Part I: Lesson

Continuing from our study of conjunctive propositions last time, within this lesson, we’ll be looking at disjunctive propositions.

• Disjunctive Proposition: a proposition consisting of at least two other, categorical propositions, linked together by “or.” The simplest form of this proposition is “p or q.”

Disjunctive Compound Propositions:

In variable notation, propositions of this form will appear as “p or q”, with “p” and “q” both being separate propositions. For example, let’s take the propositions: 

1. “A frog is an amphibian” 

and 

2. “A frog is a reptile.” 

We can form the disjunctive proposition (1 or 2): “A frog is an amphibian or a frog is a reptile”. We call the propositions (1) and (2) disjuncts of the larger disjunctive proposition. 

Truth Conditions for Disjunctive Propositions: As is the case in natural language, the “or” propositions require that at least one of the disjuncts is true. But “or” has some ambiguity in natural language, and this causes a bit of confusion in the interpretation of disjunctive propositions. As with our use of “or” in common speech, disjunctive propositions can be interpreted inclusively or exclusively. 

Intuitively speaking, “or” is inclusive if both disjuncts can be true together. Suppose your friend has been making dinner and there’s ice cream for dessert. She says “you can have chocolate ice cream or vanilla.” In many contexts, this means that you can have either chocolate or vanilla or both! Perhaps you like both ice cream flavors so much that you would rather have a mix than have one of them. The exclusive “or” doesn’t permit this. Again, here’s an intuitive example. Suppose your friend is looking for her book. You say “well, either you can take the northern route home or you can take the southern route home.” Here the assumption is that these two scenarios cannot be true together. Either your friend will take the northern road home or the summer road home, but not both.

‘Inclusive or’: In its inclusive interpretation, a disjunctive proposition will be true when either disjunct is true alone, or when both disjuncts are true together. We can look at this, like we did with conjunctive propositions, in a table:

Recall that to read this table, starting with the first row with truth-values, we say: “if p is true and q is true, then ‘p or q’ is true.” Next, starting now with the next row, we say: “if p is false and q is true, then ‘p or q’ is true.” As we can see, the only time that “p or q” is false is if both of the disjuncts, p and q, are false. 

‘Exclusive or’: In its exclusive interpretation, a disjunctive proposition will only be true when either one disjunct is true or the other is true, but not when both are. Under either interpretation, the disjunction will be false when both disjuncts are false. This could be formalized in variable notation as “p or* q”. In other words, “p or q, and not both p and q.” We can also interpret ‘exclusive or’ as saying “one and only one of either p or q is true.”

Here we can see that the ‘exclusive or’ table looks identical to the ‘inclusive or’ table, except for the highlighted row. This makes sense, because the highlighted row is the instance in which both p and q are true. Remember, we can think of the ‘exclusive or’ as “p or q, and not both p and q.”