Logical Consistency and Contradiction

G. Randolph Mayes

We say that a statement, or set of statements is logically consistent when it involves no logical contradiction.  A logical contradiction is the conjunction of a statement S and its denial not-S.   In logic, it is a fundamental law- the law of non contradiction- that a statement and its denial cannot both be true at the same time.  Here are some simple examples of contradictions.

1.  I love you and I don't love you.
2.  Butch is married to Barb but Barb is not married to Butch.
3.  I know I promised to show up today, but I don't see why I should come if I don't feel like it.
4.  The restaurant opens at five o'clock and it begins serving between four and nine.
5.  John Lasagna will be a little late for the party.  He died yesterday.

These all seem to be contradictions because they seem either explicitly to state or logically imply a certain statement and its denial. (1) is an explicit contradiction.  You can't love someone and not love someone at the same time.  (2) is an implicit contradiction.  It depends on the unstated but well known principle:  if x is married to y, then y is married to x.  (3) is also an implicit contradiction.  It depends on the unstated meaning of promising, namely, that whenever you promise to do something you thereby acquire a moral obligation to do it.

Very often contradictions are only apparent.  For example someone in a love-hate relationship might utter something like (1), meaning "I love you in some ways, but I hate you in others".  This, of course, is not a contradiction at all.  (4) also can look like a contradiction, but this may just be the result of unclarity.  Perhaps the restaurant opens at 5:00 in the morning. (5) is not literally a contradiction, since a dead person could show up at a party.  We call it a contradiction just because the statement "John will be a little late for the party," strongly suggests  that John will be alive when he shows up.

When we tell people that they aren't making any sense, it is often because we think that they are saying something contradictory.  In a Dilbert cartoon one of Dilbert's office mates is complaining that she hasn't been trained how to use the new computer. The conversation proceeds as follows:

• Dilbert:  Why don't you just read the manual?
• Office mate:  Right. Who has time to do that?
• Dilbert:  You mean you have time to go to a training session, but you don't have time to read the manual?  That doesn't add up.
Here Dilbert's point is obviously that she is contradicting herself :  She is saying that she has time to learn and she doesn't.  Of course she might not really be contradicting herself at all.  It may be that she finds computer manuals very hard to understand, so that the time it takes to be trained really is far less than the time it would take to learn from the manual.

This example shows that while it is very important to be logically consistent, it is also important to permit people to be so. When people speak in a way that seems logically contradictory  it is often just because they are not speaking completely or clearly.  So the point of exposing apparent contradictions is not, ultimately, to criticize peoples views as nonsensical, but rather to make them be clearer about what they are saying.

Even when someone really is contradicting himself we often tend to go too far in the criticism.  For example, if I were to tell you that I do not like to eat fish, and you knew that I often ate and enjoyed tuna salad sandwiches, you would be justified in pointing out that I was involved in a logical contradiction and you might reason as follows:

• "I know you like tuna sandwiches, and tuna is fish, so you are saying that you both do and don't like fish.  That makes no sense."
It is important to observe, however, that you would not have been  justified in responding as follows:
• "I know you like tuna sandwiches and since tuna is a fish you do like at least one fish."
The difference between these two responses is subtle but important.  In the first case you have informed me, correctly, that I can not both like tuna and not like tuna at the same time, but you allowed me to make up my mind about which statement I am going to revise.  In the second case you assumed that my commitment to liking tuna is stronger than my commitment to not liking fish, and instructed me to revise my view that I don't like fish.  But suppose that the information that tuna is a fish made me sick to my stomach- perhaps I had never fully appreciated this fact- I may, as a consequence, never touch tuna again.  In that case, I would have revised the statement that I like tuna and thereby preserved the generalization that I don't like fish . You might consider this to be an absurd outcome, but it's not actually logically contradictory, because its not a contradiction to like tuna at one time and not like it at another.

The point here is that exposing a logical contradiction is just the beginning of a useful criticism. If the contradiction we point out is real, then we have essentially challenged the speaker to revise his or her views in one of several possible ways. A full, fair, critique, must take account of the various possible ways that this can be done.

Consider another example: Suppose I believe that atheists are bad people, and that all my friends are good people.  But  Mr. Pheeper, my long-time friend, decides that he is an atheist.  I am now faced with accepting the following list of statements:

(a) Mr. Pheeper is my friend
(b) All my friends are good people.
(c) Mr. Pheeper is an atheist.
(d) All atheists are bad people.

