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 notes titled “Propositions and Symbols Used in Propositional or 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 p is true and q is true, then p v q is true.
- If p is true and q is false, then p v q is true.
- If p is false and q is true, then p v q is true.
- If p is false and q is false, then p v q is 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 q is false. So, if p is true and q false, then the statement p v ~q is true. To illustrate:
The illustration above says that p is true and q is false. Now, before we apply the rules in inclusive disjunction in the statement p v ~q, we need to simplify ~q first because the truth-value “false” is assigned to q and not to ~q. If we recall our discussion on the rule in negation, we learned that the negation of false is true. So, if q is false, then ~q is true. Thus, at the end of it all, p v ~q is true if p is true and q is false.
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.”