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There exists either that which is TRUE or that which is not TRUE. It cannot be TRUE that only non-TRUE things exist. Therefore TRUTH exists. I say the conclusion is valid. Here is how I would reason it through. Either T or not-T. “All things are not-T” is false. Thus, by implication, its contradictory statement is true: “Some things are not not-T”. ...


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I can't quite figure out what underlying thinking led to the formulation of your question; what's your background, exactly? And ignoring the psychological aspects, it sounds like a more suitable cs.stackexchange question. So I'd primarily recommend posting it there after editing it to remove the non-cs extraneous stuff. Anyway, with respect to the cs stuff, ...


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FORMAL FALLACIES Formal and informal fallacies are both persuasive forms of incorrect reasoning. Formal fallies are easy to identify because the structure of the formal language is what linguists called surface structure. For instance, a simple formal fallacy goes as such: P1 A implies B P2 B C Therefore A. If we want to see a natural language ...


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This question might be referring to the converse of a statement. The converse does not necessarily have the same truth value as the original. Start with “If P then Q”. The converse reverses the two terms: “If Q then P”. Here, “If I live, then my purpose will be to eat.” The second statement is the converse: “If I eat, then my purpose will be to live.” I ...


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Example: Tolerance limits in engineering design. Tolerance limits might be a good example. For example, engineers are often asked to design shelves, that hold up to a certain weight; electrical wires, that carry up to a certain electrical current; web servers, that serve up to so many web clients; towers, that withstand up to so much wind; and other ...


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It's not a fallacy, it's an adage. "I don't live to eat, I eat to live" The intent is: "I'm consuming because I need to, not for frivolous (unneeded / personal) reasons" Normally, this adage is applied to hunting or meat consumption in the United States (as far as I'm aware.) But, I've heard it applied to other aspects of buying things - like a car. ...


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In first order logic, we don't assume existence of any model, we are quantifying over all possibly existing models. If we show something follows from our axioms, it means that all models that satisfy our axioms also satisfy the conclusion. If, for some reason, there is no such model (because the axioms were inconsistent), then our proof is still valid, just ...


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If we interpret 'following logic' in the mathematical sense, i.e reasoning correctly step-by-step beginning with axioms and arriving at conclusions, then it is still possible to be biased: The bias may be built into the axioms. This is particularly pernicious as of course it is not possible to fix the axioms by purely logical reasoning (where would one ...


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I'd be tempted to say that logic is not a model that we use to reason. It's a model, a representation, of how we reason (or should reason). People don't need to learn the rules of classical logic to make inferences, but logicians are interested in how people make inferences, this is where the models of logic come from. This might apply to some extent to ...


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How do I prove, :((A ⊃ B) ⊃ C) ⊃ (B ⊃ C), using symbolic logic derivations where ⊃ represents a conditional i.e. A ⊃ B = A implies B? The first line of my derivations is the assumption, (A ⊃ B) ⊃ C. The second line is a second sub derivation, B. However I don't know how to get C out of my second sub derivation so I can return to the original assumption ...


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I think the title and body questions are subtly different. Here I'm going to address the title question, which I'll paraphrase for clarity as: What sort of "mathematical truth" can a non-Platonist make sense of? I think this is less strange than it may first appear, since there is an existing parallel: "sharp" vs. "fuzzy" referents in natural language. ...


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The decision is about formal systems. So the problem arises only within Formalism. It is possible to deny Platonism and still not anchor mathematics in axiomatic systems. The program that led to Goedel's completeness competes with more radical reactions to the gap in Frege's work. The first among these is the original form of Intuitionism proposed by ...


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first of all, @Mauro ALLEGRANZA, ⊨A is a tautology. as has been explained by @N. Bar, ⊨A means A is a consequence of the empty set, so ⊨A would be a tautology. second, here is my proof. the first statement is equivalent to, ⊨A→B ⊨¬A∨B concerning the second statement, I have to point out that the symbol "⊨" is not a logical connective, it means "...


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I'd like to add an addition to A.K.'s otherwise very good answer to address the topic of progressing to the "state of the art." A.K.'s reading list will get you the basics. That will be very doable on your own. By contrast, it's very difficult to progress to the state of the art without a mentor or PhD advisor, simply because you won't have any orientation ...


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if you'd like to teach yourself through reading, I'd sincerely recommend the following books. the prerequisite is familiarity with propositional logic and first-order logic. and I suppose you've met this requirement as a math student. First-Order Modal Logic By M. Fitting & Richard L. Mendelsohn. I learnt modal logic by reading this book. its exposition ...


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A formal logical system does not ever tell you what it means to prove something. It tells you what works in the model. The point is that you need to have faith that the model models your real thinking, and not something else. Otherwise, you can't use it to discuss the nature of thinking. Without proof of a more casual sort that the proofs you are going ...


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One way would be to follow along a syllabus for a relevant course at the MA or PhD level. I know for a fact that this is a good course, but I'm sure there are many others.


