## Inclusive Disjunction

**Inclusive Disjunction**

A **disjunction** or **disjunctive statement** is a compound statement or proposition that is connected by the words “**Either…or**” or just “**or**.” And the component statements in a disjunction are called “disjuncts.” There are two types of disjunctive statements used in symbolic logic, namely: **inclusive** and **exclusive** disjunction. In this post, I will only focus on inclusive disjunction.

As I discussed in my other post titled “Propositions and Symbols Used in Symbolic Logic (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/), the symbol for the connective “Either…or” is** v **(wedge).

**Inclusive disjunction** uses the connective “**Either…or, perhaps both**.” Consider the example below.

**Either** Jake is sleeping **or **Robert is studying, perhaps both. (**J, R**)

If we let J stand for “Jake is sleeping” and R for “Robert is studying,” then the statement “**Either** Jake is sleeping **or **Robert is studying, perhaps both”is symbolized as follows:

* **J***v** *R*

Please note that the constants J and R do not just represent Jake and Robert respectively; rather, they represent the entire statement. Thus, J represents “Jake is sleeping,” while R represents “Robert is studying.”

It must also be noted that in most cases, the phrase “**perhaps both**” in an inclusive disjunction is not written in the statement. Thus, in determining whether the statement is an inclusive or an exclusive disjunction, we just need to analyze the statement per se. Let us consider this example:

**Either** Jake is sleeping **or **Robert is studying.

As we notice, the statement does not contain the phrase “perhaps both.” But if we analyze the statement, it is clear that it is an inclusive disjunction because it is possible for the two component statements, namely, “Jake is sleeping” and “Robert is studying,” to occur at the same time. (Please note that I will discuss the nature and characteristics of an exclusive disjunction in my next post.)

Rules in Inclusive Disjunction

- An inclusive disjunction is
**true**if at least one of the disjuncts is**true**. - If both disjuncts are
**false**, then the inclusive disjunction is**false**.

In other words, the rules say that the only condition wherein the inclusive disjunction becomes **false** is when both disjuncts are **false**. This is because the connective “Either…or” directly implies that either of the disjuncts is possible. Thus, in an inclusive disjunction, we just need one disjunct to be** true** in order for the entire disjunctive statement to become **true**. The truth table below illustrates this point.

** **The truth table above says:

- If
is*p***true**andis*q***true**, then*p***v**is*q***true**. - If
is*p***true**andis*q***false**, then*p***v**is*q***true**. - If
is*p***false**andis*q***true**, then*p***v**is*q***true**. - If
is*p***false**andis*q***false**, then*p***v**is*q***false**.

Now, given the rules in inclusive disjunction, how do we, for example, determine the truth-value of the inclusive

disjunction *p ***v** *~*** q**?

Let us suppose that the truth-value of ** p** is

**true**and

**is**

*q***false**. So, if

**is true and**

*p***false, then the statement**

*q*

*p***v**

*~***is**

*q***true**. To illustrate:

The illustration above says that ** p** is true and

**is false. Now, before we apply the rules in inclusive disjunction in the statement**

*q*

*p***v**

*~***, we need to simplify**

*q*

*~***first because the truth-value “**

*q***false**” is assigned to

**and not to**

*q*

*~***. If we recall our discussion on the**

*q***rule in negation**, we learned that the negation of

**false**is

**true**. So, if

**is**

*q***false**, then

*~***is**

*q***true**. Thus, at the end of it all,

*p***v**

*~***is**

*q***true**if

**is true and**

*p***is false.**

*q*Alternatively, we can determine the truth-value of the inclusive disjunction *p ***v** *~*** q **in the following manner:

The illustration above says that if we assign the truth-value **true** for ** p**, then we can conclude right away that the inclusive disjunction is

**true**because one of the disjuncts is already

**true**. If we recall, the rule in inclusive disjunction says “An inclusive disjunction is

**true**if at least one of the disjuncts 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)