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the two different 1's in 1=1 can either refer to a single object or two objects which we consider similar No, the two 1's in 1 = 1 both refer to exactly the same thing, namely, the numerical value. There is only one numerical value denoted by the symbol '1'. we demand they have to refer to two different objects in 1+1. We don't count the same object twice. ...


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This question deals with the difference between two distinct concepts: Identity: Two symbols refer to the exact same object Class equivalence: Two symbols refer to two different objects of the same class In other words, if I have a red playground ball, it is 'identical' to itself, but it is 'equivalent' to every other object in the class 'red playground ...


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All equalities by definition require two different things. In 1+1=2, we presume that 1=1. No, they don't. In particular, equivalence relations are defined to be reflexive, allowing you to equate an object with itself. For this reason, the fact that 1=1 follows from the definition of =. Also, the "equality" relation denoted by "=" is ...


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And/or is just a way to express non-exclusive or, because or is very often used as exclusive or in natural language. I would parse "A, B, C and/or D" as "(A AND B AND C) OR D". Alternatively, if it is supposed to mean "A and/or B and/or C and/or D" then it just means "A OR B OR C OR D".


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Preface There's a deep thread in this question that asks about the philosophical basis for understanding the differences between reasoning and computation, or logic and calculation. Broadly, this question was of great philosophical significance historically to thinkers such as David Hilbert and it's fair to say that the philosophies of mathematics, logic, ...


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It is true that ⊥ is equivalent to ⊤∧⊥, so that ⊥↔(⊤&⊥) is true. However, we cannot distribute the biconditional across the conjunction; 𝐴↔(𝐵&𝐶) is not in general equivalent to (𝐴↔𝐵)&(𝐴↔𝐶), and in particular we cannot conclude from the above that the sentence (⊥↔⊤)&(⊥↔⊥) is true. Indeed, this latter sentence is false. So there is no ...


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Your proposition is undefined until you specify what logic it is under. It seems, at a glance, you want to work in propositional logic, but propositional logic does not have equality. Most logical systems don’t. The closest thing in propositional logic is the biconditional “<->”. Now, F <-> (T & F) is True, but this does not mean that F and (...


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Following from your comments, I think your question is not so much about whether it is logically necessary that something exists, but about whether probability theory can be used to demonstrate that it is highly probable. To start with the question as you have asked it, logic itself cannot tell you whether something exists or not. In the standard way that ...


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If one accepts imagination as a sufficient condition for something to exist, some counter-examples are needed to disprove it and if these examples can be formulated in such a way that they violate the laws of the universe, then we can say that imagination may involve physically impossible situation because of a contradiction. A Counter-example may be: (1) I ...


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The Tao opens up with the sentence: The way that can be walked is not the way. This is true if one understands the Tao. But it is not deductively, that is, logically true as the sentence rescinds the deductive meaning given it. A way, normatively thought, is somethimg that can be walked. But then the sentence turns around and says this is not the way. ...


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P V ~Q. P -> (V ^ T). (~V v ~Q) -> T. Therefore, R v T. First, since the conclusion has R in it and you don't have R anywhere in the premises, you must show T by itself and then use disjunction introduction to get R v T. If P is true, then V ^ T is true, therefore T is true. If ~Q is true, then (~V v ~Q) is true, therefore T is true. You know P v ~Q ...


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No, it usually wouldn't be fallacious. A fallacy is a mistake in reasoning. It occurs when the rules used to form a conclusion from a set of premises don't logically necessitate a true conclusion. If I say, "experts say X is true, therefore X is true," then that is a mistake in reasoning because the conclusion doesn't necessarily follow from the ...


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If you stick to ordinary classical logic, then it is not possible, given the way you have defined truth-functional validity. An argument in the propositional (or sentential) fragment of classical logic is valid just in case every valuation under which the premises are all true also gives the value true to the conclusion. So there is no room for a sentential ...


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Computers are the embodiment of deterministic rationality. If computers are programmed properly for rational analysis, determinism is no handicap. However, if the computer was running an app consisting of random lines of code, it would be foolish to trust its logic. Even the discipline of evolutionary programming requires careful construction of evaluation ...


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I have since learned that Dawkins addressed a similar question in another conversation he had with Lennox, this time at the Oxford Museum of Natural History: Has Science Buried God? (14:10). It supports some of what has been posted in other answers. Lennox: "It seems to me that your atheism undermines the very rationality that I assume and you assume ...


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If you assume perfect determinism (and having the necessary knowledge to confirm the question at hand), then your claim say is correct. Opinion does not matter. The simplest example of such a case is mathematics. No matter how many experts agree that 1 + 1 = 3, it doesn't matter until there is irrefutable proof. However, real life is usually not as cut-and-...


