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Any axiomatic system has serious limitations , see Godel incompleteness theorem and related results. The vagueness of language, on which philosophy is based, is its strength, not its weakness. This is the only way to circumvent the serious limitations imposed by any rigid axiomatic system (which can be modelled by a Turing machine). The notions of ...


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This is not a valid inference. The reason is that one object may satisfy Fx → Gx while another satisfies Fx. There's then no guarantee that anything satisfies Gx. Here's an example. There is a person A who, whenever he is hungry, eats ice cream. We have: ∃x(Fx → Gx). Person B is hungry. We have: ∃xFx. It doesn't follow that someone eats ice cream.


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This tree proof generator is able to find a countermodel: The countermodel considers a domain with two members, {0,1). F is true only when x = 0 and G has no element of the domain that makes it true. Then using x = 1, the conditional, F1 → G1, is true since F1 is false and G1 is also false. Using x=0 for the second premise we have F0 is true. This makes ...


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As Conifold suggests in a comment you may be having trouble because of ambiguity. I will assume what you are trying to prove is (P→(Q→R))→((P→Q)→(P→R)) without premises. What you seem to be doing wrong is assuming r on the first line of your proof. This is what you have to derive if I understand the goal correctly. Here is how this could be done in a ...


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I will use the following symbolization to fit the constraints of the proof checker associated with forallx: D: I'm doubting. (The OP uses d.) T: I'm thinking. (The OP uses t.) F: I exist. (The OP uses e) Consider the following as premises: D, D → T, and T → F we can conclude F. Line 4 is the result of using modus ponens or conditional elimination (→E) on ...


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Here are the questions about God's omnipotence and our free will: Regardless of whether these beliefs are true or not, are these claims contradictory with one another? It seems they are to me, and if this is true, under what circumstances can these claims be made non-contradictory? God's being omnipotent does not appear to violate our having free will. ...


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For logic to be respected we need to specify all the assumptions to the best of our abilities. Also there is no point in assuming that God's omniscience includes knowing something that is not knowable. In addition to assuming that God can know future events, we need to agree on a premise that there actually are future events that can be known. We can't use ...


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Well it depends a lot on what kind of philosophy you're doing. The most important split in this case is that between Analytic Philosophy and Continental Philosophy. Analytics generally are much more concerned with standards of proof, rigorous argumentation, logic, and the like. Think Bertrand Russell. For this type of philosophy, you generally actually do ...


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Mathematical logic has axioms and/or assumptions as the foundation. I am not clear what you mean by SOUND ARGUMENT. If you as a math person require sense verifiable objects and awareness of truth value, you have made a nonsensical statement. Sense verification is needed for Mathematical truth but you want SOUND ARGUMENTS? An issue is existential import ...


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According to the explanation of validity you yourself quoted, the argument with the conclusion "This is an argument" is not formally valid. Thus, it is possible for the premises of this argument to be true and the conclusion false. The argument is therefore invalid. It is not formally valid because the conclusion can be in fact interpreted as referring ...


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Nothing is better than eternal happiness. A ham sandwich is better than nothing. ∴ A ham sandwich is better than eternal happiness. This is an example of the Four Terms fallacy of syllogistic logic, in which a single word — 'nothing' in this case — is used in two different senses to produce an erroneous result. You've made this error above. 'Matter' in the ...


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It wouldn't be philosophy if it uses math. It would be math (or applied math). What is Philosophy? Go to any philosophy website (like this one, or SEP), or read philosophy text books or notes -there is one thing which is strikingly common. The usage of natural language. There are things which aren't developed enough, or not clear enough to allow us to ...


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This answer will focus on two references that may be useful to understand the issues dividing realism and anti-realism. Rather than looking at this from the perspective of logic it may be more useful to see it from the perspective of various metaphysical disputes such as platonism in mathematics versus intuitionism, realism of the physical world versus ...


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The mathematical community has proofs Please note that some of the best practitioners of mathematics disparage proofs. Lefzchetz, for example, told his students not to just present pretty new proofs (they probably already knew that they ought not to present ugly ones!). He wanted new substantiatial new ideas. Poincare was similarly disparaging in his book ...


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Cookies are not matter. Cookies are made up of matter. They are a structured form of matter. The logic you provided does not prohibit the existence of matter that existed before the epoch where these three statements hold true. The initial creation of all matter at the beginning of the universe is a complex thing. It is not clear whether "Matter cannot ...


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What is stopping the philosophical community from holding themselves to the same standard? The impression that the philosophers' "standards" are not sufficiently high, I think, is due to (1) the apparent lack of progress in solving philosophical puzzles in conjunction with (2) the deceiving simplicity of these puzzles. In fact, nothing stops the ...


