Exclusive Disjunction in Propositional Logic
In my other notes titled “Inclusive Disjunction in Propositional Logic”, I discussed the
nature and characteristics of an inclusive disjunction, including its rules and how to
determine its truth-value. In these notes, 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)
In my other notes titled “Inclusive Disjunction in Propositional Logic”, I discussed the
nature and characteristics of an inclusive disjunction, including its rules and how to
determine its truth-value. In these notes, 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)
