The Logical Absolutes = The Laws of Thought

[Petros Agapetos]

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?

[Petros Agapetos]

The Laws of Non-Contradiction and Excluded Middle Explained

Let X := anything (object) or any proposition
Let: X = something <--> ~X = nothing, or
Let X := "something exists" <---> ~X = "nothing exists" (something does not exist).

The Law of Identity

X = X {something is what it is}
X =/= ~X {something is not what it is not}

The Law of Non-Contradiction [LNC]
A proposition X and its negation ~X cannot both be true:

The conjunction (X & ~X) cannot be true = ~ (X and Not:X)
X cannot be both true and false.
Nothing can both be and not be
Something cannot both be and not be
~(X & ~X)

The Law of Excluded Middle
Something must either be or not be
Either X is true or ~X is true
Either X is true or X is false
Not neither X is true nor ~X is true
Nothing can neither be nor not be
Something cannot neither be nor not be
X or ~X

Permutations:
X ~X X nor ~X
__________________
T T F
T F F
F T F
F F T


Combinations:
X ~X X nor ~X ~(X nor ~X)
_____________________________
T F F T
F T F T

Both: T & F ---- is excluded by non-contradiction
Neither: T nor F --- is excluded by excluded middle

Neither X is true nor ~X is true means:
(X nor ~X) means: not X and not ( ~X)
Either X is true or ~X is true
outputs "true: {T}" if either X or ~X is true; includes the option in which both are true B{X} = T & B{~X} = T, where B{X} =; bivaluation of X (i.e. assignment of truth values); the principle of bivalence states X is either true or false . A bivalent proposition X can only take one truth value, that truth value being either true or false.

Therefore, for any proposition X, the following options exist:

X is true
X is not true and false (non-contradiction)
X is either true or false (excluded middle)
X is both true and false (excluded - by non-contradiction)
X is neither true nor false (excluded by excluded middle)
X is not both true and false (non-contradiction)
Let B{X}: = Bivaluation of X;

Bivaluation = assigning truth values (T, F)
Then:

The principle of bivalence states: Either B{X} = T or B{X} = F

The law of excluded middle states: B{X} = T or B{~X} = T

B{X} = T & B{~X} = T --- excluded by non-contradiction

Neither B{X} = T nor B{~X} = T --- excluded by excluded middle
B{X} = T or B{~X} = T
Not: neither [B{X} = T nor B{~X} = T]
Neither B{X} = T nor B{~X} = T --- cannot be --- excluded by excluded middle

[Petros Agapetos]

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.

[Petros Agapetos]

Members, can you think of any object, thought, or proposition that violates the logical absolutes?

Can god violate the logical absolutes? If so, can such a god be ruled out of existence?

[Petros Agapetos]

Members, can you think of any object, thought, or proposition that violates the logical absolutes?

Can god violate the logical absolutes? If so, can such a god be ruled out of existence?

X: "This statement (X) is false."

Does this violate any law of logic?

[Petros Agapetos]

Can you think of any examples that violate the logical absolutes?

[Aldaris]

X: "This statement (X) is false."

Does this violate any law of logic?

Let me rephrase it to take the mystery off.

'Let X be a proposition such that X is equivalent to ~X.'

Now by definition, X doesn't exist. It's just a simple contradiction phrased in a confusing way.

[Petros Agapetos]

If some 'thing' violates the logical absolutes, is that sufficient for us to conclude it cannot exist (no logical possibility exists)?

[Aldaris]

If some 'thing' violates the logical absolutes, is that sufficient for us to conclude it cannot exist (no logical possibility exists)?

Yeah.

[Petros Agapetos]

The Logical Absolutes

Are these principles of logic (laws of logic)

1. Inviolable (not able to be violated)?
2. Undoubtable (not able to be doubted, indubitable)?
3. A-Priori or A-Posteriori (Empirical)?
4. Analytic ? = True by virtue of its definition/meaning (?)
5. Synthetic ?= True NOT by virtue of its meaning (?) - ex.

Synthetic Empirical knowledge such as evidence for a scientific hypothesis. = an explanation of a physical phenomenon of nature.
A Scientific Theory is a body of knowledge (explanatory framework) encompassing within it hypotheses and models.
Model (in science) = a representation of an idea ex. (mathematical) description {ex. a physics equation} - of an object *ex. physical or material thing) or a process or a system that is used to describe and explain phenomena that cannot be experienced directly. [1]

Are the logical absolutes (laws of thought) also laws of science?

Is the Law of Non-Contradiction a scientific law?

The Law of Non-Contradiction: contradictions cannot exist = nothing can both be and not be = No proposition can be simultaneously true and false.

The Law of Identity: Something is what it is, and it is not what it is not [LI],
and it is not neither [LEM] or both [LNC]:
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.