Questions tagged [nonclassical-logic]

For questions about three-valued logic and other non-classical logics. Please use the more specific tags 'modal-logic' and 'fuzzy-logic' instead of this tag if they apply.

99 questions
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Is math built on assumptions?

I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals , ... One of my students just rose and asked me: Why do we assume so much in math? Is math…
Anz Joy
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10 answers

Tricks for Constructing Hilbert-Style Proofs

Several times in my studies, I've come across Hilbert-style proof systems for various systems of logic, and when an author says, "Theorem: $\varphi$ is provable in system $\cal H$," or "Theorem: the following axiomatizations of $\cal H$ are…
Alex Kocurek
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22
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Was von Neumann's 1954 ICM address "Unsolved Problems in Mathematics" outdated?

I recently tried to "explain" the generalized probability theory aspect of quantum theory (as one common part of both quantum field theory and quantum mechanics), in the sense of motivations for the different parts of Qiaochu Yuan's post on…
15
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1 answer

Semantics for minimal logic

Minimal logic is a fragment of intuitionistic logic that rejects not only the classical law of excluded middle (as intuitionistic logic does), but also the principle of explosion (ex falso quodlibet). Essentially, from proof-theoretical viewpoint,…
13
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14 answers

A proportionality puzzle: If half of $5$ is $3$, then what's one-third of $10$?

My professor gave us this problem. In a foreign country, half of 5 is 3. Based on that same proportion, what's one-third of 10? I removed my try because it's wrong.
user140581
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10
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1 answer

Defined negation in intuitionistic linear logic

Is it possible to define a negation in intuitionistic linear logic, the way one does in intuitionistic logic, i.e. $A^{\bot} \equiv A \multimap \mathbf{0}$ (or, as it would be written in intuitionistic logic, $\neg A \equiv A \to \bot$)? While I can…
7
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4 answers

Multiple Conditioning on Event Probabilities

I am trying to understand what's wrong with the following logic related to "multiple conditioning." Why is the probability of [(A given B) given C] not the same as the probability of [A given (B and C)] ? I know it's not true, but only because…
Bill
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7
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6 answers

Book about different kinds of logic.

I'm searching for a book that talks about different kinds of logic (esoteric and particular ones too) and their uses and differences. Does such a book exist?
7
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4 answers

Good book for learning and practising axiomatic logic

I want to learn axiomatic (Hilbert style ) logic. not just a book that says that it exist and is an good way to proof theorems. What is a good book to learn and practice this method? would like: - a book published after 2000 - not limited to a…
7
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3 answers

Looking for a simple proof of the independence of the law of excluded middle

I have seen a number of excellent posts on the difference between intuitionist propositional logic (IPL) and classical propositional logic (CPL), all of which state that IPL is agnostic on the law of excluded middle (LEM) or its equivalent forms. I…
Nat Kuhn
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Quicksort with Trivalued Logic

Does anyone know a way to do a quick sort with trivalued logic? The problem I’m trying to solve is this: I’m trying to display a view of a complex 3d object from a given viewing angle. I’ve broken the object into many 2d surfaces that I can draw…
6
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4 answers

What obstacles prevent three-valued logic from being used as a modal logic?

I am familiar with many of the surveys of many valued logic referenced in the SEP article on many valued logic, such as Ackermann, Rescher, Rosser and Turquette, Bolc and Borowic, and Malinowski. It is asserted in the article that "Many-valued logic…
Confutus
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How can some statements be consistent with intuitionistic logic but not classical logic, when intuitionistic logic proves not not LEM?

I've heard that some axioms, such as "all functions are continuous" or "all functions are computable", are compatible with intuitionistic type theories but not their classical equivalents. But if they aren't compatible with LEM, shouldn't that mean…
user61295
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6
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Is $(p \to q) \to (\neg q \to \neg p)$ a theorem in (Johansson's) minimal logic?

From a related question we know that $(P\to Q)\to (\neg Q \to \neg P)$ is a theorem in intuitionistic logic. I'm asking if that's also true for the positive fragment of intuitionistic logic aka minimal logic. Note that minimal logic does allow a…
Fizz
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De Morgan laws of linear logic

I find it stated, in all the resources I have searched, that the following De Morgan laws$$(A\otimes B)^{\perp}\equiv A^{\perp}\wp B^{\perp}\quad\quad\quad (A\text{&}B)^{\perp}\equiv A^\perp \oplus B^\perp$$$$(A\wp B)^{\perp}\equiv A^{\perp}\otimes…
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