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When confronted with a paradox such as the sorites, there's always the theoretical option of taking the paradoxical argument to be valid and accepting its premises. Your way of applying sorites reasoning seems to fall into exactly this camp of theoretical possibilities. But then you cannot consistently adopt supervaluationism, since this position takes one ...


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The original formula is surprising at first glance, that is true. It is known as meaning : " the true is what is implied by everything". In a parallel way, the false is what implies everything : ~ A --> (A --> B) As bonus, lets quote "consequentia mirabilis" : (~A --> A ) --> A Let me answer your question by another question : Is it nonsence to say : ...


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You won't be able to prove it, since these are not equivalent. The premise says if there exists an x such that F(x) is true then there exists an x (not necessarily the same x) such that G(x) is true. In the desired conclusion it is saying that there exists an x such that if F(x) is true then G(x) is true with that same x.


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By " logically valid " I mean here " deductively valid". This is a broad sense of logical validity which requires only one thing : namely, that it is logically impossible for the conclusion to be false in case the premise(s) is/are true. A classification of logically valid arguments (1) formally valid ( valid in virtue of its form alone, the meaning of ...


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You might enjoy F.H.Bradley's Appearance and Reality, in which he analyses and sublates a list of everyday categories and distinctions. There is also G. S. Brown's Laws of Form, in which he presents a formal 'calculus of indications' where an 'indication' is a distinction or category-of-thought. There is also C.S Peirce, who did a great deal of work on ...


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What the purpose of a speech is? Ideally, no one likes to trick another! Thus, people should ideally try to convey their ideas. This transfer of reasoning is logos: to persuade by logic. Here is my (rhetorical) question: what else is logical? Is "persuading" by touching emotions something really (logically) valid? Of course the fact that an advertisement has ...


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Ethos, pathos, and logos are a typology of modes of persuasion. They are not mutually exclusive, and in the general case one must invoke all three to be truly persuasive. It's a question of balance and proper usage. Pathos builds an emotional connection with the listener, but an excess of pathos alienates others. Pathos without ethos is manipulative; pathos ...


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Arendt distinguished between the mere observation of race and racism itself; the latter which is objectively allied with the structures of power in society. In her book, The Origins of Totalitarianism, she writes: "The historical truth of the matter is that race-thinking, with its roots deep in the eighteenth century, emerged simultaneously in all ...


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Consider this material implication statement : If The Eiffel Tower is in Paris then The Eiffel Tower is in the capital-city of France. If the antecedent and the consequent are given the truth value they have in the actual world, then the whole material conditional is true. But that does not mean that the consequent " The Eiffel Tower is in the capital-...


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First, let's review some ideas of argumentation. With deduction, we can talk about arguments about being sound and valid. Valid means the structure of the argument leads to the correct conclusion independent of the premises, whereas soundness implies the argument is not only valid, but has true premises. For instance, "If Socrates is in the kitchen, he is ...


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These rules are intuitionistically valid, but... Intuitionists accept modus tollens in the form given (2), but not {¬q→¬p, p} ⊢ q, see Can one prove by contraposition in intuitionistic logic? A contradiction out of ¬q only gives us ¬¬q, and removing the double negation is something they reject. This is because p → q is interpreted as "given a proof of p a ...


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Welcome, Tom You deliver a large and complex topic ! The following extract from an article by Matti Eklund (2011) could scarcely be fully satisfying but it does (a) relate sorites and supervaluationism and (b) start a discussion of difficulties: Vagueness, as discussed in the philosophical literature, is the phenomenon that paradigmatically rears its ...


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Think of variables as names, and a variable assignment as something that tells us what a variable names. If you have an open formula, say Px, then Px will come out true when x names an entity which falls within the extension of P. If you consider the formula ∃xPx, x is bound by the existential quantifier. Since x is bound, what x happens to name under a ...


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I'm not sure this qualifies as a question about philosophy - at least not "deep philosophy." Your question focuses largely on common sense and political correctness. The key words are attitude, context and interpretation. Identifying a bassist as "the white guy" can be perfectly fine in some circumstances and a little dodgy in others. Let's ask a pertinent ...


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The issue you are not considering is that the act of 'observation' carries two distinct modes in language: description and ascription. Description is a passive mode that merely notes and relates a characteristic of the observed: e.g., "John has hands and feet". Ascription is an active mode that imports causal relationships into a characteristic of the ...


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Hint: Your basic idea is correct, but you want to use the exists Elim on the contradiction, this will bring it from the second subproof to the first subproof, so try to find a contradiction inside the second subproof, and use exists Elim on it, this will do the work. Answer:


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A distinction between a statement which describes the essential properties of an object and one which describes only contingent properties is the key point. For instance, 'A triangle is a a plane figure with three straight sides and three angles', states all the definitional properties of a triangle - hence in this specific sense it captures everything (...


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This is just a common logical fallacy: fallacy of composition, assuming that the whole is X because a part is X.


