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Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician Luitzen Egbertus Jan Brouwer (1881–1966): he developed a very personal philosophy of mathematics that founds mathematics (partially following Kant; see Kant's Philosophy of Mathematics) on a pure intuition of time. You can see his Intuitionism and formalism (1912) : ...


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The quote describes what Brouwer calls the first act of intuitionism, the splitting off of discrete from the comprehensive intuition of which discrete and continuous are idealized poles. Here is some quick background. The basis of Brouwer's philosophizing is the non-linguistic "primordial intuition of mathematics", of a continuum without qualitative ...


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The term "material implication" was coined by Russell, who made a distinction between formal and material implication. Here's a quote from the Principia: [W]herever [...] one particular proposition is deduced from another, material implication is involved, though as a rule the material implication may be regarded as a particular instance of some formal ...


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The ethico-mathematical analogy is ancient, but it did gain some recent prominence among analytic philosophers. Clarke-Doane's Moral Epistemology: The Mathematics Analogy, Franklin's On the Parallel between Mathematics and Morals, Lear's Ethics, Mathematics and Relativism all focus on the analogy. And all of them name book VII of Plato's Republic as its ...


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Think of ∀xP(x) as an implicit conditional: ∀x(xϵU → P(x)), where U is the universe. In an empty universe the antecedent is always false, hence the conditional is vacuously true. In contrast, ∃xP(x) is an implicit conjunction ∃x(xϵU ∧ P(x)), so it is vacuously false. This is in line with the standard way of transcribing "all humans are liars" with a ...


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The second sentence of the SEP's article on mathematical intuitionism gives a pretty good explanation of why it is named as it is: Intuitionism is based on the idea that mathematics is a creation of the mind. As Mauro points out in his comment to your question, Brouwer based a lot of his ideas on Kant's views of mathematics. Central to Kant's views is ...


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Minimal logic is intuitionistic logic without ex falso quodlibet. One way to understand the difference in interpretation between minimal (ML), intuitionistic (IL) and classical logic (CL) is by considering the different way they treat the implication operator →. In ML, A → B can be interpreted as "I can show that a proof of A can be manipulated into a proof ...


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The central question This is whether human 'moral judgment is primarily given rise to by intuition' ? It is not about whether we share morality with (other) animals or how such animals come about whatever moral judgements or moral thinking if any they are capable of. How Haidt explicates intuition Haidt refers to the work of Bargh, Damasio, de Waal, and ...


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On a certain level, it's definitely true that the origin of formal logic is an attempt to formalize certain intuitions. But after the process began, it's basically become a set of rules that dictate a certain set of expectations and outcomes. Thinking of sentential deductive logic, the basic idea is that if we begin with (a) bivalence, (b) Aristotle's 3 ...


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No. You're saying that for the biconditional to have the truth table that it does, and for it to be equivalent to (A_B)&(B_A), then _ has to be → with its current truth table. But there are other options: A B A_B (A_B)&(B_A) T T T T T F T F F T F F F F T T A B A_B (A_B)&(B_A) T ...


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Here is an alternative wording which may help. If we know that something is false, then anything that implies its truth cannot itself be true. Schematically : ¬n ( if we know that n is false ) s ⇒ n ( and if we know that something implies its truth - i.e., s ) ¬s ( we know s cannot be true ) It's not a million miles from the given description. It ...


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In the passage from s → n to ¬s ← ¬n the term n does not jump over the chasm between necessity and sufficiency, ¬n does. [This preceding paragraph answers an older version of the OP.] It is the negation that does the trick. Think of a "condition" as a restriction on the class of things that satisfy it, the stronger the restriction the narrower the class. ...


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It has been argued that most of our reasoning is based on intuiting conclusions, and only then using reason to back them up https://www.skepticink.com/tippling/2013/11/14/post-hoc-rationalisation-reasoning-our-intuition-and-changing-our-minds/ From this perspective intuition faces a range of problems, as being potentially irrational biased or partisan, and ...


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Jonathan Haidt in the first part of The Righteous Mind describes the dominant idea of moral psychology in the late twentieth century as a belief in rationalism with moral themes restricted to issues of harm. Moral development in a child was not viewed as social construction (nor anything innate) but the child’s own reasoning about what led to least harm. ...


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Conjectures are often stated without any knowledge that there might be a proof, and with no intuition about a proof. Conjectures are often wrong. There are many, many conjectures that didn't require any intuition at all. Just a bit of statistics, or heuristics. "It's unlikely to be wrong" is often a good reason to state a conjecture.


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"Intuition is like instinct because you cannot do anything about it. It is part of your consciousness, just as instinct is part of your body. You cannot do anything about your instinct and you cannot do anything about your intuition. But just as you can allow your instincts to be fulfilled, you can allow and give total freedom to your intuition to ...


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How to explain philosophical disagreements ? A broad variety of considerations play their part but I am not inclined to put all or even most philosophical disagreements down to differences in intuition as will become clear. Preliminaries 'Intuition' is a term with many shades of meaning. Some at least of these must be noted : ▻ Intuition as knowledge of ...


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Do all serious disagreements in philosophy come down to these [axiomatic] differences in intuition? Intuitions are not axiomatic and logic tells us nothing about the way things are. Differences and disagreements in philosophy usually have nothing to do with logic. Usually, at least in modern philosophy in the west, disagreements stem from presumably ...


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The issues are the paradox of mans experience and his representation of the present moment in relation to the past or the future. Regarding the paradox of experience: One way of seeing this is to recognize that the notion of “present,” as sandwiched between past and future, is simply a useful hoax. After all, if the present is a moment in time without ...


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Not exactly. You are trying to map Kant into modern cognitive psychology, which is a natural thing to do, but can only give us an idea of what Kant might have been getting at from our modern perspective, not how he actually thought about it. In his own mind he was not working with introspective data, nor was he trying to build a dynamical model of mental ...


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Yes. P -> Q means either, P is false, or if true then Q is implied to be so too. Ergo ~P v (P & Q). (via Law of Excluded Middle) Also ~P v (P & Q) means that if P is true, then so must be Q. Ergo P -> Q. Thus the statements are equivalent in Classical Logic.


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One way to view this intuitively is to create a graph of the relationship of P and Q when P → Q. Here is one such graph: The domain is the part within the square. Each of the regions is represented by a conjunction of P and Q or their negations. The question would be whether ¬P ∨ (P ∧ Q) covers the entire domain or not in this case where P implies Q. One ...


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From your referenced answer: However, one way of thinking about it is that "P implies Q" is logically equivalent to "(P and Q) or (not-P)", which is at least somewhat intuitive--either P is true and therefore Q is true as well, or P is false so it doesn't matter if Q is true or false. ...the author here is making note that the only case where P is ...


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B → S is more intuitive; most people more or less have the feeling that if you say this then you also mean that without any intervention of "not" and "or"; this is the origin of implication. B → S only mandates that it cannot be the case that B is true and S is false, i.e. ¬(B.¬S) (See PM ✳1 Primitive Ideas, Definition of Implication). Thus ¬B ∨ S is just ...


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Maybe the text you quote is a bit misleading in that it uses counterfactuals ("if we had f...") which we could be tempted to interpret in terms of possibility and necessity as you do (since you talk about necessary conditions). However the right interpretation to have of the text is not in terms of possibilities but of hypothesis: "if we had f" means "let us ...


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