Combining Propositions

In our analysis in previous sections we allowed ourselves to refer to specific propositions with single letters, such as p, q, and r and so forth,with the restriction that we do not use t, f, or v. We will continue to use this convention throughout this text, indeed, when we start our formal analysis of arguments such single letters will dominate our exposition, together with other symbols which do not stand for propositions but instead words like, "and", "or", "if. . .then" and others.

To anticipate such usage, let us informally stipulate that two propositions p and q are equivalent only if p and q are true (or false) under the exact same conditions or circumstances. For example, if p stands for, "Exactly 24 hours have passed since we put the petri dish in the incubator", and if q stands for, "Exactly one day has passed since we put the petri dish in this incubator", then p is equivalent to q since one day is 24 hours by definition.

Keeping this in mind, suppose we have two propositions, p and q, which are not equivalent, can we form another proposition that implies both p and q but not equivalent to either alone? The answer is yes!

To illustrate how, consider two specific propositions: let p be, "John is a student at Pima Community College" and let q be, "Mary is a student at the University of Arizona". Suppose that John and Mary are not connected in any way, (in other words p and q are independent propositions). We want to create another proposition, which we will call R, which implies both p and q.

The way we do this is to join p and q together with the conjunction, "and" and in doing so create another propositions R which implies both. A specific example will be helpful: using the p and q given above we create R by connecting p and q with the word "and", hence R becomes, "John is a student at Pima Community College and Mary is a student at the University of Arizona" . Symbolically we might say that R = "p and q". Clearly R is not equivalent to p or to q alone since if just one of p or q is false then R is false (which means R could be false while p is true making it impossible for the two to be equivalent). But the truth of R implies the truth of both p and q individually, since R is true only when both p and q are true (if you have become lost with the p's and q's at this point, just go back and re-read the above replacing p, q and R with the given propositions).

The process of joining two propositions with the word "and" is called conjunction. In the above example, we restricted the use of conjunction to two propositions which are not logically equivalent. This restriction assured us that the resulting proposition R was not equivalent to either of the original two. In general, we can ignore this restriction and can use conjunction freely between any number of propositions of any type to create another proposition whose truth implies the truth of each of the individual propositions.

To summarize, we can take any number of propositions, join them all together with multiple uses of the word "and" to form another proposition which is the conjunction of all of the individual ones with the result that the truth of the new proposition implies that each individual proposition be true.

Example 2.1.1

Let R be "John is 28 and John is a student at San Antonio College and John is a philosophy major", then R is true only if the individual propositions, "John is 28", "John is a student at San Antonio College" and "John is a philosophy major" are all true.

Example 2.1.2

Let p, q, s, m and n all be premises to an argument whose conclusion we denote as c. Let R be the proposition created by the conjunction of p, q, s, n and m. In other words, R is equivalent to the proposition "p and q and s and n and m".

We will learn more about the process of conjunction when we study formally the truth table definitions where words like "and' will play an important role.

For now we introduce the concept of inference which will take up a long way in our goal to classify arguments in the sense of deciding which arguments give good reasons to accept the truth of their conclusions.