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Traditionally, mathematics was far less formalized than now, and even when rigor was shockingly lacking (like in the beginning of calculus) and mathematicians didn’t yet possess crisp concepts it still produced insanely much more consensus and progress to virtually undeniable truths than philosophy. I guess it has something to do with the fact that ...


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I find the question a little awkward but important. In metaphysics 'making sense' would mean being logically coherent. So, where the existence of a thing would not be logically coherent we would assume its non-existence. The usual measure is contradiction, such that if the existence of a thing would cause a formal contradiction then we would assume it does ...


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I'm reminded by an inaugural lecture by a Professor of Physics (A. B. Pippard) who posed the question "what is physics", and ended up defining it as "that subset of science which is completely understood", pointing out that when parts of chemistry or astronomy become sufficiently well understood, they get reclassified as physics. Arguably the subset of ...


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The formalist mathematician David Hilbert wrote: (page 185) I myself have always supposed that only statements, and hypotheses insofar as they lead through deductions to statements, could contradict one another. The view that facts and events could themselves be in contradiction seems to me to be a prime example of careless thinking. What Hilbert is ...


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As well as concurring with @Carduus's answer, let's just look at this one statement: The same general principles that apply to thinking about the abstract objective universe should also apply to the concrete objective universe. "concrete objective universe": The sheer existence of this is itself a question not just for philosophy, but physics as well ...


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Often, when a useful higher standard for arguments is discovered, a brand new field of study is created based on that higher standard, or maybe a subfield/subtopic of an existing field of study. For example, the study of physical sciences replaced the study of natural philosophy, and the study of the infinite is a mathematical topic. Since the arguments ...


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The mistake here, I think, is that the question assumes that philosophy is simply another field of research on par with mathematics, physics, or whatnot. But philosophy is actually the superset: the basic mode of reasoning and logic that other fields implement to create their more exacting and specific rule-sets. In this sense, mathematics is the philosophy ...


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The term subalterantion explains why one can derive a particular proposition from a universal proposition. This can also be referred to as the dictum de omni et nullo. There is a wiki page on it that I cant apply here for some reason. Basically what can be denied of a whole must also apply to members of the whole simultaneously. We can't have exceptions ...


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Because it would then cease to be philosophy. Philosophy sees itself as the progenitor of all the sciences, as its questions lead to the paradigm shifts upon which branches of science are founded. To limit itself to a predetermined set of rules would be to strip itself of the flexibility needed to come up with the next new thing. In other words, it is ...


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Another thing I would add is that proofs are built on strong axioms, but also on precise definitions. It's hard to find a precise and universally accepted definition for any complex concept in philosophy. What is life? Soul? What is a cause, an action? What is truth? Those are a much harder to define than a point, a circle or a function (not that they're ...


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Philosophical theories are more like scientific theories than mathematical theories, in that they have empirical content. As such, there aren't any (universally agreed upon) "first principles" that must be respected. Any potential first principles might get discarded if the reasons for doing so are compelling enough. And even if there are some such ...


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A proof is only as strong as the axioms it is built upon. Mathematics works over a very limited number of strong axioms to work with, which gives it a limited number* of things that can be proven, but the proofs are very strong thanks to the axioms they work with (and prior proofs relying on the same axioms). Philosophy works with much broader field of ...


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If I'm understanding your question correctly, then you're basically asking "why doesn't philosophy have the same level of rigor as mathematical proof?" I think there's two parts involved in answering this. First, one aspect of philosophy for many philosophers (arguably all) is that philosophy is actually a form of history, meaning we are studying ideas ...


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Do you have a proof that we don't hold ourselves to higher standards? There's actually a rather interesting little corner of mathematics called "proof theory." It deals with the question of what a proof is and how can we use them. It starts to look like philosophy from time to time. I think the real difference is that mathematics typically starts with a ...


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This is a formal description of causality. The text specifies a "domain of discourse" or Universe (U). U = {The set of causes and effects} This isn't explicitly written but then the universe is basically divided into two sets of objects. X = {the set of effects} Y = {the set of causes} Note: the difference in case is important. x = an arbitrary element ...


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1). ∀x ∃y xCy Literally: for every x, there exists some y such that x is caused by y. This means that every object x has at least one object y that causes it. 2).∃y ∀x xCy Literally: there exists some y such that every x is caused by y. This means that there is some object y that is the cause of every object x. I'm not sure what your text is asking for ...


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The concepts 'cause' and 'effect' are predicated on the concept 'change': an 'effect' is a change in the state of an object brought on by a 'cause'. The concept 'change' implies at least two non-identical states for the object in question. ∴ Unless a single object can exist in two places, or otherwise maintain two non-identical states, change can only occur ...


