Translating a statement to logic.

Jan 28, 2012

lizarton

Homework Statement

Translate the following sentences into propositional or predicate logic. Use the shorthand
symbols (e.g. [itex]\vee[/itex]) and define the meaning of each of your predicates and propositional variables. Be sure to include a domain (aka replacement set) for each quantified variable. You may need to rewrite the sentences slightly so as to make a variable more explicit.

"Irrational numbers have decimal expansions that neither terminate nor become
periodic".

Homework Equations

The Attempt at a Solution

Since the predicate refers to the decimal expansion of irrational numbers, I don't know whether to (a) declare/quantify a variable that represents the decimal expansion of an irrational number, or to (b) declare/quantify a variable that represents an irrational number.

(a)
T(x) is "d terminates".
P(x) is "d becomes periodic".
D is the set of all decimal expansions of irrational numbers.

[itex]\left(\forall d \in D\right)\left(\neg T(d) \wedge \neg P(d)\right)[/itex]

...versus:
(b)
D(x) is “x has a decimal expansion”.
T(D(x)) is “the decimal expansion of x terminates”.
P(D(x)) is “the decimal expansion of x becomes periodic”.
I is the set of all irrational numbers.

[itex]\left(\forall x \in I\right)\left( D(x) \wedge \neg T(D(x)) \wedge \neg P(D(x)) \right)[/itex]

1. What is the purpose of translating a statement to logic?

Translating a statement to logic is a fundamental skill in the field of logic and reasoning. It allows us to understand and analyze complex ideas and arguments by breaking them down into simpler, logical components. This process helps to eliminate ambiguity and make the statement easier to evaluate.

2. How do you translate a statement to logic?

The process of translating a statement to logic involves identifying the main components of the statement, including the subject, predicate, and any logical operators (such as "and," "or," and "not"). These components are then represented using symbols and logical notation, following established rules and conventions.

3. Why is it important to translate statements to logic?

Translating statements to logic is important because it allows us to analyze and evaluate arguments in a systematic and rigorous manner. By breaking down a statement into its logical components, we can identify any flaws or inconsistencies in the argument and make more informed decisions based on sound reasoning.

4. What are some common mistakes to avoid when translating a statement to logic?

One common mistake when translating a statement to logic is failing to accurately represent the logical operators. For example, using "and" instead of "or" can completely change the meaning of a statement. It is also important to avoid introducing any new information or assumptions that are not explicitly stated in the original statement.

5. Can a statement always be translated to logic?

In most cases, yes, a statement can be translated to logic. However, there are some statements that are inherently ambiguous or subjective, making it difficult to translate them into a logical form. Additionally, statements that involve emotions or values may not be easily translated to logic, as these concepts are difficult to quantify using logical notation.