Truth Tables

One of the advantages of symbolic logic is that it gives us a handy way to examine and compare the patterns of true-false conditions inherent in any standard logical operation or combination of operations.  A truth table shows us, in convenient shorthand notation, how we can expect any logical relationship to turn out under various conditions, and also detect when the relationship is being misused.  Every logical statement, as we've already learned, has a truth value for which there are only two possibilities: either "true" or "false."  So, if we set up a table listing the possible truth values of each statement in a related group of statements, we can tell whether the truth value of the group as a whole is true or false—that is, whether or not it makes logical sense.  Let's use p and q as our representative statements, and put them into different relationships by subjecting them to various logical operations.  To figure the number of horizontal rows we'll need in the table, we must raise 2 (the number of truth-value possibilities for each statement) to the power equal to the number of statements we want to analyze, in this case also 2 (p and q).  Thus, we'll need 2², or 2 x 2 = 4 lines in the table to accommodate all possible combinations.  (If we had three statements (p, q, and r), we'd need 2³ or 2 x 2 x 2 = 8 lines; for four statements (p, q, r, and s), we'd need 2⁴, or 2 x 2 x 2 x 2 = 16 lines, and so on.)  Plus, to keep things clear, we'll need a top row to label clearly what each column contains, and a left column to identify each set of truth values by number.

In the two leftmost columns (not counting the number column), representing p and q by themselves, we begin by filling in all possible combinations of "true" (T) and "false" (F) for each of these two statements.  In row 1, we assume both p and q statements are true.  In row 2, we assume p true and q false.  In row 3, p false and q true.  And in row 4, both p and q false.  In so doing, we'll be ready to cover all possible combinations of truth values for the two statements.

Because this is a learning exercise, we'll take advantage of color in our examples to help illustrate the process.  Since the two variables p and q are under our direct control, we'll shade the top cells of their columns aqua to identify them as our source assumptions.  For any column we use to record the results of an evaluation, we'll shade the top cell yellow to identify it as a result column.

 Analyzing AND  Moving to the first result column, p*q (p AND q), we can consider whether this relationship is true or false for each case of p and q taken separately.  If we're not yet comfortable thinking in the abstract, we can substitute a couple of concrete statements for p and q.  Let's say p represents "Paisley likes to party" and q represents "Quigley likes quiet."  If each of these statements (also called conjuncts with respect to the AND or conjunction relationship) is true individually, then the combined statement, "Paisley likes to party AND Quigley likes quiet," is also true.  This is the logical nature of the AND relationship: when both of its conjuncts are true, then the AND relationship as a whole is also true.  So on row 1, where p is assumed true and q is also assumed true, we can also mark T for a true truth value in the column for p*q.

On row 2, we note that statement p (Paisley likes to party) is still assumed to be true.  However, statement q (Quigley likes quiet) is false.  If it isn't true that Quigley likes quiet, then the conjunction (the AND relationship) that Paisley likes to party and Quigley likes quiet isn't true either, because an AND relationship claims that both of its conjuncts are true, and in this case that claim is false.  If either individual conjunct is false, then the conjunction as a whole is also false.  So we can see that the evaluations on rows 2 and 3 will both turn out false for p * q, since on row 3 the other conjunct, p, "Paisley likes to party," is assumed false.  On row 4, both conjuncts p and q are deemed false, and this simply nails the lid on the AND relationship's coffin, so to speak.  So for the p * q column, the relationship evaluates as "true" only on row 1, where both conjuncts are considered true, and the p * q relationship is false in the other three cases, where either one or both of the conjuncts are false.

Now, just what do we mean by a "false logical relationship"?  Using the AND relationship as an example, we see that an instance in which a relationship doesn't hold true represents an absurdity, a logical impossibility, when a combination of truth values of the individual statements that make up a relationship are inconsistent with the logical sense of the relationship itself.  The absurdity renders the relationship either meaningless or impossible with the particular combination of truth values in question.  This pattern—the relationship is true (possible and meaningful) only if all of its premises are true—is consistent for all AND statements.  We'll find that the other basic relationships have consistent patterns as well, though the pattern for each of the five types of relationships is unique.  Oddly enough, absurdities can be very useful tools in logic.  For instance, when we're sifting through a variety of ideas and weeding out the ones that are obviously illogical, telltale absurdities enable us to reject bogus notions straightaway, and narrow our attention to only those ideas worthy of serious consideration.  Or, if we're trying to make the point that some idea is false, we can sometimes use a reduction to absurdity to demonstrate that the idea isn't logically consistent.

