even". Here we are interested in not any arbitrary object(number) but a specific type of objects, that is integers. But if we write ∀x it means "for any object in the universe". So we must say "For any object, if it is integer .." to narrow it down to integers.

"Some integers are odd" can be restated as "There are objects that are integers and odd", which is expressed as

∃x [ I(x) ⋀E(x)]

"A number is even only if it is integer" becomes

∀x [ E(x)→I(x)]

"Only integers are even" is equivalent to "If it is even, then it is integer". Thus it is translated to

∀x [ E(x)→I(x)]

Reasoning with predicate logic

Reasoning

Predicate logic is more powerful than propositional logic. It allows one to reason about properties and relationships of individual objects. In predicate logic, one can use some additional inference rules, which are discussed below, as well as those for propositional logic such as the equivalences, implications and inference rules.

The following four rules describe when and how the universal and existential quantifiers can be added to or deleted from an assertion.

1. Universal Instantiation: ∀x P(x)

-----------------

P(c)

where c is some arbitrary element of the universe.

2. Universal Generalization:

P(c)

--------------

∀x P(x)

where P(c) holds for every element c of the universe of discourse.

3. Existential Instantiation:

∃x P(x)

---------------

P(c)

where c is some element of the universe of discourse. It is not arbitrary but must be one for which P(c) is true.

4. Existential Generalization:

P(c)

----------------

∃x P(x)

where c is an element of the universe.

Negation of quantified statement

Another important inference rule is the following:

¬∃x P(x) ⇔∀x ¬P(x)

This, for example, shows that if P(x) represents x is happy and the universe is the set of people, then "There does not exist a person who is happy" is equivalent to "Everyone is not happy".