1: Consistency and Validity

The subject matter of Logic

The Logic we are dealing with is concerned with the sorts of things which can be true or false, such as beliefs and declarative sentences.

Truth and falsehood are truth-values.

Logic is not, however, concerned much with the actual truth-values of beliefs and sentences, but rather with such questions as:

            Is such and such a set of beliefs or sentences consistent?

            Is such and such an argument valid?

Consistency/Inconsistency

A set of beliefs is consistent just if it would be possible for them all to be true together: that is, if they are either in fact all true or could all have been true.

 

A set of beliefs is inconsistent just if it would be impossible for them all to be true.

 

A single belief can also be said to be consistent (if it is possible for it to be true) or inconsistent (if it is not possible). An inconsistent belief is said to be self-contradictory, or a contradiction.

 

A single belief which could not be false is said to express a necessary truth.

 

A single belief which is not inconsistent and does not  express a necessary truth is said to be contingent.

Note: we are dealing with a very weak sort of possibility, or a very strong sort of impossibility. For the purpose these definitions, it is possible for a human being to run a mile in under a minute (humans might have been stronger; the laws of nature might have been different) but it is not possible for 2 and 2 to make 5; it is not possible for a person to be both a bachelor and married; it is not possible for it to be both raining and not raining in the same place at the same time.

A set of sentences is consistent just if it would be possible for the sentences as being used all to be true.

 

A set of sentences is inconsistent just if it would be impossible for the sentences as being used all to be true.

 

A single sentence may also be inconsistent; in which case it is said to be self-contradictory, or a contradiction.

 

A single sentence which could not be false is said to be a necessary truth.

 

A single sentence which is neither inconsistent not a necessary truth is said to be contingent.

Notice that we have added "as being used". This is to take care of the fact that, for instance, the sentences "It is raining" and "It is not raining" could be used to refer to different times or places, or to the same time and place. In the latter case they would be inconsistent, but in the former case they would be consistent. It is very common for the truth of a sentence to depend on who or what is being referred to. So, whether "John is tall" is true or not will depend on which person is being referred to by "John". For the same reason, whether "John is tall" is consistent with "John is not tall" will depend on whether "John" is being used to refer to the same person each time (and, whether the time reference is the same, of course). Given this, we may often be unable to tell whether a set on sentences is consistent or not, without knowing how they are (or were) being used. Suppose, then, that we are faced with the question of whether the following set is consistent:

            {Socrates is a man, All men are mortal, Socrates is not mortal}

What are we to say? Well we won't find it difficult to say that the set is inconsistent, but we should no doubt add (if we want to make things absolutely clear) that we are assuming that "Socrates" is being used to refer to the same person throughout".