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The data you provide are a mixed bag. so it's not quite clear what you are aiming at. The purpose-sentence is pragmatically quite odd, since the definite noun phrase pragmatically conveys that there's only one purpose. Equally, the lack of restricting modifiers of the adjective 'blue' indicates that all proper parts of the flag are blue. There may be ...


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It depends on what the meaning of "is" is. But seriously, the problem here is not so much one of logic but rather of clarity of intended meaning. "The" and "is" in your presented uses are at best ambiguous (at worst purposefully misleading and false). "The purpose" can be interpreted as "The [only/main] purpose" or better expressed as "A purpose". "is ...


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If everything is possible is it possible for something to be impossible? Yes.. Everything is possible because it is possible for something to be impossible.. It is not a contradiction.. it is a loopback..


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Generally speaking, a classification system is an inclusion system. The ideal is that if we have a stream of objects in front of us, we can sort those objects into appropriate buckets based on the characteristics of the objects themselves. There may, of course, be buckets within buckets, but the buckets are meant to be mutually exclusive. With that in mind, ...


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You've hit on an important philosophical issue. If we define every-thing as a set then 'everything' cannot be a set. Hence fundamental theories must transcend sets and the categories of thought in the manner of Kant or the Perennial philosophy. If we do not do this we cannot have a fundamental theory. The mystics call this place the 'world of opposites' ...


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Beware of conflating individual attributes themselves with objects that can have those attributes. For instance, "happy" and "angry" are two different emotions. Their individual definitions might be mutually exclusive, but that doesn't mean that their existence is mutually exclusive. It is possible for someone to experience both emotions at the same time. ...


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My understanding of the use-mention distinction is that the former refers to a disposition (behavior) or proposition (meaning bearer) while the latter is merely a reference (syntactical expression such as a string). In this way, dispositions correspond to correspondent truths (combining two individual cookies in results in a state of affairs that a box has a ...


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1+1=2 is a formula (an expression of mathematical language that express a statement) and "1+1=2" is the way to refer to the expression: correct. 1+1 is a term, i.e. an expression that denotes a number. Thus, it is not a formula. The principle of the Indiscernibility of Identicals (the converse of the Identity of Indiscernibles) in its predicate logic ...


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Reference : Russell, Introduction to Mathematical Philosophy. As to what belongs to "logic" : See Alonzo Church's article " logic , formal - " in Rune's Dictionary Of Philosophy . ( at Archive.org). You will see that the algebra of classes ( set algebra) , the algebra of relations, and even " set theory" is considered as a part of logic. Brief answer : ...


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Frege considered the notion of functions to be logically primitive and so to be undefinable. He tried to give some elucidations of his idea of functions by saying that functions comprise all and only the unsaturated or incomplete things. So Frege's logicism simply takes functions for granted as it does regarding objects (the complement of functions). As a ...


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I know it’s a formal logic question with a mechanical proof, but maybe taking a semantic approach might help you see what’s going on. Firstly, we’re talking about a constant object C. (In practice C might be a parameter for any object, but let’s think of it in specific terms first) We have three premises about C to start with. Firstly, the last one is the ...


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Lines 6 and 7 should not be assumptions. They should be derived from Premise 1 by conjunction elimination; and thus do not raise contexts. The disjunction you should be eliminating is ~Mythical(c) v Mythical(c), derived by LEM. If you cannot use a TautCon to derive ~Mythical(c) v Mythical(c) you can use nested proof by contradiction. |_ : | |_ ~...


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It follows from how the formal semantics of the quantifiers and the connectives used with these quantifiers is defined. An existential statement There is a P which is Q is formalized as ∃x(P(x) ∧ Q(x)) A universal statement All P are Q is formalized as ∀x(P(x) → Q(x)) The semantics of the quantifiers themselves and the logical constants ∧ ...


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What might seem odd to you is that Russell treats the description operator in a syncategorematic way. That is, the operator itself is not associated with an explicitly defined operation, but formulas containing the operator are associated with satisfaction conditions. The problem with syncategorematic treatments is that the syntax of the formula interpreting ...


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@GrahamKemp is correct. The statement says that if P(x) holds for no member of the universe of discourse, then P(a)->Q(a). The fact that we don't have a def for P(x) or Q(x) is is irrelevant because of the truth-table for the material conditional, which tells us that anytime the antecedent --here P(a)-- is false, the material conditional is true. [Remember,...


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The purpose of Russell’s Theory of Descriptions is precisely to give meaning (i.e. truth value) to a statement concerning a non-existent entity. The basic assumtpion is that names of individuals must refer to existing objects (individuals). Thus, what does it mean to assert something about a non-existing objects referring to it with a sort of "name" ? The ...


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¬∃xP(x)→(P(a)→Q(a)) What does it mean? I tried to understand what it means before proof but am totally clueless It says: P(a)→Q(a) is true, if P(x) holds for no x. So why would P(a)→Q(a) be true when that is assumed?


