Exclusive Disjunction

Symbolic Logic

 

 

 

Exclusive Disjunction

In my previous post titled “Inclusive Disjunction” (see http://philonotes.com/index.php/2018/02/06/inclusive-disjunction/), I discussed the nature and characteristics of an inclusive disjunction, including its rules and how to determine its truth-value. In this post, I will focus on exclusive disjunction.

An exclusive disjunction is a type of disjunction that is connected by the words “Either…or, but not both.” As we already know, the symbol for the connective of a disjunctive statement is v (wedge). However, an exclusive disjunction is symbolized differently from an inclusive disjunction. Consider the following examples below:

  1. Either John is singing or he is dancing, but not both.
  2. Either John is sleeping or he is studying.

Example #1 is clearly an exclusive disjunction because of the words “but not both.” Please note that it is possible for John to be singing and dancing at the same time (hence, inclusive), but because of the qualifier “but not both,” which clearly emphasized the point that John is not singing and dancing at the same time, then the statement is clearly an exclusive one.

Now, if we let p stand for “John is singing” and q for “He is dancing,” then the statement “Either John is singing or he is dancing, but not both” maybe symbolized as p v q. However, this is faulty because it does not clearly specify what the statement “Either John is singing or he is dancing, but not both” states. So, how do we symbolize example #1 above?

As already mentioned, if we let p stand for “John is singing” and q for “He is dancing,” then we can come up with p v q.  But it’s not yet complete. We need to take into consideration the phrase “but not both.” If we recall the discussion on conjunctive statements, we know that the symbol for “but” is (dot), and in the discussion on negative statements, we learned that the symbol for a negation is ~ (tilde). Now, the word “both” in the statement refers to “John is singing (p)” and “He is dancing (q).” Thus, the phrase “but not both” is symbolized as follows: ~ (p q). If we add this symbol to the previous statement p v q, then we arrived at
(p v q) ~ (p q).

Thus, the symbol for the exclusive disjunction “Either John is singing or he is dancing, but not both” is:

(p v q) ~ (p q)

However, logicians used a more simplified symbol for the phrase “but not both.” They used the underlined wedge v to symbolize “but not both.” Thus, the exclusive disjunction “Either John is singing or he is dancing, but not both” is symbolized as follows:

p v q

Please note that the symbol p v q is read as follows: “p or q, but not p and q.”

In some cases, the exclusive disjunction does not contain the phrase “but not both,” but if we analyze the statement, it denotes exclusivity. Let us consider example #2, which reads:

Either John is sleeping or he is studying.

Although the statement does not contain the phrase “but not both,” it is pretty obvious that it is not possible for John to be sleeping and studying at the same time. Hence, example #2 above is an exclusive disjunction.

If we let p stand for “John is sleeping” and q for “He is studying,” then the statement “Either John is sleeping or he is studying” is symbolized as follows:

(p v q) ~ (p q)

or, simply,

p v q

 


Rules in Exclusive Disjunction

  1. An exclusive disjunction is false if both disjuncts have the same truth-value.
  2. Thus, for an exclusive disjunction to be true, one disjunct must true and the other false, and vice versa.

The truth table below illustrates this point.

The truth table above says:

  1. If p is true and q is true, then p v q is false.
  2. If p is true and q is false, then p v q is true.
  3. If p is false and q is true, then p v q is true.
  4. If p is false and q is false, then p v q is false.

Now, given the rule in exclusive disjunction, how do we, for example, determine the truth-value of the
exclusive disjunction ~ p v q?

Let us suppose that the truth-value of p is true and q is true. So, if p is true and q true, then the statement ~ p v q is true. To illustrate:

 

The illustration says that p is true and q is true. Now, before we apply the rule in exclusive disjunction in the statement ~p v q, we need to simplify ~p first because the truth-value “true” is assigned to p and not to ~p. If we recall our discussion on the rule in negation, we learned that the negation of true is false. So, if p is true, then ~p is false. Thus, at the end of it all, ~p v q is true if p is true and q is true.

 

 

We have provided a video for 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)

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