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The answer to this turns on your views on propositional granularity. You say a proposition is, or can be defined as, an equivalence class of propositions. This sounds right but is underinformative, since to know when two propositions are identical we need to know what equivalence relation over propositions you have in mind. Some kind of translatability ...


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Preliminary note : " iff " ( material implication) is not the same thing as " is logically equivalent to " ; " The Eiffel Tower is in Paris iff The Eiffel Tower is in the capital of France" is a true sentence , because the two component propositions have the same truth value. Now, it is not correct to say that "The Eiffel ...


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The short answer is that no English word is truth functional, unless it is given an artificially specified meaning. Truth functions are a feature of formal logic, and formal logic only approximates the meanings of words in natural languages. Take, for example, the word 'and'. This is the connective that tends to be the least argued about by logicians. If A ...


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No, I don't think so. Suppose A, B, C are three true propositions. Nevertheless, we might have a situation where A&B is true, whereas A&C is false. Therefore, B and C can't mean the same thing. ___ E d i t / E x a m p l e   R e l a t i n g   t o   C o m m e n t s ___ Are you guys aware of "resource awareness" (e.g., as in linear logic)? The ...


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The answer depends on who the audience for your paper is, as follows. If the intended audience is a roomful of ecology professors and postdocs and grad students who work with physical laws all the time, I will bet that substantively citing/footnoting/referencing Lange in your paper will be a waste of time and may subject you to intense ridicule. This is ...


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by answering your last question of what “makes the trick” the rest of your questions should become lighter. The trick is logical/linguistic. Chess/preparing hotdogs. that’s first. using logic, which of the two are logically easier just based on common being? chess? or preparing hotdogs? which one is easier to pronounce/linguistic? Chess or preparing hot dogs?...


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The material biconditional "P ↔ Q" expresses only that P and Q have the same truth value. It does not express logical equivalence, which is a much stronger relationship. Logical equivalence can be understood syntactically as P and Q are inter-derivable, or semantically as every model of P is a model of Q and vice versa. If your book is using "...


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Here is one more way of looking at this: Consider the argument Premises Cube(c) | Dodec(c) Tet(b) Conclusion ~(b=c) This argument has a general form of Premises: A, B Conclusion: C If this argument is valid then it is impossible for A^B to be true and C to be false. The argument can be represented with the connective "material conditional": A & ...


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I haven't seen an example of exactly what you're looking for, but if you're interested in trying to design such a graph on your own terms, I do have some suggested reading for you. One general theory about the relationship between logic and graphs is Alessio Moretti's NOT theory. "The Geometry of Logical Opposition" has an emphatically ...


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Here's a slightly opinionated but probably useful simplification. The single, most important logic is classical first-order logic (henceforth FOL). It's used all over the place and there are many important first-order theories, including but not limited to ZFC and Peano Arithmetic. This answer is an extremely brief overview of things that are more expressive ...


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Why is any sentence a logical consequence of a set of inconsistent premises? It is not true that any sentence is a logical consequence of a set of inconsistent premises. This is not how humans reason logically and not how human logic works. This is only what mathematical logic says. This is what mathematical logic says essentially because the philosophers ...


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Consider an open formula F(x). In case that the variable x is bound to an element of the domain of discourse, say z, it becomes a closed formula F(z). We may treat a closed formula as a proposition of propositional calculus. For the semantic values of propositions, we assign the logical predicates truth and falsity; in this case, the question is whether any ...


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I would just like to point out, and I'm doing so in an answer and not in a comment because I do not have sufficient reputation, that the standard "least undefinable number" strategy that @Mozibur Ullah mentions does not work for the real numbers. In order for this to work, the structure needs to be not just ordered, but well-ordered. JDH (mentioned ...


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So if I've understood you correctly, based on your original question and your replies in comments, then I think the short answer here is that you are constructing a biconditional; so the inference can go, so to speak, both ways. In your analysis, you say that you have an analysandum, here, the complete proposition: (C) The circulatory system is able to ...


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To contrast "studying logic mathematically" with "studying logic philosophically" is perhaps not looking at the issue quite straight. Logic overlaps with mathematics in a two-way fashion. We can use mathematical methods to make logic more rigorous, and we can use logic to study the foundations of mathematics. These activities are ...


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The first question was answered in the comments. The short answer to the second is that they assume that the reader can figure out how the truth about there being many incompatible maximal consistent ways to extend PA with T-bivalences follows once it has been pointed out that PA proves that for every φ in L_T, there is an equivalent T-bivalence_φ, plus they ...


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So, the conditional theorem below is correct. P ⊢ S Q ⊢ S --------- P∨Q ⊢ S The conditional theorem is only false if there's a case where the premises are true and conclusion is false. And similarly, in order for the conclusion to be false, P∨Q must be true. You are examining all the cases you need to examine by looking at P,Q, P,¬Q, and ¬P,Q. To be ...