 Analyzing OR  Now we'll do the same sort of assessments for the p ˅ q (p OR q) column.  First, though, we need to address a potential conflict.  In the real world, the expression OR can have either of two meanings.  In some cases it can mean "one or the other, but not both," as in "You may have your steak either rare or well done (but not both rare and well done)," which is referred to as an exclusive OR relationship.  In others, it can mean "one or the other, or both," as in "You can have cream or sugar (or both cream and sugar) in your coffee," which is an inclusive OR relationship.  This inclusive relationship is always the sense implied by the OR operator in symbolic logic.  (It's possible to express an exclusive OR in symbolic logic, but it requires a combination of operations rather than a simple one.)

Now back to our analysis.  In the OR relationship, also known as disjunction, the individual statements are known as disjuncts (or as alternates, depending on whom we ask).  The rule for OR is as follows:  If either of the disjuncts is true, then the disjunction as a whole is true.  Thus, the disjunction as a whole is false only if both disjuncts are false.

If this sounds pretty flat, let's recall our pals Paisley and Quigley to demonstrate.  Statement p represents "Paisley likes to party," and statement q represents "Quigley likes quiet."  If either statement is true, then the OR relationship is true.  In row 1 we note that both p and q are true, so without a hitch we can put a T in row 1 of the p ˅ q column.  In row 2 we see that p is still true, so that's another T for line 2, regardless of the value of q.  In row 3, p is false, but q is true, and that means we can count the p ˅ q relationship true yet again.  However, in row 4, we see that neither p nor q is true.  In this case, p ˅ q cannot be true either, since an OR relationship claims that at least one of its disjuncts is true.  Or to use our concrete examples, if it's false both that Paisley likes to party and that Quigley likes quiet, then it's also false that Paisley likes to party OR Quigley likes quiet.  But if either statement about either of these characters is true, then the OR relationship is also true.

 Analyzing IF  Now things begin to get a bit more complicated.  It seems that Paisley has fallen on hard times, and has moved in with his old friend, Quigley.  However, their different personalities put a strain on the relationship.  Paisley insists on inviting noisy groups of people of questionable repute to the apartment at all hours of the night.  After just a week or two, Quigley, in his misery, finds himself speculating that his beloved serenity might be restored if Paisley's erratic behavior gets him arrested and hauled off to jail.  We can symbolize these thoughts thus: p represents "Paisley goes to prison," q represents "Quigley gets his quiet back," and the whole musing is summed up as p → q (IF p THEN q): "IF Paisley goes to prison, THEN Quigley gets his quiet back."

So now we consider what happens when one or both of these statements are true, or one or both are false, as stipulated in rows 1 through 4 of the p and q columns of our table.  Recall that in the AND (conjunctive) relationship, both individual statements are called conjuncts, and that in the OR (disjunctive) relationship, both statements are called disjuncts.  There's no logical distinction between first and second conjuncts, or between first and second disjuncts.  That's because in both AND and OR relationships, the truth or falseness of each statement stands on its own, independent of the other statement.  In an IF...THEN (conditional) relationship, this is not the case; rather, the relationship implies that the truth value of one of the statements depends in some way on the truth value of the other statement.  To distinguish what controls what, the statement governed by IF is called the antecedent of the condition, and the statement governed by THEN is called the consequent of the condition.  While the statements of a conjunction or a disjunction are functionally interchangeable, the statements of a condition are not.  ("IF Paisley goes to prison, THEN Quigley gets his quiet back" is not logically equivalent to "IF Quigley gets his quiet back, THEN Paisley goes to prison.")

According to the logical nature of an IF...THEN condition, IF the antecedent is true, THEN the consequent must also be true.  So, since in row 1 both p and q are assumed true, then this requirement is fulfilled, and the IF...THEN relationship holds up: p → q, "IF Paisley goes to prison, THEN Quigley gets his quiet back," is true. 

In row 2, we've again assumed that the antecedent p is true.  However, we've also assumed that the consequent q is false.  Note that this particular combination is contrary to the already stated logical nature of the IF...THEN condition, and thus in this case the relationship p → q doesn't hold.  It's a logical impossibility for the consequent of a condition to be false when the antecedent is true.  So if this were actually to happen, it would mean that the condition itself is false, an absurdity.

