Consistency and Inconsistency

Logic is a study of the consistency of beliefs. A belief is part of a psychological state in which a person thinks, is under an impression or believes that the universe has some property. They are readily represented by sentences, a linguistic interpretation. Such sentences can then be evaluated for truth.

Truth

What is truth? What sort of thing could be considered true?

Shut that door!
Are you going to the party?
Would that I were not hungry.
A panda eats shoots and leaves.

Of the above sentences, only one, it seems, is capable of being assigned a truth value (considered true or false). The last sentence is true, but what of the others? The first is a command. It can be obeyed or disobeyed, but cannot be true (or false). The second is a question and again cannot be true or false. The third expresses a wish; it may be true that the person uttering it wishes he or she were not hungry, but it itself cannot be evaluated for truth. It shall be said that only declarative statements are evaluable for truth.

A test, perhaps, for determining if a sentence is declarative or not is to ask if the following question is a meaningful, grammatical question in English:

Is it true that P?

where P is the sentence being analysed.

Is it true that shut that door?
Is it true that are you going to the party?
Is it true that would that I were hungry?

None of the above is meaningful English. These therefore fail the test.

So what is it for a declarative sentence to be true. There are many truth theories, each defining truth slightly differently. The definition of truth adopted in this book comes from the correspondence theory, which states that a sentence P is true if and only if the state of affairs designated by P obtains; "it is true that P" is true if P. Falsehood is similarly defined for a sentence: a sentence is false if the state of affairs it designates do not exist. Although this seems straightforward and platitudinous, there are complications, explanation of which roams outside the scope of this book. The reader is urged to investigate these!

For the sake of simplicity it will be assumed that all declarative sentences are true or false and so, if a sentence is true, its negation is false. The negation of a sentence P is any sentence that shares its meaning precisely with the sentence

it is not the case that P.

Definition of Consistency

Logic is concerned with the consistency of sets of sentences (used to represent beliefs) and with consistent outcomes to processing logical formulae containing operands and operatives. A set of sentences is said to be consistent if and only if there is at least one possible situation in which they are all true. So, the following set of sentences is consistent:

Grass is green.
It is not the case that the moon is green.
Most humans have ten fingers.
Grasshoppers have six legs and dogs have four.
Earth is a cube.

Note that not all of the sentences are true (sentence 5). However, if the Earth were a cube, it need not be that any of the other sentences are false. There is a possible situation in which they are all true. In this way, logicians are not interested as such in the actual truth value of a sentence (whether it is in fact true or false), but more in a sentence's possible truth values.

Consider the following English sentences:

The moon is entirely constituted from cheese.
The moon is only partly constituted from cheese.

There is no possible situation in which 1 and 2 are both true, that is to say, it is impossible for them both to be true at the same moment. Such a pair of sentences are said to be inconsistent. A set of sentences is said to be inconsistent if there is no possible situation in which they are all true.

A set of sentences may consist of only one sentence. In which case, if there is a possible situation in which that sentence is true, the sentence is said to be consistent. If there is no such situation, the sentence is said to be inconsistent.

For example, "grass is green" and "snow is blue" are examples of consistent sentences (and the set containing them and only them is consistent). The sentences "grass is green" and "grass is blue" (when taken individually) are consistent sentences, but any set containing them both is inconsistent.

The sentence "2+2=5" is inconsistent. There is no situation in which it is true.

If two sentences cannot be both true and cannot be both false, they are said to be contradictory. For example, "Socrates was a philosopher" and "Socrates was not a philosopher" are contradictory statements.

If two sentences cannot be both true at the same time (form an inconsistent set), they are said to be contrary. For example, "I have exactly 10 fingers" and "I have exactly 9 fingers" are contrary (both statements cannot be true, but both could be false). All contradictory statements are contrary.

Logical Possibility and Necessity

What exactly is meant by possible where it is used above? The type of possibility being considered is logical. If something is considered logically possible, it could have been the case that it was true, regardless of whether it is actually true or not. So, it could have been true that Charles I was not beheaded, for example if he had evaded the executioner, or that the moon is made of cheese, if, for example, the universe was slightly different to how it actually is.

Necessity can be defined in terms of possibility. It is, for example, impossible that Charles I was both beheaded and not beheaded, i.e. it is not possible that he was beheaded and not beheaded. A statement about impossibility is a statement about necessity, i.e. the way things must be. To say something is impossible is to say that no possible situation exists where it is true. If no situation exists where it is true, it must be false - it is necessarily false and so its negation is necessarily true.

It is not the case that P is possible.
It must be the case that it is not the case that P is true.

The reader should strive to understand that 2 follows from 1 (and indeed that 1 follows from 2) before continuing. Some examples will clarify this (and extend the idea):

*It is not possible that two plus two makes five.

 *It must be that two plus two does not make five.

*It is not possible that four is not the square of two.

 *It must be that four is the square of two.

*According to the definition above, it must be that a sentence is either true or false.

 *According to the definition above, it could not be that a sentence were neither true nor false.

A sentence that describes a state of affairs that is not possible is said to be necessarily false. A sentence the negation of which describes a state of affairs that is not possible is said to be necessarily true.