Truth Tables: Part 1

Logicians are typically not ones for images and visualization. Logic is a highly abstract field. Even whole statements - or propositions, in other words - are symbolized using a single letter, placed into strings of symbols and letters in a formal structure, from which a conclusion is easily(?) derived. These proofs, especially when they are long, can make one's head spin. They can also lead to frustration, just as a lengthy mathematical proof satisfies none but the most dedicated to the acquisition of mathematical knowledge.

Because of this, as we begin this section in the blog on "Doing Logic Well," we're going to take things slow. These posts are meant to be easy to read, even if the concepts can get challenging at times. They are also meant to build on one another. Therefore, these posts are designed such that you can read from one post to another and gain greater understanding of how to think well as a whole. You should, if I do things well, be able to see a movement throughout the posts in this section from less complex ideas to more complex ideas.

In spite of the abstractness of logic, sometimes you'll find an image or helpful visualization in the field. Truth tables are one such example. A truth table is a visual device used in logic to display the truth-conditions for a proposition. A proposition is the information content of a declarative statement. Declarative statements are assertive - they assert the truth of a claim. To give an example, consider the following proposition, which I will symbolize with the letter S:

S: The sky is blue.

(Typically, the letter used to represent the proposition is italicized, but there is no rule requiring this use. It is just what I'm used to.) S is expressed by the statement, in English, that "the sky is blue." Quotation marks are used to indicate that what's contained within them is an utterance; hence, it is in English. A different statement, which expresses the same proposition (S) is "Der Himmel ist blau." This statement, which is uttered in German, is not identical to the statement in English, but they express the truth of one and the same proposition, which is S. Just in case it seems like the distinction between a statement and a proposition is what we might call a difference without a distinction, consider the fact that the statement, "The sky is blue," is in English, but the proposition S is not in English. S is information content, and no information content is "in English." But statements, as utterances, are stated in a language.

In formal symbolic logic, we symbolize propositions and their negations. A negation is the denial of the truth of a proposition. When we want to symbolize a negation, we use the same letter as the proposition negated but use the symbol ~ to modify the proposition by negating it. Thus, the negation of S is the following:

~S: The sky is not blue.

As we can see, the options here are exhaustive. Either the sky is blue, or it is not. For every proposition, it is the case that the proposition is true or false. This is called the Law of Excluded Middle, and it is one of the most important laws in all of logic. There is no halfway house between truth and falsity. If you've found a proposition, there are only two options.

Some might be reading this and thinking, "We sometimes say of a statement that it is 'true and false.' What do we say about those instances?" The answer is that, typically, when we say of a statement that it is "true and false," we mean that it is true in one sense and false in a different sense. This typically indicates that the statement being considered is ambiguous. By ambiguous, I mean that the statement does not clearly point to one - and only one - proposition. One unique attribute of proposition is that they are the only truth-bearers. If a statement - or utterance - does not point precisely to only one proposition, then the statement cannot be clearly affirmed or denied as either true or false. Statements are true or false only insofar as they point to propositions, which are direct truth-bearers.

Some logicians, using physics as a metaphor, also describe propositions as either atomic or molecular. Atomic propositions are those propositions whose symbolization includes no operators or other propositions. (I'll define "operator" in a moment.) S is an atomic proposition, but ~S is not an atomic proposition because it includes the ~ operator, which modifies the proposition S. Atomic propositions cannot be divided into anything else, and in this sense, they are simple and primitive.

Molecular propositions combine operators and other propositions to form a more complex proposition. Operators are tools in logic for combining or modifying propositions. We've already seen the operator for negating a proposition (~). Here are some other examples, which will include S and another proposition Q, which I'll leave undefined because it is not important to define it right now:

Conjunction:

S ∧ Q

Translation: "S is true, and Q is true."

Disjunction:

S ∨ Q

Translation: "Either S is true, or Q is true."

Conditional:

S → Q

Translation: "If S is true, then Q is true."

(The locution "S is true" is technically redundant. Simply writing "S" is enough, but I added "is true" for the sake of clarity.)

In each of the examples above, the operators combine propositions to form new propositions. Therefore, (S ∧ Q), for example, is now itself a new molecular proposition formed by combining the atomic propositions S and Q with the operator for conjunction (∧). If we wanted to form other new molecular propositions, we could bring in the operator for negation (~) and use it to modify either the atomic parts of the proposition or the whole molecular proposition itself, thereby negating it. Here are those new molecular propositions and their translations:

Modification 1:

~S ∧ Q

Translation: "S is false, and Q is true."