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Question 1: No, a proof with premises A, B, and C doesn't prove that A, B, and C are true. The conclusion of a proof is guaranteed to be true if you started with true premises and used only valid methods of reasoning. You need to look elsewhere for justification of the premises and the methods of reasoning. As to whether the intermediate statements are true, ...


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The explanation that you quoted in your question is correct. This is the notion of vacuous truth. Something is a logical consequence of something else if the former is true in every interpretation in which the latter is true. If there are no interpretations at all in which the latter holds, then the logical consequence relation is vacuously true. Let's ...


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By "universal generalization", I assume you mean the rule that if you can derive P(x), then you can infer forall x.P(x). There are two ways to understand this. First, you can view it as a formal property of the logic. If a logic allows you to introduce and reason with free variables, then it needs a rule to say what a proposition with a free ...


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If we go down to the philosophical level, what we're dealing with is the issue of class inclusion, and it's a thorny subject. Think of it as these three tendencies: The tendency to restrict class inclusion on frivolous or tangential grounds, creating an unnecessary bias. 'No True Scotsman' is an informal fallacy (rhetorical ploy) when it restricts the ...


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You have a ∩ (denoting intersection), not a ∧ (denoting AND) here. Thus, in the third statement, it could be that sentences present in both Γ and Δ do not necessarily entail the formula φ. Γ and Δ could have an empty intersection. For example, given two distinct formulas ψ1 and ψ2,let Γ be {(φ AND ψ1 )} and let Δ be {(φ AND ψ2)}. Γ |= φ and Δ |= φ, but Γ ∩...


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Intuitively a logical truth and a tautology are more or less the same thing. However, in modern logic they have become distinguished: A logic is a language with a set of axioms and a deductive system. A logical truth is a sentence that can be deduced via the deductive system starting out from the axioms. A model of a logic is a world where we can interpret ...


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In the Aristotelian tradition, a logical truth is an implication which is self-evidently true, such as for example the modus ponens, transposition, the hypothetical syllogism, or even more basic implications such as A ∧ B ⊢ B and A ⊢ A ∨ B, and indeed A ⊢ A. Thus, logical truths are not identified through formal proofs but through our innate logical ...


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I am following along the course "Language, Proof, and Logic" from Stanford on EdX. I was confused by the question of the distinction between tautology and logical truth (aka logical necessity). One particular explanation from a lecture, and then a lecture from the textbook cleared up a lot for. I'm going to try to give some intuitive ideas here, ...


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As mentioned in the comments, Venn diagrams and Euler diagrams look similar and are sometimes confused. A Venn diagram always shows shows areas of intersections of sets, even if those areas cannot have members. It then marks possible/impossible areas with color. In the Venn diagram, the circle labelled M for Mammals is show to be a subset of the circle A for ...


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Short Answer Attacking the opponent directly is clearly ad hominem. The more specific question is 'is there a sub-type such that snobbery is a key characterization?'. According to WP, this may qualify as an appeal to motive since by characterizing the opponent as elitist or snobbish, one is purporting to speak to the motive of the critic instead of the ...


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True and false have meaning only within a truth table, that is within a context of having defined them - and establishing whether rules apply or not, like the law of the excluded middle. Is the set of odd numbers the 'opposite' of the set of even numbers? Is negatively curved spacetime the opposite of positively curved? They are as artifacts of language only....


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Short Answer Is it possible to get a probable proof using the method of philosophical argument? Yes, there are several methods of argumentation that lead to uncertain conclusions such as induction, abduction, and statistical argumentation that are used by philosophers, particularly mathematically savvy ones. Deduction is the gold standard if you are ...


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Short Answer By logical standards, mathematical induction is a form of deduction. From WP: Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a ...


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As conifold points out in the comments, it makes little sense to consider names as correct or incorrect. However, induction as a mathematical proof technique clearly is a kind of deductive reasoning, not of inductive reasoning. The mathematical concept most closely corresponding to inductive reasoning would be extrapolation.


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And there is 'the unreasonable ineffectiveness of mathematics in the biological sciences' discussed here: Does reality have axioms? It's just symmetry. Continuous symmetries, and symmetries under transformation, are what makes physics work, and what make mathematics apply well to the part of science which focuses on the phenomena with relatively simple ...


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Yes. Contradiction just means that not both can be true at the same time. Opposite means the other extreme of the scale. (Of course, if there is no scale and just two possible states, contradiction and opposite coincide.) Also, the notion of contradiction is always applicable whenever there are several mutually exclusive states, whereas the notion of ...


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Short Answer Opposite in logic is generally taken to mean logical negation. A contradiction in logic is generally taken to be two propositions whose truth values are opposites. Example: P and ~P are opposites, but the set {P, ~P} is a contradiction. Long Answer If you're familiar with formal set theory, you know there are different between naive or ...


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