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.

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Czelakowski matrix semantics vs ordinary matrix semantics vs Co-Czelakowski matrix semantics

The kind of matrix semantics that I'm accustomed to seeing is something like $\langle A, F \rangle$ where $A$ is an algebra in some signature and $F$ is a subset of the carrier of $A$. A Czelakowski ...
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Proof of the Local Deduction Theorem, for one of many logics

(I also asked this in MathOverflow) I'm looking for a proof of the Local Deduction Theorem for any one of many logics. That is, a proof of the statement: $\Sigma \bigcup \{\phi\} \models \psi$ iff for ...
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An exercise in fuzzy logics built from a t-norm

(I also asked this in MathOverflow) Consider the following t-norm: $ a * b = \begin{cases} \text{$2ab,$} &\quad\text{if $a, b$}\le1/2\\ \text{$min\{a, b\}$} &\quad\text{...
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What Tarski-Lindenbaum algebra is associated with Angell's four-valued logic?

This talk by Hitoshi Omori on connexive logic, at around 11:50 shows the truth table for a four-valued logic created by Richard Angell to support a proof system called PA1. (I think. I'm guessing the ...
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Which logic do you get by combining the topological semantics for IPC with a binary accessibility relation?

Which logic do you get by combining the topological semantics for IPC with a binary accessibility relation? I'm trying to come up with a semantics for a relevance logic that's simpler than a Routley-...
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Do the inference rules of the implication-free fragment of classical logic without disjunction elimination form a complete Hilbert calculus for LP?

If I take the inference rules for the implication-free fragment of classical logic (listed below) and drop the pair of rules for disjunction elimination, do I get a complete Hilbert calculus for LP (...
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7 votes
3 answers
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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 ...
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Question on Logic

Consider the following (complete and distributive) three-element lattice $\mathbf{A}_3= \langle \{1,a,0\}, \wedge, \vee, \Rightarrow, ^*,1, 0 \rangle$ where $1^*=0, a^*=1, 0=1^*$ and $x \Rightarrow y =...
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Can a theory with plural quantification have elementary equivalent structures that use different length sequences to satisfy sentences?

Suppose we have a language $\mathcal{L}$ that is first order but allows for plural quantification, and let $T$ be an $\mathcal{L}$-theory that includes a sentence of the form $$ \exists\bar{x}\;\phi(\...
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Is a paraconsistent model a type of boolean-valued model?

A boolean-valued model is a generalization of an ordinary model where the set of truth values is any complete boolean lattice, instead of just the smallest one $\{ \bot, \top \}$. I will nonstandardly ...
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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 ...
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Classical Tautologies in a Deviant Logic of Paradox

I am stuck with the following problem. The Logic of Paradox (LP) has the truth values {f,p,t}, designated values {p,t} and the following truth tables: ...
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Logic without the three traditional laws of thought

The wikipedia page on the laws of thought mentions the The three traditional laws: The Law of the Excluded Middle, Law of Non-contradiction and The Law of the Identity. These laws are embedded in a ...
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Logic without the Law of Identity

The wikipedia page on the laws of thought mentions the The three traditional laws: The Law of the Excluded Middle, Law of Non-contradiction and The Law of Identity. These laws are accepted and used by ...
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5 votes
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What are the various respects under which a logic can deviate from classical logic, thus being " non-classical"?

Is it possible to get a synoptic view of the ways in which a logic can, so to say, deviate from classical logic? I think one can find rather easily a list ( though maybe incomplete) of non-classical ...
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Type theory and constructivist mathematics with paraconsistent logic?

Type theory, together with the Curry-Howard correspondence is a formal system for stating formal proofs of intuitionistic logic, which is used in constructive mathematics. Intuitionistic logic differs ...
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What is a semi-valuation in paraconsistent/paracomplete logic?