Modification 2:

S ∧ ~Q

Translation: "S is true, and Q is false."

Modification 3:

~(S ∧ Q)

Translation: "It is not the case that both S and Q are true."

(In translating symbolized propositions into English, the locution "it is not the case that..." is used to translate the negation operator and to show that the operator is intended to modify the entire molecular proposition.)

These modifications are distinct molecular propositions, and the conditions under which they are true or false are distinct.

Speaking of which, let's move on to truth-conditions, which I've mentioned a couple of times already. Truth-conditions are, as the name suggests, conditions under which a given proposition is true or false. Given the Law of Excluded Middle, any distinct atomic proposition must have one of two possible "truth values": true or false. We can symbolize these truth values with the letter T and F, unitalicized to distinguish them from the symbols we give to distinct propositions. Because of the Law of Excluded Middle, the truth-conditions for atomic propositions are not terribly interesting; there are just two options.

But what happens when we combine atomic propositions and operators to yield molecular propositions? Here, the matter is a little more complex because the operators determine the truth-conditions for the molecular proposition. The truth-conditions for an "or-proposition" (otherwise called a disjunction, symbolized with the operator ∨) is not the same as the truth-conditions for an "and-proposition" (otherwise called a conjunction, symbolized with the operator ∧). It would be helpful if we had a way to visualize and distinguish these truth-conditions.

That is where truth tables become very helpful for logicians. As I said above, a truth table is a visual device used in logic to display the truth-conditions for a proposition. Once you become proficient at building these truth tables, it becomes easy to use them to see under what conditions a proposition is true or false. When used in actual argumentation, the results can be enlightening.

However, steps must be closely followed to correctly build a truth table. Here are those steps:

1. For any molecular proposition whose truth-conditions you want to find, divide the molecular proposition into its parts. We are going to begin with the conjunction (p ∧ q). (The lowercase, italicized letters p and q are used in logic to represent generic, non-specific atomic propositions.) No operator can stand on its own, but the two atomic propositions, p and q, can be separated.
   

2. Form a column for each of your atomic propositions and the molecular proposition, with enough rows to list all of the possible truth-conditions for the molecular proposition. If you're working on a sheet of paper, work from the left to the right. To determine how many rows you need, use the formula 2^x, where x represents the number of atomic propositions. Since this truth table maps a molecular proposition containing two atomic propositions (p and q), the number of rows needed is 2^2, which equals 4. So, you will need four rows (under the rows with the propositions) to map out all of the possible truth values.
   

3. Working from left to right, for each atomic proposition, list all of the possible truth values. Working in the proper order is essential here. Starting in the leftmost column, the first two rows will be labeled "T" for the truth value "true." The bottom two rows will be labeled "F" for the truth value "false." Then, in the middle column, you will alternate with the truth values, writing "T" on the first row, "F" on the second, and so on. We do this, so that all of the possible combinations of values are represented. Do not label anything under the molecular proposition yet.
   

4. For the molecular proposition, read each of the values in a row, left to right, to determine the truth value for the molecular proposition, given the truth values for the atomic propositions. For example, starting with the first row, notice that on the first row p and q are both labeled "T". This means that p and q are both true. Notice as well that the molecular proposition (p ∧ q) means, in English, "p and q are both true." So, if both p and q are true alone, then the molecular proposition is also true. Therefore, on the first row below (p ∧ q), we can write a "T". We'll continue to determine the truth values in the other rows in the same way.
   

After filling out the truth table, the final result can be found below.

p	q	p ∧ q
T	T	T
T	F	F
F	T	F
F	F	F

Now, we have a visual representation of the truth conditions for the molecular proposition (p ∧ q). In other words, we know under what circumstances this proposition is either true or false. Notice that conjunctions are difficult to prove true, since they are true only if all of their conjuncts is true. Since it is generally more difficult to prove a proposition true than false, conjunctions tend to be more difficult to prove. (This is a problem, for instance, in certain apologetic approaches.)

As a helpful exercise on your own, you can attempt to draw the truth tables for disjunction (p ∨ q) and conditional (p → q). In Part 2, I will discuss these truth tables and further applications of truth tables.

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