## Transcribing English to Predicate Logic wffs

### Subjects to be Learned

• Translating English sentences to wff

### Contents

English sentences appearing in logical reasoning can be expressed as a wff. This makes the expressions compact and precise. It thus eliminates possibilities of misinterpretation of sentences. The use of symbolic logic also makes reasoning formal and mechanical, contributing to the simplification of the reasoning and making it less prone to errors.

Transcribing English sentences into wffs is sometimes a non-trivial task. In this course we are concerned with the transcription using given predicate symbols and the universe.
To transcribe a proposition stated in English using a given set of predicate symbols, first restate in English the proposition using the predicates, connectives, and quantifiers. Then replace the English phrases with the corresponding symbols.

Example: Given the sentence "Not every integer is even", the predicate "E(x)" meaning x is even, and that the universe is the set of integers,
first restate it as "It is not the case that every integer is even" or "It is not the case that for every object x in the universe, x is even."
Then "it is not the case" can be represented by the connective "", "every object x in the universe" by " x", and "x is even" by E(x).
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 even". 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) O(x)]

For another interpretation of this sentence see a note

"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)]

### Test Your Understanding of Translation

Indicate which of the following statements are correct and which are not.

Click Yes or No , then Submit. There are two sets of questions.

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