These four statements are logically contradictory, because they jointly imply that Mr. Pheeper is both a good person and a bad person. Logic requires some sort of revision to my set of beliefs, but logic does not demand one particular revision. I could (a) decide that Mr. Pheeper is no longer my friend, (b) decide that atheists aren't necessarily bad people, (c) decide that not all my friends are good people,  or (d) decide Mr. Pheeper  is not an atheist even though he says that he is. The point is that these are all possible solutions, each of which must be examined on their own merits.

Everyone understands at some level that contradictions must be avoided because they don't make any sense; so it is rare for people who understand what they are saying to contradict themselves explicitly. Usually contradictions are implicit in (i.e., logically implied by) what someone says. This means that there is usually a generalization, or principle, that the person is committed to which implies a statement that is inconsistent with (a) particular facts (like the tuna example) or (b) statements implied by other principles that the person also accepts (like the atheist example.)

Most people find it difficult to identify contradictions in an explicit way, but it is important to learn to do so.   Here is an explicit definition of a contradiction together with the proper method for identifying one.

Def.:  To be logically committed to the assertion of some statement, S, and  its denial, not-S, at the same time.
ID.:  Identify the statement that being both asserted and denied.

To see how to identify contradictions properly, consider the following conversational example.
• Mrs. Beeble:  You have been absent from class 11 times this month.  You fail.  Goodbye.
• Butch:  What?  That's impossible!   The class only meets twice a week.
• Mrs. Beeble:  True, but you have missed it 11 times nevertheless.
• Butch:  That doesn't make any sense.  You can't be absent from a class on days that it doesn't meet.
• Mrs.  Beeble:  You're saying you weren't absent on those days?
• Butch:  That's right..
• Mrs. Beeble:  So you were here?
• Butch:  No!
• Mrs. Beeble:  You are wasting my time.  If you weren't here, then you were absent.  Goodbye.
Notice here that Butch, while perfectly aware that Mrs. Beeble is talking nonsense, has not successfully identified the contradiction.  The closest he comes is to say  "You can't be absent from a class on days that it doesn't meet."  But this statement can have several different meanings.  Butch is trying to say that it is logically contradictory  to mark someone absent from a class that doesn't meet.  But all he is succeeded in communicating is that it is somehow an unfair or unreasonable requirement, which is open to interpretation.  Here then is a better way for Butch to proceed:

• Mrs. Beeble:  You have been absent from class 11 times this month.  You fail.  Goodbye.
• Butch:  What?  That's impossible!   The class only meets twice a week.
• Mrs. Beeble:  True, but you have missed it 11 times nevertheless.
• Butch:  So you are saying that I was absent on days that the class didn't meet.
• Mrs.  Beeble:  That's right.
• Butch:  Well, then you have committed a logical error.  "X was absent from event Y,"  logically implies that event Y occurred.  So you are saying that on the days that I was absent, the class both met and did not meet. That is nonsense.
• Mrs. Beeble:  Sit down and shut up you little bastard.
• Actually, the truth is that proper logical analysis does not compel agreement anymore than Butch's first approach.  Many people find it tiresome and offensive, but logic doesn't concern itself with things like that.

Consistency and Deductive Implication

Logical consistency is essential to good reasoning, but it is by no means sufficient.  Completely invalid reasoning will be logically consistent if the statements simply have nothing to do with each other.  For example the following argument

1.  Some dogs have fleas.
2.  Therefore I want a Reese's Peanut Butter Cup.

is logically consistent.  So, like deductive implication, the fact that an argument is logically consistent isn't always interesting.

The concept of contradiction does, however, give us an interesting way of defining the idea of deductive implication.  We know that reasoning is deductively valid whenever it is impossible for the premise(s) to be true and the conclusion false.  Another way of saying this is that a logical contradiction arises when we assume that the premises are true but the conclusion is false.  For example the following reasoning is valid

(1) All philosophers are transvestites.
(2) Melvin is a philosopher.
(3) Therefore Melvin is a transvestite.

because  if we assume that Melvin is not a transvestite this contradicts (1) and (2) which jointly imply that he is a transvestite.

On the other hand

(1) Everyone over the age of 30 is a liar.
(2) Mr. Pheepher is a liar.
(3) Therefore Mr. Pheepher is over the age of 30.

strikes many people as deductively valid.  It is not, however, because if we assume that Mr. Pheepher is not over the age of 30, no contradiction arises.  This sort of thing can be difficult to see intuitively, so in order to show that no contradiction arises we can offer a counterexample.  A counterexample is a state of affairs in which the premises are true and the conclusion is false.  A counterexample to the above situation would be a world in which everyone lies. In this case everyone over the age of 30 remains a liar, but so is everyone 30 and under, so there is no contradiction in assuming that Mr. Pheepher is 30 or under.