In these notes, I’ll briefly focus on tautologies and contradictions in propositional or symbolic logic. However please be aware that that is simply an introductory dialogue on tautologies and contradictions as my essential intention right here is simply to make college students in logic change into conversant in the subject underneath investigation.
On the one hand, a tautology is outlined as a propositional method that’s true underneath any circumstance. In different phrases, a propositional expression is a tautology if and provided that for all attainable assignments of reality values to its variables its reality worth is at all times true.
Thus, a tautology is a proposition that’s at all times true. Contemplate the next instance:
Both the accused is responsible or the accused will not be responsible. (p)
Clearly, the proposition is a disjunction; but each disjuncts will be represented by the variable p. Therefore, the proposition is symbolized as follows:
p v ~p
Now, in what sense that this proposition is at all times true? The reality desk under will show this level.
As we will see within the reality desk above, if p is true, then ~p is false; and if p is false, then ~p is true. And if we apply the foundations in each inclusive and unique disjunction, the results of p v ~p is at all times true. If we recall our dialogue on inclusive and unique disjunction, we discovered that an inclusive disjunction is true if a minimum of one of many disjuncts is true; and an unique disjunction is true if one disjunct is true and the opposite is false, or one disjunct is false and the opposite is true.
Therefore, there is no such thing as a means that p v ~p will change into false. Certainly, the propositional kind p v ~p is at all times true.
Then again, a contradiction is outlined as a propositional method that’s at all times false underneath any circumstance. In different phrases, a propositional expression is a contradiction if and provided that for all attainable assignments of reality values to its variables its reality worth is at all times false. Thus, once more, a contradiction is a proposition that’s at all times false. Allow us to think about the examples under.
Man is each mortal and immortal. (p)
Clearly, the proposition is a conjunction; but each conjuncts will be represented by the variable p. Therefore, the proposition is symbolized as follows:
p • ~p
Now, in what sense that this proposition is at all times false? The reality desk under will show this level.
As we will see within the reality desk 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 right here, which says that “A conjunction is true if and provided that each conjuncts are true,” then absolutely there is no such thing as a means that the proposition “Man is each mortal and immortal” or p • ~p will change into true. Certainly, the propositional kind p • ~p is at all times false.
