## Tautologies and Contradictions

**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

**is**

*~p***false**; and if

**is**

*p***false**, then

**is**

*~p***true**. And if we apply the rules in both

**inclusive**and

**exclusive**disjunction, the result of

*p***v**

**is always**

*~p***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**

**will become false. Indeed, the propositional form**

*~p*

*p***v**

**is always**

*~p***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

**is**

*~p***false**; and if

**is**

*p***false**, then

**is**

*~p***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***•**

**will become true. Indeed, the propositional form**

*~p*

*p***•**

**is always**

*~p***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/)