An interesting twist: Most people seem almost instinctively to understand the nature of the IF...THEN conditional relationship in this straightforward way: IF the antecedent is true, THEN the consequent must also be true.  But if we stop to think it over for a moment, it might occur to us that this also implies something else: Since the consequent of a valid condition must be true whenever the antecedent is true, then if we find that the consequent is false, we also know that the antecedent must be false (or else the conditional relationship itself is false).  Observe that this is also borne out by the results on row 2!  (If this doesn't quite sink in on the first try, read it again a few times, until your brain latches onto it.  It's an important logical implication that comes in very handy at times!)  Thus, the IF...THEN conditional relationship can be used in two ways, depending on what we happen to know about the truth value of the antecedent and the consequent:

• if we know that the antecedent of a valid condition is true, then the consequent must also be true; or

• if we know that the consequent of a valid condition is false, then the antecedent must also be false.

Now, moving on to row 3, we see that p, the antecedent, is presumed false.  What does this imply about the consequent q?  Nothing!  If "Paisley goes to prison" is false (i.e., Paisley does not go to prison), this says nothing at all about whether "Quigley gets his quiet back" is true or false.  Quigley's quiet existence could be restored even if Paisley doesn't go to prison, if instead Paisley is killed in a street brawl, or dies of a drug overdose, or resolves to mend his wild ways, or simply moves out of Quigley's apartment.  In any of these cases (or any others that in one way or another remove Paisley's noisy lifestyle from the picture), statement q, "Quigley gets his quiet back," would be true.  On the other hand, it might not work out that way at all; if Paisley doesn't go to prison and nothing else happens, it's possible that Quigley could be stuck with Paisley's noisy partying for quite some time, and q would be false.  In other words, if the antecedent of a condition is false, then the truth value of the consequent could be either true or false, and the condition itself would still be valid.  In any case of a false antecedent, there's nothing about the condition that is violated; it holds, so we can mark it true, both in row 3, where p is false and q is true, and in row 4, where p is false and q is false.

From this, we can see that knowing that a condition's antecedent is false tells us nothing about its consequent, and knowing that the consequent is true tells us nothing about the antecedent.  If we try to draw any inferences from either a false antecedent or a true consequent, then we commit what's called a formal error.  It's a very common error to make, through either innocent mistake or deliberate effort to mislead.  Thus, we can generally say that a valid IF...THEN condition yields two valid inferences based on either a true antecedent or a false consequent, but that if we reverse these inferences our reasoning will become nonsensical (even if the results accidentally happen to be true in some cases).  We can also generally say that a condition evaluates as false—invalid or absurd—only if the antecedent is true and the consequent false; with any other combination of truth values, a condition evaluates as true, or valid.

 Analyzing IFF (IF AND ONLY IF)  Now let's change the scene once again.  Suppose, for a change, that Paisley comes home alone and in an unusually mellow mood one evening.  Surprised by this change, Quigley engages him in conversation.  As the two get to talking, they discover that both of them are musicians.  Paisley goes to his room to fetch his tenor saxophone, while Quigley wrestles his old double-bass out of the closet.  After tuning up, they get into a little jam session.  As the evening progresses, it's further discovered that each man also knows how to play the other's instrument, and they decide to switch off, with Paisley on bass and Quigley playing sax—maybe not musically as pleasing as the other way around, but well enough that both have a good time, punctuated by a few laughs.

Now obviously, neither man can play both instruments at once.  If one opts for the sax, then the other must play the bass, or vice versa.  So whenever they agree to play together, IF Quigley plays sax, THEN Paisley plays bass.  Now, recall that in a usual IF...THEN situation, the two statements are not interchangeable.  But in this case, it turns out that they are.  It's also true that IF Paisley plays bass, THEN Quigley plays sax.  So here we find we have a conditional relationship that works in both directions.  This is called a bi-condition, and is expressed as IF AND ONLY IF...THEN.  Now, this might seem to be getting ever more complicated and confusing.  But we'll be relieved to find that it really isn't, as we'll see when we fill in the p ↔ q column of our table.  (And we'll be overjoyed to learn that logicians allow us to abbreviate IF AND ONLY IF as simply IFF.)

