And that is the very definition of
special pleading.Your personal definition of what constitutes logic, is not congruent with more established sources.
Wikipedia (yes, I see your eyes rolling) has this to say.
Logic (from the Ancient Greek: λογική, logike)[1] is the use and study of valid reasoning.[2][3] The study of logic features most prominently in the subjects of philosophy, mathematics, and computer science.
Logic was studied in several ancient civilizations, including India,[4] China,[5] Persia and Greece. In the West, logic was established as a formal discipline by Aristotle, who gave it a fundamental place in philosophy. The study of logic was part of the classical trivium, which also included grammar and rhetoric. Logic was further extended by Al-Farabi who categorized it into two separate groups (idea and proof). Later, Avicenna revived the study of logic and developed relationship between temporalis and the implication. In the East, logic was developed by Buddhists and Jains.
Logic is often divided into three parts: inductive reasoning, abductive reasoning, and deductive reasoning.
Since we are discussing an axiom of modal logic, let's get the established definition of what it is.
Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. Modals—words that express modalities—qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is usually happy, in which case the term "usually" is functioning as a modal. The traditional alethic modalities, or modalities of truth, include possibility ("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("Impossibly, p", "It is impossible that p").[1] Other modalities that have been formalized in modal logic include temporal modalities, or modalities of time (notably, "It was the case that p", "It has always been that p", "It will be that p", "It will always be that p"),[2][3] deontic modalities (notably, "It is obligatory that p", and "It is permissible that p"), epistemic modalities, or modalities of knowledge ("It is known that p")[4] and doxastic modalities, or modalities of belief ("It is believed that p").[5]
A formal modal logic represents modalities using modal operators. For example, "It might rain today" and "It is possible that rain will fall today" both contain the notion of possibility. In a modal logic this is represented as an operator, Possibly, attached to the sentence "It will rain today".
So in essence, Modal Logic is a structured and rigorous exploration of possible states in formal logic.
As pointed out eariler, formal logic is the foundation for Modal logic. Formal logic was created as a basis to establish reason, and one of the foundational types of proof that can be given for logic is a physical one. I just provided a physical proof.
Specifially, I used deductive reasoning to arrive at an uncertainty about this axiom. Since there is uncertainty, the axiom is not necessarily true, and thus cannot be called true.
Deductive reasoning, also deductive logic or logical deduction or, informally, "top-down" logic,[1] is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.[2] It differs from inductive reasoning or abductive reasoning.
Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.
Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from initial information. As a result, induction can be used even in an open domain, one where there is epistemic uncertainty. Note, however, that the inductive reasoning mentioned here is not the same as induction used in mathematical proofs – mathematical induction is actually a form of deductive reasoning.
I noted a closed domain-- "Our universe"-- and a feature-- "Time". Time is both possible and impossible within that closed domain, as I pointed out. For the axiom expressed to be true, then this event cannot occur. It DOES occur, therefore the axiom must be false.
You responded with special pleading.
Basically, I am showing that there is uncertainty about the necessity of the axiom, and thus showing that the axiom can be false, which removes it from candidacy as an axiom.
One could twist this around in a knot, and say that because time can be shown to be possible and also impossible in the same universe, then time must be possible and impossible in all universes, but that is a useless statement. It just means that black holes must exist in all universes, or at least, that all universes must have this kind of inconsistency. This is not true, as mathematically cogent models of universes without these features exist-- which creates a contradiction with this axiom.
If we use the other form you quoted:
"If A is possibly necessary, A is necessary" (=> A is true in reality)
We can say that singuarities are possibly necessary, Singularities ARE necessary ==> Singularities exist
But we also have things like Godel spacetime models that fully satisfy the same mathematical fundaments of our spacetime, which are not able to have these kinds of features.
BOOM. Contradiction.
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Topic: Argument against Modal logic axiom S5. (Read 7325 times)