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Part 1: Relations in Logic (Samples)

By philobean | June 26, 2007

Understanding truth tables is key to the next steps in logical deduction. This is so simply because all that follows are based on the truth relations shown in the truth tables, that is, what follows are simply derivative rules that were designed to ease analysis of validity and , consequently, soundness.

The truth or falsity of all complex statements can be deduced using truth tables. For example, take the following complex statement:

Manila is the capital of the Philippines and Bangkok that of Thailand, if and only if the world is flat or the sun is the center of the solar system.

There are four simple statements herein–(1) Manila is the capital of the Philippines, (2) Bangkok is the capital of Thailand, (3) the world is flat, and (4) the sun is the center of the solar system. These are true, true, false and true, respectively. Thus, we can represent all true statements with ‘T’ and all false statements with ‘F’ and present the statement in this wise:

T AND T, IF AND ONLY IF F OR T.

Using ‘^’ to represent the AND relation, ‘v’ to represent the OR relation, ‘:’ to represent the CONDITONAL relation, and ‘::’ to represent the EQUIVALENCE relation, we have the following: (T ^ T) :: (F v T). We use brackets, braces and parenthesis to represent proximity of relationship. We can then simplify the analysis of the above as follows:

(T ^ T) :: (F v T)
=> T :: (F v T)
=> T :: T
=> T

We can also represent statements with small roman letters. Taking the same example above, we’d have the following representation: (a ^ b) :: (c v d).

It is when we reduce complex statements in common language to symbols that we are able to easily determine a statement’s truth value and an argument’s validity and soundness. Using truth tables may become cumbersome when we deal with lengthy argumentation. Thus, a standard set of the rules of replacement and the rules of inference were developed as methods to facilitate our analyses.

Topics: Academic |

2 Responses to “Part 1: Relations in Logic (Samples)”

  1. wini^^ Says:
    June 29th, 2007 at 12:35 pm

    remind me of my basic digital logic theories..T and F..0 and 1..can we apply boolean law for arguments too?

  2. philobean Says:
    June 29th, 2007 at 12:38 pm

    Well, yes. If you read on, you’ll soon realize that the rules of replacement and the rules of inference quite cover boolean law.

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