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I have a feeling that we need to use ∀Intro for the second conjunct of conclusion as well as ∀z (Cube(z) → (z = x v z = y)) part of premise. Yes, you need to apply Universal Elimination thrice, once to each arbitrary term. You can then derive three disjunctions through Conditional Eliminations. Up next is a nested disjunction elimination, where you ...


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The symbol F stands for 'False' or 'Contradiction'. Negation is at times defined as "this implies a contradiction". ~P is equivalent to P => F Your system reuses the Conditional Elimination and Introduction symbols (=>E, =>I), but the inferences are more commonly known as Negation Elimination and Introduction. Just as we may infer F from P and ...


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I'm getting a PhD in philosophy, and I'm friends/acquaintances with many professional philosophers (i.e. philosophy professors), and I guess I'll weigh in here. The answer depends on what you think a philosophical question looks like and what would count as an answer to one. Most philosopher think philosophical questions have answers (assuming the questions ...


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I don't see "pattern" or "prediction" in any of the answers so far, and only one instance of "explain". Unlike deduction, which is always at least as correct as the given facts, induction and abduction both rely on probabilities. But those probabilities are used in quite different ways. With inductive reasoning, one takes patterns and by interpolation or ...


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Fundamental reference : Chance , Love and Logic by C.S. Peirce ( at archive.org). See: Part I, chapter 6 " Deduction, induction and hypothesis"). Peirce is the one who coined this term " abduction". See also : article " reasoning" ( by Peirce) in Baldwin's dictionary of philosophy and psychology. Short answer Both are "ampliative"; but while induction ...


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As far as I understand it, an inductive inference is any inference that is non deductive, essentially meaning an argument in which the premisses can be true whilst the conclusion is false as the premisses do not necessarily entail the conclusion, whereas in a deductive inference the premises must always entail the conclusion. An abductive inference is simply ...


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Assume ~ [ (p-->q) v (q -->p)]. Using DeMorgan, infer : ~ (p-->q) & ~ (q-->p) Using &-elim, infer (1) ~ (p-->q) (2) ~ (q-->p) Using ~ ( A& ~B) as a definition of material implication, and Double Negation infer from (1) : p & ~Q from (2) : q& ~p Using &-intro, & commutativity and &-associativity, and finally &...


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My question is if it's fine to make statements like "all swans are things that are probably white". Because it also seems like this is the same thing as saying "some swans are white". All swans are things that are probably white" and "some swans are white" differ in quantity -- "all" vs "some." Moreover, the predicate-terms "things that are probably white" ...


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(1) All the swans observed until today were white. (2) Therefore, all swans are white. This is a strong inductive reasoning ( although the conclusion is false). When I reason like this, I'm not looking for the "probable color" of swans, but about the color of swans. I mean I am not looking for a probability. It seems to me that the probability is ...


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You have asked about using the term 'probably' in this post. Yes, you can use any word at all in a proposition if it's meaningful. There are different types of logic such as syllogistic, sentential, FOPC, and modal for starters, and some are more sophisticated than others. Ultimately, when analyzing language, one chooses a logic depending on what is ...


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Hint: Universal elimination can be to any term in the context, which includes the assumed witness for the existential. |_ Ex Ay (Cube(y)<->y=x) Premise | |_ [a] Ay (Cube(y)<->y=a) Assumption | | Cube(a)<->a=a Universal Elimination | | : | | : | | Ex (Cube(x) & Ay (Cube(y)->y=x)) ...


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I tried ∀x∀y ((x ≠ y ∧ Larger(x,y)) → Dodec(x)) and ∀x∀y (Larger(x,y) → Dodec(x)), Translate it a bit at a time. "Only dodecahdera are larger than everything else." ∀x ("larger than everything else"(x) → Dodec(x)) ∀x (∀y (x ≠ y → Larger(x,y)) → Dodec(x)) If you need it in prenex form, use contraposition and duality, so you may apply null quantification. ∀...


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Sentence (2) is a version of Curry's paradox, while (1) is simple instance of the liar paradox. Both of course are close cousins: They involve self-reference as well as semantical predicates like 'is true'. However, there are some differences: Curry paradoxes involve principles concerning conditional reasoning, while liars don't. This is quite obvious when ...


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Your first proposition is that statement 1 = statement 2. However, that is false. This becomes clearer if we negate both propositions. 1) This statement is true 2) This statement is untrue if true and not true 1) is always true, while 2) just says it is untrue if it is contradictory. 1) is always true, while 2) doesn't always have to be true. Negate both ...


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Many predicate language sentences come in one of two forms: "All A's are B's" and "Some A's are B's". There's also "No A's are B's", but that's just a negation of "Some A's are B's". They are translated as: All A's are B's: ∀x(Ax -> Bx) and Some A's are B's: ∃x(Ax & Bx) All of your sentences in one way or another have one of those two basic forms....


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