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The underlying picture science currently uses, conceived of by Newton as 'fluxion' and institutionalized in the integral and differential calculus, really is not one of these three options. It discards the Aristotelian distinction between potentialities and actualities, and does not move backward into one of the two earlier positions. Instead, it creates ...


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I do not see a fallacy in the reasoning, although the absence of a fallacy does not mean that the reasoning is correct. The problem looks to me like an inductive conclusion (a prediction of the future) has been drawn from a database (one failed event) much too small to ever justify it. A larger bank of observations, even including the failed event, might ...


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Before Frege, axiomatic systems were not a focus of philosophy, and Goedel is pursuing the immediate upshot of Frege's failure. So, in some sense, no. Nobody cared. Mathematics was grounded in some internal, perfect mental reality and not really based on axioms. Axioms just helped keep things clear. Paradoxes abound throughout the history of philosophy. ...


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Godel himself said that all he did was formalise the Cretan liar paradox into a formal system. So the idea or notion of undecidable statements was already apparent a long time before Godel but obviously not phrased in such terms. A simple parallel is with arithmetic. Its easy enough to notice that putting things into a group has certain properties which is ...


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While it’s not quite a perfect parallel, a related concern about decision problems had been phrased some years before Gödel’s work: see https://en.m.wikipedia.org/wiki/Entscheidungsproblem . David Hilbert was pretty convinced as to the decidability of arithmetic prior to the Incompleteness/Incomputability proofs, but was aware that the problem remained open.


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A formal fallacy is one which is made independent of the meaning of the claims and is a function of the logic applied to the terms itself. These fallacies are syntactical. For instance, affirming the consequent: A -> B B Therefore A. The fallacy of which you give examples would be considered informal because the lack of warrant derives from dubious ...


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Technically speaking, the Quinean criterion of ontological commitment to prevent philosophical "double-talk" is straight forward. He stated in On What There Is "we are convicted of a particular ontological presupposition if, and only if, the alleged presuppositum has to be reckoned among the entities over which our variables range in order to render one of ...


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Deductive proofs in first-order logic are essentially transformations of one statement of the language into another. You start with some statement (or several or none at all) and then produce from it an ordered sequence of new statements derived by successive applications of the established rules of the system. You can end this sequence at any time and any ...


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Other contributors gave really interesting answers, yet no one mentioned the link to the classical confusion of necessity and sufficiency - the X is necessary for being an element of A, i.e. A ⊆ T, where T = {t ∈ U | X is true for t}. Yet no one guarantees that A = T (property is not a criterion), thus, ¬X is not necessary for being an element of A' (unless ...


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I think it's a fallacy concerning negations. By using first-order logic, let's suppose "B" means "being bad", "T" means "being a teenager". Then the first proposition can be formalized as follows, ∀x(Tx⊃Bx) ≡∀x(¬Tx∨Bx) ≡∀x¬(Tx∧¬Bx) ≡¬∃x(Tx∧¬Bx) While the other proposition is formalized as follows, ∀x(¬Tx⊃¬Bx) ≡∀x(Tx∨¬Bx) ≡∀x¬(¬Tx∧Bx) ≡¬∃x(¬Tx∧Bx) The terms ...


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@willross1 currently writing a thesis on logical fallacies, new to this forum. I am confused about your account of modus tollens. I believe it is a simple clerical error in which you have accidentally forgotten a parentheses? There are too many close parentheses per open parentheses. Is this simply a keyboard error or am I too prone to misunderstand MT? {[(p➡...


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The assumption "A ⊨ B" does not mean that A is a tautology. You have assumed f(A)=1, that means that A is true for an interpretation f. The same for the second case : "When ⊨A → B, B must be true since ⊨A is a tautology" is wrong. A → B is a tautology: this means that, whenever A is true also B is, and this is enough to conclude with A⊨B.


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John of St. Thomas’ following remark, introducing his comparative treatment of Suárez and St. Thomas, captures a distinctly Catholic principle: “Non Contrariatur Principiis Luminis Naturalis Hoc Mysterium.” We may also say of the doctrine of the Most Blessed Trinity—inspired by the same principle: Supra luminis naturalis rationis sed non adversus eam. ...


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The quote you link to (here's a link for those in the US since google books links are sometimes country-specific) doesn't seem to be basing this on any other writings of Hawkings besides the one they quote on p. 121, rather they seem to be inferring that Hawking thought the laws of physics were needed to protect the laws of logic based on his argument about ...


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I recommend the paper: B. Buldt, The Scope of Godel’s First Incompleteness Theorem, Log. Univers. 8 (2014), 499–552 , especially pages 530 - 531.


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Perhaps we can here begin to show, without specifically answering the question posted on this page, that a regular use of two-valued Aristotelian logic, useful though it is in its own right and realm of specific operation, fails to apply to the powerful doctrine of the Most Blessed Trinity. The intrinsic relationality of the One divine essence is self-...