I have been reading Carnielli and Rodrigues' "An epistemic approach to paraconsistency: a logic of evidence and truth" when they discuss the notion of semi-valuation (BLE stands for Basic ...
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On Axiom of Choice and the principle of Explosion (ex contradictione sequitur quodlibet)

My question is on the relation of this two principles. For instance, in the of Intuitionistic set theory we can prove Zorns Lemma (but Zorns Lemma will not be equivalent to the axiom of choice, since ...
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2 votes
1 answer
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Semantic explanation for converting intuitionistic logic into classical logic by adding LEM as an axiom

I have a question about converting intuitionistic logic (IL) into classical logic (CL) by adding LEM as an axiom. IL is usually understood as a logic without LEM. $$\textrm{LEM}:=A\vee\neg A.$$ In ...
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Attempt to make a meaningful ternary logic

Remember when I asked a question about ternary logic? It was my first question here. Let $F$, $U$, $T$ be the truth values, where $F$ is designated for false and $T$ is designated for true. Though ...
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Reading Chapter 14 of Goldblatt's, "Topoi: A Categorial Analysis of Logic."

I'm aware that things could get too broad if I'm not specific & careful enough, so please bear with me! Having just "read" $\S 14.7$ of the titular book, I'm exhausted. It has $71$ ...
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Why is $\lozenge (p\to p)$ not valid in system $K$ of modal logic?

Why is $\lozenge (p\to p)$ not valid in system $K$ of modal logic? How could this formula be false in any accessibility relation or setting of values in worlds? Even if it was a dead end world, ...
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1 answer
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Sudokus and the Distance to a Contradiction.

Consider a sudoku puzzle for which there is a unique solution. In solving the puzzle, one enters in pencil what, a priori, each of the $81$ small squares could be, given the (at least $17$) clues that ...
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Do non-monotonicity / higher-order / probabilistics / modalities / connectives exhaust all possible features of logical reasoning?

I am searching for all the possible features of reasoning (all of them can be expressed in logic), so far I have found the following features: non-monotonicity, defeasible reasoning (expressed by ...
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Non-distributive lattices with chains of different lengths

Consider the following Hasse diagramme. We will dub such lattices $\mathbf{Mkn}$. Assume a propositional language over $\{\wedge,\vee,\neg\}$. Let $v$ be a mapping from the set of all propositional ...
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3 votes
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Is it possible that A ∪ B ⊢ α ∧ ¬α, but there be no β such that A ⊢ β and B ⊢ ¬β? [duplicate]

In other words, is it possible in classical logic that two sets of sentences together ($\mathbf{A} \cup \mathbf{B}$) imply a contradictory proposition ($\alpha \wedge \neg \alpha$), but they fail to ...
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What's the difference between $\vdash$ and $\vDash$? [duplicate]

This appears to be a new question to MSE according to Approach0 (surprisingly). I'm looking for a detailed description of the difference between $\vdash$ and $\vDash$ in the study of logics, even ...
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Priest's nonstandard $N$: showing $\not\vdash_N \square p\supset p$.

I'm reading up on nonclassical-logic. In Priest's nonstandard $N$ of his "Introduction to Nonclassical Logic [. . .], Second Edition", it is an exercise to show $$\not\vdash_N \square p\...
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5 votes
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What references should I follow if I want to learn more about semantic entailment in multi-valued logics?

I am simply curious about this subject, and would like to learn more about it. I am an undergraduate student and in our studies we've always tackled classical logic and simply mentioned that other "...
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What are the automorphisms on the strucuture consisting of the nonzero vectors of a Hilbert space with the orthogonality relation?

Let $V$ be an infinite-dimensional complex Hilbert space. With this space we can associate a relational structure $V^+ = (V^+, \bot)$, where $V^+$ is the set of non-zero vectors in $V$, and $\bot$ ...
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" Logic does not allow you to say this": is this assertion outdated?

I think one cannot say nowadays without further qualification " geometry does not allow you to say that the sum of a triangle's angles is less than 180 degrees". The sentence concerning the sum of ...
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2 votes
2 answers
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Relative strength and propositional indistinguishability of non-distributive lattices

Consider the class of bounded non-distributive lattices $\mathbf{Mn}$ ($n\geqslant 3$). From left to right: M3, M4, Mn Now consider a propositional language over $\{\wedge,\vee,\neg\}$ with the ...
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2 votes
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Expansion postulates in Belief revision logic

I've just started looking into epistemic logic and belief revision approaches, and I'm struggling already with proving some basic properties. I am given the following definition: Let $\mathcal{L}_0$...
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Paradox vs Tautology.

