**The Laws of Thought**
Laws of Identity, Non-Contradiction, Excluded Middle
Something is what it is,
and it is not what it is not,
and it is not neither or both:
what it is and what it is not.
Let: X be:
(i) Something (X): i.e., a ‘thing’ (i.e., an object of thought)
(ii) A proposition (X): a declarative statement capable of being
either true or false.(iii) A thought X (itself): a thought about an ‘object of thought’: ex., a belief or affirmation that a given proposition X is true.
Let: X: = something or a ‘thing’ or anything (i.e. some thing)
Then: ~X: = nothing or not a thing or not any thing (i.e., no thing)
**The Laws of Thought** Something is what it is [LI], and it is not what it is not [LI] and it is not neither [LEM] or both [LNC]: what it is (X) and what it is not (~X).LI: = Law of Identity = [X = X]
LNC: = Law of Non-Contradiction = ~ [X & ~X]
LEM: = Law of Excluded Middle = [X V ~X]
LI: = Something is what it is, and it is not what it is not.
LNC: = Something is not both what it is and what it is not.
LEM: = Something is not neither what it is nor what it is not.
Therefore, the laws of thought can be summarized as:
LI: = Identity = Something (X) is what it is (X), and it is not (~) what it is not (~X).
LNC: = Non-Contradiction = Something (X) cannot be both what it is (X) and what it is not (~X); that is, nothing (i.e., no thing) can both be what it is (X) and not be what it is (X).
LEM: = Excluded Middle = Something (X) either is or is not (what it is: X), and it cannot be neither what it is nor what it is not. In other words, something must either be or not be, and it cannot neither be nor not be: nothing can neither be nor not be.
The Laws of Thought Applied to Propositions
Let: X: = a proposition; then ~X = the negation (“not” operation) of X.
The Law of Identity: (X = X) & (X =|= ~X)
[LI]: A proposition X is identical to and implies itself and is not identical to and does not imply its negation ~X:
The Law of Non-Contradiction (LNC): ~ (X & ~X)
[LNC]: A proposition X and its negation ~X cannot both be true.
LNC can be restated as the joint affirmation of contradictories (X, ~X) is denied, since this constitutes a contradiction, and LNC states contradictions cannot be.
That is, a proposition X cannot be both true and false (simultaneously, at the same time, in the same sense): no proposition can both be true and not be true (i.e., be false):
The Law of Excluded Middle: X V ~X, where V = inclusive disjunction (“or”)
[LEM]: Either a proposition X is true or its negation ~X is true. The former statement of LEM can be recast in the form: Either X is true, or X is not true (i.e. false);
that is, X is either true or false.
Note however, that in LEM, the “or” operator is an inclusive disjunction.
Therefore, LEM can be reformulated as follows: A proposition X and its negation ~X cannot both be false together; that is,
It cannot be the case that neither X is true nor ~X is true, at least one of the two (X, ~X) must be true, including the option in which both X and ~X are both true together, but excluding the option in which both X and ~X are both false together.
Therefore, LEM can be restated as the joint denial of contradictories is denied!
Questions:
- Can god be so defined as to violate the laws of thought, also called "logical absolutes"?
- Is such a god logically possible, or is it a necessary falsehood (falsum) that god exists in some possible world? Can one logically exclude the possibility of god's existence if god creates logically impossible things, such a pen which is not a pen, or a pen that is both a pen and not a pen, or a pen that neither is a pen not is not a pen?
- Are the laws of thought logical absolutes?
- If so, in what sense are they absolute (logically)?
- Can god violate the logical absolutes?
- If so, can such a god be ruled out of existence?
