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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1answer
32 views

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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48 views

qualify on metalanguage will cause what?

Add quantificational expressive power to metalanguage will cause what? In Teller's A Modern Formal Logic Primer, for example, (∀I) { [ Mod ( I , X ) & Mod (I , Y) ] -> Mod ( I, W ) }, which ...
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58 views

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 ...
2
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1answer
44 views

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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148 views

A ‘canonical’ bounded lattice with proper de Morgan negation?

Call a lattice negation $\neg$ proper de Morgan negation iff it satisfies the following conditions. $\neg\neg a=a$. $\neg(a\vee b)=\neg a\wedge\neg b$ and $\neg(a\wedge b)=\neg a\vee\neg b$. $a\...
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1answer
114 views

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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1answer
111 views

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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2answers
74 views

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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1answer
103 views

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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39 views

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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1answer
204 views

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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49 views

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 ...
3
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1answer
194 views

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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1answer
151 views

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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3answers
72 views

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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1answer
59 views

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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1answer
244 views

“ 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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2answers
168 views

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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1answer
46 views

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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2answers
263 views

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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1answer
50 views

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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1answer
206 views

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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0answers
49 views

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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1answer
283 views

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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1answer
700 views

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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1answer
133 views

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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1answer
104 views

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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2answers
226 views

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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2answers
607 views

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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50 views

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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87 views

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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0answers
410 views

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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145 views

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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1answer
67 views

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 ...
3
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1answer
211 views

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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2answers
71 views

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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1answer
216 views

What may be the use of Quantum Logic

Is there any particular problem or scenario where quantum logic may be applied? If so, what is the benefit of using quantum logic instead of classical logic? I've been reading quite a lot on this ...
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1answer
699 views

Logics in which bi-implication and equivalence comes apart

Where $\phi, \psi$ are variables over formulas, are there logics for which we have either (or both) of $(1)$ and $(2)\thinspace$? $$(1) \hspace{0.5cm}(\phi \rightarrow \psi)\hspace{0.3cm} \land \...
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1answer
71 views

Can I derive $ \vdash \Gamma $ from $ \vdash \Gamma, A, A^\bot $?

The Wikipedia article on linear logic mentions the following as an initial sequent: $$ \over \vdash A, A^\bot $$ As far as I can understand from informal descriptions of linear-logic semantics, this ...
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1answer
170 views

Is it possible to formalize mathematics by using a different logic system (higher order, non-classical logics, model theory…)?

Is it possible to formalize mathematics by using a different logic system (higher order, non-classical logics, model theory…)? Hilbert's dream (against Gödel). The axiomatic system of all ...
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1answer
85 views

Basic equivalences in linear logic

How do we obtain the equivalence $A \otimes 0 \equiv 0$ and its dual in linear logic? Are they a consequence of cut-elimination? I found them listed as basic equivalences in the following resource: ...
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1answer
149 views

Are there any examples of consistent proper axiomatic extensions of classical logic?

By a proper axiomatic extension, I mean a logic with the same set of well formed formulas as classical logic, but with the set of theorems of the logic a proper superset of the theorems of classical ...
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1answer
111 views

Interpretation of relations in varying-domain models of F.O. modal logic

I am studying the book "First Order Modal Logic" By Fitting and Mendelsohn. In their definition of interpretation for varying domain models (def 4.7.3 pg 103), the interpretation of a relation in a ...
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2answers
134 views

Logical equivalence implication between Kleene and Classical logic

For any propositional assertions, $\phi$ and $\psi$, expressed using only the standard propositional logical connectives $\{\lnot,\land,\lor,\rightarrow,\iff\}$, if $\phi$ and $\psi$ are logically ...
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42 views

“Relative unsatisfiability” of SAT instances

There's a natural way to view any SAT instance as a variety: just replace the Boolean algebra $2$ of truth values with the corresponding Boolean ring $\mathbb{Z}/2\mathbb{Z}$. (See my answer to Is ...
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2answers
687 views

Is ¬¬(¬¬P → P) provable in intuitionistic logic?

I have a feeling it's not, because ¬¬P → P is not provable. If it is, I'm not sure what kind of reductio I'd need to negate ¬(¬¬P → P). I believe a textbook somewhere said it was provable in ...
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2answers
1k views

How to proofs work in three-valued Kleene logic?

In three-valued logics such as Kleene logic, there is a third truth value U, which represents "undefined", or "who knows?". It behaves like "either true or false", ...
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2answers
200 views

Does the Law of the Excluded Middle imply syntactical completeness?

The Law of the Excluded Middle (LEM) states that for any proposition $p$, we have $\vdash p \lor \lnot p $. Syntactic completeness (a.k.a negation completeness) states that for any proposition $p$, ...
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3answers
646 views

Logic & Reality [closed]

Maybe just a quick preface first before the question. I recently started a YouTube channel where I'm trying to clear up confusions I see on various (usually philosophical topics). In my 2nd video, the ...
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1answer
410 views

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 ...