How important is it to be logically consistent? To answer this question it seems prudent to first explain what it means to be logically consistent. In simple terms we would describe a series of statements as being logically consistent if it is possible for all statements to be true at the same time. In the same way we would describe a person to be logically consistent if it is possible for all their beliefs (and statements) to be true at the same time.
Opposed to consistency is inconsistency. Using the above definition of consistency in reverse, we can describe a person to be logically inconsistent if it is impossible for all their beliefs to be true at the same time.
There are two major views on the importance of logical consistency (or in reverse: how bad it is to be logically inconsistent). The first view is that logical consistency is of primary importance and that inconsistency is the worst intellectual crime of all. In the second view, the importance of consistency is degraded; consistency is still considered desirable but inconsistency isn't all that bad.
Traditional logical discourse is based on three laws of thought.
The law of Non-contradiction
No statement can be both true and false
(For a statement cannot have both itself and its negation true at once)
The law of Identity
Everything is what it is and not another thing
The law of Excluded Middle
Every statement is either true or false
(either it, or its negation must be true)
A person cannot possibly follow those laws if they are logically inconsistent. This means that they are using a non-traditional form of logic. Avicenna took a very strong view against those who objected to the law of non-contraction...
""Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned."[ Avicenna, Metaphysics, I]
Yet alternative systems have been established, merely rejecting the traditional system is not bad in itself. It becomes a problem when a person is acting under a flawed or an imperfect system (a lay person may call them illogical... or insane) which could lead to the person performing actions detrimental to their life and happiness.
This would seem to concur with the first view on the importance of consistency. Since inconsistency can lead to self destructive behaviour; being logically consistent is of critical importance!
However this is clearly very extreme! Performing the occasional contradiction does not mean a person will accept all inconsistencies, nor does it mean they will engage in self-destructive behaviour. At the least all it means is that one of their beliefs turns out to be wrong.
Walt Whitman is supporting of this view...
"Do I contradict myself?
Very well then I contradict myself,
(I am large, I contain multitudes.)" [Walt Whitman, Song of myself, 1867]
A contradiction does not break the human mind and is easily shrugged off. This supports the second view on the importance of consistency. Although it is desirable to be consistent (I mean, who wants to be wrong?) it is not that important.
Yet again it is not quite this simple: Ex falso quodlibet 'the explosion principle'. .[The Internet Encyclopaedia of Philosophy]If one claims something is both true and not true, one can logically derive any conclusion. By accepting even one inconsistency we can logically derive any conclusion. If we accept this principle then being inconsistent suddenly becomes very bad. From a strictly logical perspective it means that any and all conclusions we reach inferred by the contradiction (and if it involves a person's beliefs, this means all) are logical. The explosion principle is evidence supporting the first view, if by making a contradiction we can logically reach any conclusion it is very important to be consistent! Otherwise we may make conclusions our common sense rejects completely... (bad).
A common argument against inconsistency is that contradictions are meaningless. But the argument form 'reduction ad absurdum' relies on contradictions having meaning to find proof.[The Internet Encyclopaedia of Philosophy] If we assert that contradictions have no meaning then we can no longer use this argument for any proof or purpose. Since the utility derived from reduction ad absurdum is much greater than any derived from having meaningless contradictions, it is much more sensible to give contradictions meaning. Of course, just giving meaning to contradictions does not make them good or bad: but it does mean that (comparatively) they are not as bad as if we didn't assign them any meaning at all. This supports the former view then, inconsistency is not (all) that important.
There has been one longstanding contradiction in modern physics: it has now been solved but we may still look at it when interpreting the importance of consistency. In the theory of Quantum Mechanics we demand that electrons jump immediately from one orbit to another instantly; the quantum leap. [Bohr model, 1913] Electrons cannot ever occupy an intermediate orbit. Now for the contradictions...In the general theory of relativity [general theory of relativity , 1920] we demand that nothing can travel faster than light (300,000 km/sec). As we can see we cannot possible accept that both theories are true, but all physicists do! This shows that consistency is not all that important; physicists did not find themselves unable to work because these theories contradicted.
Time to draw this essay to a conclusion...Inconsistency in its extreme can lead to self-destruction, yet common sense tells us that this is not probable. Common sense also tells us that contradictions are easy to make and have no big consequences. Strictly logically, consistency is much more important; if we are not consistent then the conclusions we logically derive are much different than those which common sense allows us to accept. Looking at 'real world' examples in modern physics of inconsistencies also demonstrates that is not that important, Physicists were able to operate under two apparently conflicting theories for years with no great drawback. In short, I will conclude that consistency is (although certainly desirable) not that important.