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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.

reference

1. Statements and Logical Operators

Ex. 1 Propositions

The sentence "2+2 = 4" is a statement, since it can be either true or false. Since it happens to be a true statement, its truth value is T.

The sentence "1 = 0" is also a statement, but its truth value is F.

"It will rain tomorrow" is a proposition. For its truth value we shall have to wait for tomorrow.

The following statement might well be uttered by a Zen Master to a puzzled disciple: "If I am Buddha, then I am not Buddha." This is a statement which, we shall see later on, really amounts to the simpler statement "I am not Buddha." As long as the speaker is not Buddha, this is true.

"Solve the following equation for x" is not a statement, as it cannot be assigned any truth value whatsoever. (It is an imperative, or command, rather than a declarative sentence.)

"The number 5" is not a proposition, since it is not even a complete sentence.


Ex. 1B Self-Referential Sentences

"This statement is false" gets us into a bind: If it were true, then, since it is declaring itself to be false, it must be false. On the other hand, if it were false, then its declaring itself false is a lie, so it is true! In other words, if it is true, then it is false, and if it is false, then it is true, and we go around in circles. We get out of this bind by refusing to accord it the privileges of statementhood. In other words, it is not a statement. An equivalent pseudo-statement is: "I am lying," so we call this the liar's paradox.

"This statement is true" may seem like a statement, but there is no way that its truth value can ever be determined: is it true, or is it false? We thus disqualify it as well. (In fact, it is the negation of the liar's paradox; see below for a discussion of negation.)

Sentences such as these are called self-referential sentences, since they refer to themselves.

Here are some rather amusing (and slightly disturbing) examples of self-referential senatences, the first two being taken from Douglas R. Hofstadter's Metamagical Themas:

• "This sentence no verb."
• "This sentence was in the past tense."
• "This sentence asserts absolutely nothing."
• "While the last sentence had nothing to say, this sentence says a lot."
• "This sentence has more to say than the last two sentences combined, if you count the number of words."

reference: The Logic Book - Fifth Edition

Sentences are things that are capable of being True or False. Truth Values {T, F}

Pittsburgh is the capital of PA. a sentence
Where is Pittsburgh? in terms of logic, NOT a sentence

An argument is a set of two or more sentences, one of which is designated as the conclusion and the others as the premises.

An argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion to be false. An argument is deductively invalid if and only if it is not deductively valid.

Example of a valid deductive argument:

There are three, and only three, people in the room: Juarez, Sloan, and Wang.
Juarez is left-handed.
Sloan is left-handed.
Wang is left-handed.
_________________________________________________________________________________________
All the people in the room are left-handed.


Invalid deductive argument:
Sloan is left-handed.
Wang is left-handed.
_______________________________
Everyone is is left-handed.

It is invalid because, whereas the premises may be true, the conclusion is false.

A set of sentences is logically consistent iff it is possible for every member of the set to be true (together).

Consistent: {Joe is silly (T), Joe is smart (T)}

Not consistent: {All humans are mortal (T), Socrates is human (T), Socrates is not mortal (F)}