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The apostle Paul says in 1 Corinthians 15:19: "If in this life only we have hope in Christ, we are of all men the most pitiable". In other words if Pascal were to lose his wager, Paul would regard him as a very great loser. His one life would have been wasted in pointless service and sacrifice of a God that doesn't exist. In a (hypothetical) universe where ...


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To add to @Jon's answer, denying the antecedent often comes up due to confusion with the valid argument form Modus Tollens ((p → q) ∧ ¬q) → ¬p) or ((if p then q) and not q) then not p) Which is equivalent to (if p then q. Therefore, if not q then not p), i.e. If a person is a teenager, then they are bad Therefore If a person is not ...


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After some thought I realized this is a denying the antecedent fallacy. Put another way we have If the person is a teenager then they are bad therefore If the person is a non teenager then they are not bad


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Allow me to start with something that will sound like a quibble: there is a difference between 'evidence' and 'data.' The world is filled with data: sense data, analytical facts and figures, mathematical axioms, etc. But data by itself is meaningless, substance-less. For data to have meaning and substance, it must be interpreted so it fits within a ...


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Long comment : at first sight, I agree with you. We say that B is a necessary condition for A to mean: "if A, then B". And we say also that A is a sufficient condition for B to mean: "if A, then B". Thus, the Example: Sufficient Condition of A+ MUST MEAN Necessary Condition of Studying occured, amounts to : "if you received an A+, then you must ...


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Statement "X" must be true because it is clearly laid out as such in mutually reliable source "Y". That sounds like an appeal to authority, spiced with some "weasel words" ("mutually reliable source"). Joe, the statement "X" seems outlandish and therefore it ought to be rejected in its own right. It's hard to know if that's a logical statement ...


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Let's see. You say the exchange is logical, and that there lies substantive evidence of why Joe is right—which essentially is the conclusion. The term I can think of is Syllogism. According to MW, the term is defined as: 1 : a deductive scheme of a formal argument consisting of a major and a minor premise and a conclusion (as in "every virtue is ...


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There is a philosophical branch which is all about limits of understanding. It is called Hermeneutics and Gadamer's Truth and Method may be a start. Basically, you are talking about what he calls understanding The Other, ie. real understanding involves being able to conceptually grasp what something is able to "tell" you about its history of effect (...


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Francisco Suárez Suárez discusses the applicability to the Persons of the Trinity of this form of the principle of non-contradiction, A = C B = C ∴ A = B in On the Various Kinds of Distinctions p. 59 (Disputationes Metaphysicæ, Disputatio VII, De Variis Distinctionum Generibus), §2 The Signs or Norms for Discerning Various Grades of Distinction in ...


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Yes, there is a logical fallacy to identity politics. The fallacy has two parts. Identity politics begins with a great big ad hominem argument. The site LogicallyFallacious describes the ad hominem technique as Attacking the person making the argument, rather than the argument itself, when the attack on the person is completely irrelevant to the ...


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Suarez has commented on Aquinas's Summa Prima Pars, and he also wrote a theological summa called De Deo Uno et Trino (cf. Suarez in Latin online). The image below shows that (1) Pater est Deus (2) Filius est Deus (3) Pater non est Filius However, from (1) and (2) ( with the symmetry and the transitivity property of equality) one can infer that (4) ...


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IEP, " Logical Consequence" ( online) Stanford Encyclopedia, " First Order Logic" ( online) Hardegree, Introduction to Logic ( https://courses.umass.edu/phil110-gmh/MAIN/IHome-5.htm) Papineau, Philosophical Devices. ( For a first introduction to metalogical concepts). Nolt, Logics. Stoll, Set Theory And Logic. ( at archive.org) Enderton, ...


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First of all, note that nothing prevents us from using multiple different kinds of implication at once; indeed, nobody (I think!) would quibble with the claim that most of the time when we use conditionals in natural language we are not invoking the material conditional. Even in mathematical contexts we can consider non-material implications. That said, ...


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See Principle C': Let A be any nominalistically statable assertion, and N any body of such assertions.Then A∗ isn’t a consequence of N∗ + S unless it is a consequence of N∗ alone. If S is a mathematical theory, it "speaks about" mathematical objects while A∗ and N∗ do not. Thus, IMO; we have to understand it as saying: in order to derive statements about ...


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This is at best only a quasi-answer, but I think it is in the ballpark of what you are contemplating... You stated, "(p->q)->(~pVq) is not as intuitively necessarily true as maybe (pVq)->(~p->q)". I AGREE, especially when you consider the rendering of wedge as "unless", right? I think it would totally work in a system that eliminates double negation (...


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Short answer: (1) amongst our 16 possible binary operators, we have an operator at hand that never gives as output "Truth" when the input is ( Truth, Falsity) (2) we use it because this operator is the best tool we have to model the notion of logical implication (3) this is the reason we also call it implication with the lower status of " material ...


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One attractive feature of the material conditional is that, in conjunction with the universal quantifier, it offers a natural way to translate classical logical statements of the form "all A are B" into propositions in first-order logic, like "for all x, A(x) -> B(x)". Consider for example the classic example "all men are mortal", translated as "for all x, ...


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