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Roy Sorensen offers the follow description of vagueness: Vagueness is standardly defined as the possession of borderline cases. For example, ‘tall’ is vague because a man who is 1.8 meters in height is neither clearly tall nor clearly non-tall. No amount of conceptual analysis or empirical investigation can settle whether a 1.8 meter man is tall. He ...


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The question is whether the following argument is valid: If A, then B ~ A So, possible that B Whether this could be a valid argument depends on what one means by "possible that B". If it means the modal sentence ◊B then the argument would not be valid. The counter model would be A is false and B is false. One could not even derive ◊~B since ...


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Both examples draw a particular conclusion (some... not) from two universal premises (all, none). Both syllogisms are valid, but are “weakened” forms because in each case the premises support the universal conclusion. The term “weakened” originated with medieval logicians, who “thought it pointless to get a particular conclusion when one could get the ...


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Celaront was not in the original Aristotle's list of valid syllogistic figures (or : moods). It was added later (during the Middles Ages ?) as one of the two subalternate moods in the first figure (Barbari and Celaront). If we agree (as Aristotle does) that every term has reference, from Cesare : No reptiles have fur. (MeP) All snakes are reptiles....


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Attacking your opponent for hypocrisy is, at its base, attacking your partner, which is the very definition ad hominem and thus a well-known fallacy. However, their statement, instead of hypocrisy, might also be viewable as an admission of of a cultural valuation, which can then be used against them. Example: A Republican says Hillary should go to jail ...


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We can't scientifically determine causality's dependence on time until we can measure the smallest amount of time (if it exists). However, some quantum physicists believe quantum entangled particles interact instantaneously over large distances. Implying causality independent of time. According to some interpretations of quantum mechanics, the effect of ...


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Well it may be true and seems logical or sensible and as a matter... Of course, science can not touch upon everything there is and isn't and this leaves open nearly an infinity of chances (possibly, literally, an infinity of infinities^i) that there is a distinct personality that exists outside of time and within this being's "mind" exists the "our" ...


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One case where it is not a fallacy is in comparisons with the claiming side. Eg "don't vote for Red because he stole a sheep when he was a lad." "that's very nice, Green, but I recall you stole a camel." That is, if the issue is less "A is bad" and more "A is worse than me" then it is relevant. Otherwise, it is typically taken to be a special case of a red ...


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I will interpret the question as suggested by Philip: whether an argument to this effect has been discussed in philosophy. It was. A philosophizing mathematician/computer scientist/physicist David Wolpert formalized just such an argument in Physical limits of inference. Wolpert formalizes measurement, observation of a phenomenon, memory of past information ...


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Here are the definitions that require clarification: Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true. Logical entailment: A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence to ...


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The “whatabout” argument remains nothing more than the tu quoque fallacy, even in complex or difficult comparisons. Each situation must stand or fall on its own merit. That said, when there is a comparison that seems to draw a distinction without a difference, it is legitimate to question the standard that is supposedly being applied.


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Consistency is a relation defined between any two sentences (or statements, propositions, formulas etc.). Two sentences are consistent if they are not contradictory (to each other). Two sentences are contradictory if any of the two implies (entails etc.) the negation of the other. This is extended to any set of more than two sentences as follows: A set ...


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Logical entailment means that every truth assignment which satisfies statement ɸ also satisfies statement ψ. If we say "All English people drive on the left side of the road," then the statement "someone is English" logically entails "drives on the left." Note that other truth conditions might satisfy "drives on the left" (Scots and Welsh do it as well). ...


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They are two different notions that are strictly related : if a set Γ of sentences and a sentence A are not consistent, then Γ logically implies (or entails) ¬A. Consistency is a property of a set of sentences [see The Logic Book, page 92]. It can be defined either sintactically (a set Γ of sentences is inconsistent iff we can derive a contradiction ...


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You are misreading the "sufficient (but not necessary)" comment of page 22. We have to consider also the following parenthetical comment : (As we shall see in Ch.4, we obtain a complete characterization of logical consequence by omitting the restriction to sentential interpretations). Consider now Ch.4 : General Theory of Inference, page 68 : Using ...


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The following description of the two criteria in the OP's post may not be accurate: The chapter starts with asserting that every rule of inference must satisfy two criteria: Note that it is not that every rule of inference must satisfy the two criteria, but the set of rules of inference as a whole must satisfy those two criteria. Some sets of rules may ...


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If you want to avoid the ambiguities of words like 'any' and 'some,' think in terms of categories, instances, and properties. In this case we have two categories: People and Criminals (where Criminals are a subcategory of People that conform to the property criminality). With that in mind, your original phrase becomes: Members of category People who ...


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