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                                             **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:

  1. Can god be so defined as to violate the laws of thought, also called "logical absolutes"?
  2. 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?
  3. Are the laws of thought logical absolutes?
  4. If so, in what sense are they absolute (logically)?
  5. Can god violate the logical absolutes?
  6. If so, can such a god be ruled out of existence?
  • If God can violate "logical absolutes" what is the point of asking if he is logically possible? He can then be logically impossible, but still be. And what is the point of asking if he can be defined this or that way? Things exist or do not exist regardless of how we happen to define them. – Conifold Jul 8 at 4:13
  • Can God violates the laws of logic ? Maybe... maybe not. See also Logic and Ontology: according to some points of view, God can "create" different logics but, when created, he is subject to them. – Mauro ALLEGRANZA Jul 8 at 6:44
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Taking your questions in a different order.

#3. Are the laws of thought logical absolutes? #4. If so, in what sense are they absolute (logically)?

The idea that logic is concerned with the 'laws of thought' is quite old-fashioned. Since Frege it has become more common to think of logic as being concerned with the relationships of consequence between propositions or sentences. Thought and thinking is the subject of psychology. I would say that logics are human creations; there are many of them; and none can lay claim to being exclusively or absolutely correct. Even the three laws that you quote are open to dispute. Constructive logics reject LEM. Dialethic logics allow that some contradictions are true. The identity of indiscernibles is disputed, as is the possibility of cross-world identity.

#1. Can god be so defined as to violate the laws of thought, also called "logical absolutes"? #2. Is such a god logically possible?

I'm not sure why one would want to define god as something that violates logical principles. If the idea behind the question is that some accounts of god attribute the property of omnipotence and that this is paradoxical, then this is certainly an issue. There are several ways that theists have used to circumvent this problem, e.g. that omnipotence means only that god can do anything that is logically possible.

#5. Can god violate the logical absolutes? #6. If so, can such a god be ruled out of existence?

If there are no logical absolutes the question is moot. According to Descartes, god could create true contradictions, but if god did so we would be unable to understand them. On such a view, the existence of a god cannot be ruled out using logic. Alternatively, a theist might claim that god creates logic and that it is absolute for us but not for god themselves. Or one might hold that our best account of logic does not allow for true contradictions and therefore the existence of a god that does logically impossible things is ruled out. Even then, one might say that it is not so much that logic precludes the existence of something, but rather that the existence of things, and our perception of them, constrains how we may coherently describe them. Another more pessimistic possibility is that natural selection has not equipped us to reason logically about gods.

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  • Historically, most well-known theologians and theistic philosophers who have talked about this issue have taken the view that God cannot change the laws of logic, and perhaps can't change the laws of mathematics either (examples include Aquinas, Maimonides, Avicenna, Averroes and al-Ghazali). Descartes is a rare exception to the rule. Many have however taken the view that mathematical truths are somehow ontologically dependent on God, eternal ideas which are a necessary part of God's nature just like omnipotence and omniscience (a view sometimes called 'divine conceptualism'), but uncreated. – Hypnosifl Jul 8 at 0:53

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