In row 1, we're now getting used to the fact that p and q are both true.  Or in concrete terms, it's true that Paisley plays bass, and also true that Quigley plays sax.  So when we plug these two statements into the IFF relationship (and assuming that whenever this happens, the two will play together), we get "IF AND ONLY IF Paisley plays bass, THEN Quigley plays sax."  Since this is an entirely plausible situation, we can mark the relationship true in row 1.

In row 2, again p is assumed true: Paisley plays bass.  However, Quigley's lips are getting tired, and he refuses to play sax.  He'd rather play bass for a while.  Since both men can't play the bass at the same time, we indicate this as an impossible situation by marking the p ↔ q relationship false in this instance.  The same occurs if, as in row 3, Paisley and Quigley both insist on playing the sax at the same time.  So the relationship is false for row 3 as well.

Now we come to row 4, where both p and q are false.  Paisley won't play bass because he wants his own sax back, and Quigley is tired of playing sax and would rather go back to his bass.  In this case, it's false that Paisley plays bass (because he's playing sax), and also false that Quigley plays sax (because he's switched back to bass).  But this works out perfectly.  Since each one again has his own instrument to play, the jam session goes on, and so in this case it turns out that the p ↔ q relationship is true!

Now we can summarize the possible results of the four binary logical operations as follows:

• AND (conjunction): true only if both of its conjuncts are true, and false in all other cases.

• OR (disjunction): false only if both of its disjuncts are false, and true in all other cases.

• IF (condition): false only if its antecedent is true and its consequent false, and true in all other cases.

• IFF (bi-condition): true whenever its statements are both true or both false, and false if one statement is true and the other false.

 Analyzing NOT  Now it remains only to analyze the unary NOT operation, which is the easiest of all.  NOT p (~p) is simply a declaration that the statement p is not true.  So if, for example, p represents the statement "Paisley likes to party," then ~p represents its negation, "Paisley does not like to party."  (It does not represent "Paisley likes not to party" or "Paisley likes quiet" or "Paisley likes something else," but only the direct negation of p, "Paisley likes to party," the direct negation of that claim being "Paisley does not like to party.")  As long as we don't fool around with it and try to make it mean something that it doesn't, the truth-value matter of negation is about as simple as it can get:  If the original statement is true, then its negation is false; or if the original statement is false, then its negation is true.  So now we're ready to fill in the last two columns of our table.  In any row where p is true, ~p (NOT p) is false in that row, and in any row where p is false, ~p is true, without regard to q.

  The same goes for q and ~q, without regard to p.

Furthermore, since NOT is a unary operator, we can turn the reasoning for it around without any difficulty.  If ~p is true, then p is false, or if ~p is false, then p is true.  The truth values of any logical statement and its negation must always be opposite to each other at any given time and place.  If we find that they aren't, then we've discovered an absurdity, and we know something is wrong with the reasoning.

The nice thing about all this is that the pattern of each of these five basic relationships is logically consistent in every case, regardless of whether we're discussing p's and q's or any other pairings of letters or ideas—as in planting asters AND zinnias, or choosing between red OR white wine, or ruling out antibiotic treatment IF the diagnosis is a virus, or buying a Hummer IF AND ONLY IF we do NOT have a practical sense of the purpose of a personal vehicle.  We don't have to switch to a new set of rules for each situation.  Because there are few rules in basic logic, and because they're all consistent, and because their applications are virtually universal, it's clearly in our interest to devote a little time and effort to learning and understanding them.

We'll use truth tables again as an aid to understanding other concepts; at those times, however, we'll simply present the completed table, rather than working through it step by step each time.  Some readers may wish to work through truth table analyses on their own, either to verify the writer's results or to develop a firmer grasp of logical methods; this isn't mandatory, but it can be very helpful and is certainly encouraged.

Although we probably won't ever make a habit of translating plain language into symbols, it can sometimes help with analyzing a complex argument to ensure that it's internally consistent.  Then, when we also verify the accuracy of the claims made, we'll have a very firm basis on which to judge whether the argument leads to a truly justified conclusion—or else is mostly a load of rhetorical fluff without adequate substance.

TERMS

• truth table

• conjunction (AND)

• conjunct

• disjunction (OR)

• disjunct

• condition (IF)

• antecedent

• consequent

• bi-condition (IFF / IF AND ONLY IF)

• condition

• negation (NOT)

In the next lesson, we'll use truth tables to analyze groups of statements as they are combined into complex claims.

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