Tautologies and Contradictions

Symbolic Logic

 

 

Tautologies and Contradictions

In this post, I will briefly discuss tautologies and contradictions in symbolic logic. But please note that this is just an introductory discussion on tautologies and contradictions as my main intention here is just to make students in logic become familiar with the topic under investigation.

On the one hand, a tautology is defined as a propositional formula that is true under any circumstance. In other words, a propositional expression is a tautology if and only if for all possible assignments of truth values to its variables its truth value is always true. Thus, a tautology is a proposition that is always true. Consider the following example:

Either the accused is guilty or the accused is not guilty. (p)

Obviously, the proposition is a disjunction; yet both disjuncts can be represented by the variable p. Hence, the proposition is symbolized as follows:

p v ~p

Now, in what sense that this proposition is always true? The truth table below will prove this point.

As we can see in the truth table above, if p is true, then ~p is false; and if p is false, then ~p is true. And if we apply the rules in both inclusive and exclusive disjunction, the result of p v ~p is always true. If we recall our discussion on inclusive and exclusive disjunction, we learned that an inclusive disjunction is true if at least one of the disjuncts is true; and an exclusive disjunction is true if one disjunct is true and the other is false, or one disjunct is false and the other is true. Hence, there is no way that p v ~p will become false. Indeed, the propositional form p v ~p is always true.

On the other hand, a contradiction is defined as a propositional formula that is always false under any circumstance. In other words, a propositional expression is a contradiction if and only if for all possible assignments of truth values to its variables its truth value is always false. Thus, again, a contradiction is a proposition that is always false. Let us consider the examples below.

Man is both mortal and immortal. (p)

Obviously, the proposition is a conjunction; yet both conjuncts can be represented by the variable p. Hence, the proposition is symbolized as follows:

p ~p

Now, in what sense that this proposition is always false? The truth table below will prove this point.

As we can see in the truth table above, if p is true, then ~p is false; and if p is false, then ~p is true. And if we apply the rule in conjunction here, which says that “A conjunction is true if and only if both conjuncts are true,” then surely there is no way that the proposition “Man is both mortal and immortal” or p ~p will become true. Indeed, the propositional form p ~p is always false.

 

Note:

We have provided a video of all our posts in Symbolic Logic. If you are interested, please visit the following:

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)

3. Negations of Statements (see https://www.youtube.com/watch?v=hHrmrh7qYnA)

The following posts may also be of great help in understanding the discussion above:

1. Propositions and Symbols Used in Symbolic Logic (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/)

2. Negation of Propositions in Symbolic Logic (see http://philonotes.com/index.php/2018/02/03/negation-of-propositions/)

3. Conjunctive Statements (see http://philonotes.com/index.php/2018/02/03/conjunctive-statements/)

 

 

 

 

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