The Logical Absolutes = The Laws of Thought
The Logical Absolutes
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 = something or some proposition
Then the following options exist:
I. X is X; [LI]
II. X is not non-X; [LI]
III. X is not neither X nor non-X; [LEM]
IV. X is not both X and non-X; [LNC]
LI = Law of Identity
LNC = Law of Non-Contradiction
LEM = Law of Excluded Middle
[Note: ~X = not X, where X is a any proposition or anything]
1. The Law of Identity [LI]:
Something is what it is, and it is not what it is not: (X = X) <=> (X =|= ~X).
2. The Law of Non-Contradiction [LNC]:
X and ~X cannot both be true (at the same time, in the same sense) <=> X cannot be both true and false <=> Nothing can both be and not be: ~(X ∧ ~X), where ∧ = ‘and’ (logical operator).
3. The Law of Excluded Middle [LEM]:
Either X or ~X is true, which is logically equivalent to the principle of bivalence, which states: either X is true or X is false: X ∨ ~X, where V = or (logical operator).
Thus, the logical absolutes taken together state that for any proposition X:
X = X [LI];
X =|= not ~X [LI];
Either X is true or ~X (is true) [LEM];
Not neither X nor ~X [LEM];
Not both X and ~X [LNC].
For proposition X, there exist the following positions:
1. X is true
2. ~X is true
3. Both X and ~X are true
4. Neither X nor ~X is true
The above four positions can be reformulated as follows:
1. X is true
2. X is false
3. X is both true and false
4. X is neither true nor false
Position (3) is logically impermissible due to the law of non-contradiction, which states: there exists no X such that both X and ~X are both simultaneously true. Or equivalently stated, there exists no X such that X is both true and false simultaneously.
Position (4) is logically impermissible due to the law of excluded middle, which states: there exists no X such that neither X is true nor ~X is true. Or equivalently stated, there exists no such X that X is neither true nor false, thus excluding the middle option in between true and false.
The logical absolutes (the laws of thought) are brute facts of reason that establish true dichotomies (set of only two possible options exhausting all possibilities). The logical absolutes are rudimentary laws of logic which are indubitable, that cannot be doubted. In order for one to be able to attempt to disprove them, one must make use of them. Therefore there can be no deductive non-circular or non-question-begging justification for the use of the logical absolutes.
The Logical Absolutes Applied to God’s Existence
Let X be the proposition "god exists". A bivalent (two-valued) proposition is a declarative statement capable of bearing only one truth value: either true or false.
Let X = god exists.
Then ~X = god does not exist.
— Law of Identity: X=X.
1. (i) God is god
which is logically equivalent to (<=>):
—Law of Identity: X=|=~X.
1. (ii) God is not non-god
— Law of Non-contradiction:
~(X & ~X):
2. God is not both god and non-god. X is not both true and false: god cannot both be and not be. “God exists” cannot both be true and false.
— Law of Excluded Middle
3. (i) Either god exists or god does not exist: X or ~X. Either X (is true) or ~X (is true) <=> X is either true or false (no middle option exists in between).
<=>
— Law of Excluded Middle
3. (ii) God is not neither god nor nor non-god (but a third / middle option): god cannot neither be nor not be. “God exists” cannot be neither true nor false.
Therefore,
• God is god and cannot be non-god. — Law of Identity
• God cannot be both god and not god. God cannot both exist and not exist. Something cannot be both god and non-god. Nothing can be both god and non-god. The proposition: G:”God exists” cannot be both true and false.
— Law of Non-Contradiction
• God either exists or does not exist. Something/everything is either god or non-god. The proposition G:= “God exists” can either be true or false. God cannot neither exist nor not exist. The proposition G cannot neither be true nor false.
— Law of Excluded Middle
Questions to consider:
— Can god violate the logical absolutes?
-- If so, can such a god be ruled out of existence?
The Logical Absolutes & The Liar’s Paradox!
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.
The Law of Identity: X = X
[Let ~X := not X = non-X]
1. X = X (something is what it is)
2. There exists no X such that X =|= X.
3. X=|=~X (something is not what it is not)
The Law of Non-Contradiction:
1.~(X & ~X)
2. X and ~X cannot both be true:
3. There exists no X such that X = X and X = ~X (simultaneously)
4. X cannot both be true and false.
5. Something cannot both be and not be (simultaneously).
6. Nothing can both be and not be
The Law of Excluded Middle
1. X V ~X
2. Either X is true or ~X is true
3. There exists no X such that X is neither X nor ~X.
4. Either X is true or X is false
5. Something must either be or not be.
6. Everything is either X or ~X.
7. Something cannot neither be nor not be
8. Nothing can neither be nor not be.
Let X: be [A = B],
Then, not-X: is [A =|=B].
Since X and not-X are mutually exclusive, they cannot both be true otherwise leading to a contradiction, namely that A can be both equal to B and not equal to B, thus violating the law of non-contradiction, which states A cannot be both A and not A.
The Liar’s Paradox:
Let: P = “This statement [P] is false.”
If P is true: then it is true that ‘this statement [P] is false’, which leads to [P] being false.
Likewise, If [P] is false, then “this statement is false” is false which leads to [P] being true, which sets up a paradox due to the fact that [P] refers to itself and that [P] being assumed true leads to a contradiction, in fact, an internal (self-referential) contradiction, in which [P] is both true and false, thus violating the law of non-contradiction, which states:
No proposition can both be true and false,
Or stated alternatively: Nothing can both be and not be!
The law of non-contradiction is a logical absolute, also known as a law of thought! It is one of the foundations of reason that applies to everything we think about: every proposition, every object of thought, as well to every thought itself.