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The superlative 'best' is an extension of the term 'better', which compares values on some measurable dimension or dimensions. A community of experts has access to acumen, skills, tools, and methods for making such comparisons analytically; as such, their assessment of 'best' within their particular domain will be far more useful and accurate than the ...


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Determinism is only a theoretical idea, a simplified model of reality. It has no truth value. It is logically impossible to claim or believe that determinism is true, as determinism excludes concepts like claim or belief. Determinism has no effect on reason or logic. But reason and logic categorically deny determinism.


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This is an appeal to authority. Whether or not it's fallacious would depend on the details (and whether it's fallacious is also fairly subjective). In a debate I would probably say it's fallacious if it's a core part of the point you're trying to make in a debate. If you were arguing, for example, in favour of veganism, someone might say "meat is ...


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"If hard determinism is true, and our thoughts are merely the results of a causal chain of atomic interactions, are reason and logic illusory?" Lennox is not speaking about "hard determinism" in the section you've referenced. He speaks of a process that is "unguided" and "random", calling attention to the blindness of ...


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"If hard determinism is true, and our thoughts are merely the results of a causal chain of atomic interactions, are reason and logic illusory?" In the sense Lennox is talking about, no. Reason and logic can arise deterministically if you set up a system that deterministically evolves by pruning out anything irrational and illogical. Lennox relies ...


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This is less a question about logic (which the term "fallacy" would indicate) than a question about the theory of science; perhaps it would be better to ask "how reliable is an opinion of experts?" Typically, of course, the answer will be more reliable than any single opinion because the chance that a majority of any number of experts is ...


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First, a little correction. It is best to think of propositions as being logical truths (or not) and arguments as being valid (or invalid). An argument is not a logical truth. If a proposition is a logical truth of S5, and not a logical truth of K, T or S4, then of those four logics, S5 is the only one in which it is a logical truth. Hence it is the weakest, ...


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Ullah's answer gets to the heart of the question. The term fallacious implies the use of a standard of logical conclusion and in that strict sense, it is false to argue that a conclusion by experts can by a link in a syllogism. I would add that once we are thinking about how reliable experts are, it is interesting to consider how any opinion was arrived at. ...


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Argument from Authority is generally fallacious if used on its own. But reliance on authorities can sometimes be justified in conjunction with other evidence, which aims to demonstrate that there is more objective evidence for the conclusion than against it, that the experts have considered both sides, that the experts are competent, free, and motivated to ...


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John Lennox's idea is that we would have no reason to believe our own logical conclusions if nature, and therefore the brain itself, was deterministic. He says (40:15 - 40:48) that there would be no reason to trust our own logic if was based on the "unguided" and "random" processes of a materialistic world. First, John Lennox, during the ...


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Well, if I asked a community of non-experts how to perform key-hole surgery and I also asked a community of doctors, I am more likely to get a better answer from the second group. But of course, these doctors may not be surgeons as so they might plead ignorance. The point is the advice is more likely to be correct. It's a question of probability and not ...


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Take an inconsistent system S and create another system S’ by adding the axiom “S’ is consistent”. Now you have a very simple proof that S’ is consistent. But you still have some statement X with s proof for both X and not X, so S’ is just as inconsistent as S, plus you have a direct contradiction to the added axiom! Instead add an axiom “S’ is complete”. ...


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Now, Meyer in the 70s, looked at Peanos Axioms (PA) in Relevance Logic, and showed that contra Godel that it was provably consistent. The Australasian Journal of Logic has published recently a special issue devoted to Meyer's work on the Relevant Arithmetic R#. There was published among others, Meyer's paper "The Consistency of Arithmetic" (https:/...


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The first version starts of with Everything has a cause, and ends in "God". According to the bible, God (doesn't matter if this is the same god) does not have a beginning or an end. So therefore "God" can't be a part of everything, as he does not have a beginning (cause). You now have the issue of everything not encompassing everything, ...


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The first argument is logically invalid, because the premise that there is a first uncaused cause contradicts the premise that everything has a cause, while the second argument at least is logically valid, so it is definitely an improvement. The gain is substantial because the second argument helps us pinpoint what we need to prove to make the argument "...


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I think what you're missing is how our analysis of grammar and logic evolved historically. It is based on notions introduced by Aristotle. To quote - following Wikipedia - the Internet Encyclopaedia of Philosophy: Aristotelian logic identifies a categorical proposition as a sentence which affirms or denies a predicate of a subject, optionally with the help ...


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This could potentially be argument from ignorance, as they’re suggesting there is no definitive proof that vaccines don’t cause autism.


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