The expression(~p or p )is a Tautology. Consider this statement(p): This statement is false. Now here, Statement p is paradoxical. My question is :- Can we define paradoxes like this as statements ...
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2 votes
1 answer
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Motivations for using Strong-K3 in Sentential Logic.

What motivations might one have for choosing a Strong-K3 interpretation in sentential logic (over other three-valued logics). I'm looking at the functional outputs of each of the connectives on L3, ...
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3 votes
1 answer
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Lukasiewicz logic, deduction theorem and complete calculi

3-valued Lukasiewicz logics is known to lack a “semantical" deduction theorem, i.e. $$A\vDash B\not\Leftrightarrow\vDash A\supset B$$ if we define $A\vDash B$ as $\forall v(v(A)=1\Rightarrow v(B)=1)$ ...
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1 vote
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Downward Lowenheim-Skolem-Tarski theorem for Cofinality logic

I was trying to find the precise and strongest statement of the Downward Lowenheim-Skolem-Tarski theorem for Cofinality logic $\mathcal{L}(Q^{cf}_\lambda)$- first order logic enhanced with the ...
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Is minimal logic equivalent to intuitionistic?

Is minimal logic equivalent to intuitionistic? Obviously this is false if we interpret the symbol $\bot_M$ from minimal logic as meaning the same thing as the symbol $\bot_I$ of intuitionistic logic. ...
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15 votes
1 answer
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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). ...
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2 votes
1 answer
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Why there is no classification theorem for logics, if there are classification theorems for groups and algebras?

Why there is no classification theorem for logics, if there are classification theorems for groups and algebras? Why the logics are different, if there are connections between logics and category ...
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How can we interpret that $A, B \vdash A, B$ is unprovable with resource interpretation in Linear Logic?

In Linear logic (LL), it is unprovable but when considering the resource interpretation it seems to me that from the resources $A, B$ we can produce the resources $A, B$. By $A, B \vdash A, B$ I mean ...
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Fully truth-functional version of modal logic?

Though the 'square' and 'rhombus' operators from modal logic are not truth-functional, it's because modal logic is interpreted as a binary logic, isn't it? What if modal logic is considered many-...
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Are there logics without modus ponens?

The question doesn't go beyond the title. And I don't mean logics that merely just don't have it as a primitive rule - I'm interested in logic where you can't actually use it. I've searched around ...
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2 votes
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Continuous vs classical structures (2)

I am moving my first steps in continuous model theory (tl;dr). This is one of two soft questions on the relation between a continuous and a classical structure. Let $M$ be a classical first-order ...
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Continuous vs classical structures (1)

I am moving my first steps in continuous model theory (tl;dr). This is one of two soft questions on the relation between a continuous and a classical structure. What can be said about continuous ...
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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 ...
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1 vote
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Free Logic; what is the relation between 'Fa' and ' □Fa'?

I am reading Rod Girle's Modal Logics Philosophy section 8.6 on Free Logic. And I have a problem understanding how truth values of modal logic formulas are determined in free logic. In Rod Girle's ...
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3 votes
1 answer
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Free Logic; how to make (∃x)(a ≠ x) (n) true?

I am reading Rod Girle's Modal Logics and Philosophy. And I have a problem with one of the answers of the exercises. In the exercise question, the reader has to provide a counterexample showing the ...
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3 votes
1 answer
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Logics in which the substitution of logical equivalents fails

In classical propositional logic and in first order logic, if two formulas are logically equivalent then they are substitutable. That is, if we can prove $A \leftrightarrow B$, then we can substitute $...
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Does the identity $(p \lor (q \land \neg p) ) \iff (p \lor q)$ hold in intuitionistic logic?

This is an identity that comes up very often while I'm working with Boolean logic, and recently, it got me thinking. The method I use usually to prove it (at least) seems to rely on the Law of the ...
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