Lesson 16a: Compound Propositions: Conjunctive Propositions

In many cases categorical propositions—for example, “All crocodiles are reptiles”—themselves may become elements of larger propositions. These larger propositions are known as compound propositions, and there are three types: conjunctive propositions, disjunctive propositions, and hypothetical propositions.

For the purposes of the next few lessons, we will use something known as variable notation to showcase the forms of these propositions. We briefly saw this when we represented the Universal Affirmative proposition, for instance, as “All A’s are B’s.” 

• Variable Notation: a short-hand notation in which we represent the instances of a proposition in an argument with variables (common examples of which are p, q, and r).  

Just like math uses x or y to represent unknown quantities, logic sometimes uses variables to represent propositions. For example, we might represent all the instances of the proposition “Socrates was mortal” in an argument with the variable p, so that (1) we may more easily represent complex statements that involve that proposition and (2) we can more easily discern the formal qualities of that argument. Let’s also represent “All men are mortal” as q, and “Socrates was a man” as r. Thus, instead of “If all men are mortal and Socrates was a man, then Socrates was mortal,” we could represent this statement more succinctly as “If q and r, then p.” In the following lessons, we will see why this kind of notation is useful.

Within this lesson, we’ll deal with the first of the three kinds of compound propositions: conjunctive propositions.

• Conjunctive Propositions: a proposition consisting of two (or more) categorical propositions, linked together by “and.” Sticking to the case of two categorical propositions, the logical form of such a proposition, in variable notation, is “p and q.”

Conjunctive propositions are formed through a process known as conjunction, in which two propositions are linked together by “and.” The conjunction of the two propositions “p” and “q” will take the form “p and q” in variable notation. Take the two propositions “A frog is an amphibian” and “A crocodile is a reptile.” The conjunction of these two propositions —the conjunctive compound proposition — would be “A frog is an amphibian and a crocodile is a reptile.”

Truth Conditions for Conjunctive Propositions: In determining the truth of a compound proposition, we have to consider the truth of each proposition in the compound. We call these propositions linked by “and” the conjuncts of a conjunctive proposition. For instance, if we have the conjunctive proposition, “Tables are pieces of furniture and all dogs are friendly,” the two conjuncts would be (i) “Tables are pieces of furniture” and (ii) “all dogs are friendly.” For the conjunctive proposition “and,” the entire proposition is true if and only if both of the propositions within it are true. In other words, “p and q” is true if and only if p is true and q is true. For example, the proposition “a frog is an amphibian and a crocodile is a reptile” is true because “a frog is an amphibian” is true and “a crocodile is a reptile” is true. If either of the two conjuncts turns out false, the conjunction as a whole will be false. We can exactly how this works in the table below:

We can read this by saying (starting with the first row that assigns truth-values) “If p is true and q is true, then ‘p and q’ is true.” Secondly (now with the next row), “if p is false and q is true, then ‘p and q’ is false.” We can continue this for the fourth and fifth rows.  As we see above, the only situation when the compound proposition “p and q” is true is when both p and q are independently true.

In our everyday usage of “and,” we understand an “and”-sentence to be true only if both of its conjuncts are true. This holds for conjunctive propositions. If someone stated “1 + 1 = 3 and 3 + 3 = 6” or “1 + 1 = 2 and 2 +2 = 5,” we would immediately say that person is wrong — in the first utterance, because the first proposition was false; and in the second utterance, because the second proposition was false. But, if someone stated “1 + 1 = 2 and 2 + 2 = 4,” we would call that statement correct, because both the propositions in the conjunctive statement are correct.