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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
224 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“ Logic does not allow you to say this”: is this assertion outdated?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 ...
2
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1answer
100 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
39 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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0answers
33 views

Can temporal logic be included in the framework of modal logics K, 4, D, T? Substructural temporal logic?

I am reading the article "A uniform framework for substructural logics with modalities" https://easychair.org/publications/paper/d5zT which gives impression that K, 4, D, T (and some possible ...
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2answers
134 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 ...
2
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1answer
45 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, ...
3
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1answer
156 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
38 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
196 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. ...
6
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1answer
426 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
89 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
85 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
476 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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0answers
40 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 ...
2
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1answer
68 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
305 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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0answers
70 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
61 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
143 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
61 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
164 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
447 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
57 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
154 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
74 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
110 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
72 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
114 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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0answers
40 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 ...
3
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2answers
503 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
666 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
150 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
570 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
327 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 ...
6
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3answers
304 views

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 \...
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2answers
175 views

Concatenation and contraposition laws in modified Stalnaker system

The first four axioms of Stalnaker conditional logic, which adds $\mapsto$, the counterfactual conditional symbol, to the operator symbols of classical logic, are $A\mapsto A$ $(A\mapsto B)\...
2
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1answer
76 views

Conditions for total orders in temporal logic

Let $(T,>)$ be a frame of minimal temporal logic, i.e. a frame as defined in Kripke semantics where the relation is a partial order relation $>$ defined on the set $T$ of worlds, called instants....
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1answer
39 views

$A\land FB\rightarrow F(PA\land B)$ in temporal logic

Temporal minimal logic $\mathbf{K_T}$ calculus is characterised by the following axioms, where $F=_{\text{def}} \lnot G\lnot$ and $P=_{\text{def}} \lnot H\lnot$: $G(A\rightarrow B)\rightarrow (GA\...
3
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1answer
91 views

Equivalence between $\mathbf{KT_4}$ and Lewis' $\mathbf{S_4}$

Let us define modal logic system $\mathbf{KT_4}$ by adding the following axioms to classic propositional logic $\diamond A\leftrightarrow\lnot\square\lnot A$ $\square(A\rightarrow B)\rightarrow(\...
5
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1answer
211 views

Symmetric relations and $\varphi\rightarrow\square\diamond\varphi$

I read that the schema $$\varphi\rightarrow\square\diamond\varphi$$ corresponds to the symmetric property (D. Palladino, C. Palladino, Logiche non classiche, 'non-classical logics', 2007) of the ...
2
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1answer
443 views

Euclidean relations and $\diamond P\rightarrow\square\diamond P$

I read* that the formula $$\diamond \varphi\rightarrow\square\diamond\varphi$$is valid in a structure $(W,R)$, intended as in Kripke semantics, -i.e. that it is true for any interpretation $I$ and in ...
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2answers
291 views

Logical consequence in all structures in Kripke semantics

I read* the following definition of logical consequence in all structures within Kripke semantics:$$X\models A\iff\text{ for every } (W,R),\text{ if }(W,R)\models X,\text{ then }(W,R)\models A$$ $$\...
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3answers
542 views

Intuitionistic Logic and Classical Logic on the proof of (A or B)

In intuitionist logic, a proof of (A or B) means a proof of A, or a proof of B, whereas in Classical logic, a proof of (A or B) may be done withouth either proving A or proving B. I'm trying to ...
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0answers
131 views

Is there a logic to formalize the concept of “understanding” [closed]

The question may seem little bit weird given that philosophers have been struggling to have a full grasp on the concept of "understanding". But I'm wondering if there are any logics (modal-based or ...
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2answers
306 views

What are the different approachs to logic?

I have studied a little of logic (namely, FOL and propositional logic) using so-called "Hilbert style". I've heared that there are different approachs to logics like , deductions, trees, natural ...
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13answers
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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.
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4answers
386 views

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 ...
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3answers
657 views

Derive by modus ponens $[A\rightarrow(B\rightarrow C)]\rightarrow[(A\rightarrow B)\rightarrow(A\rightarrow C)]$

How could I derive by modus ponens the formula $$[A\rightarrow(B\rightarrow C)]\rightarrow[(A\rightarrow B)\rightarrow(A\rightarrow C)]$$ from, and just from, the following axiom schemata? $(A\lor A)\...
5
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2answers
173 views

Non-upper bounds without excluded middle

Motivated by an earlier question, I'm curious if we can prove the following statement without the law of excluded middle: Let $E$ be a set of real numbers. A number $x$ is said to be an upper bound ...
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1answer
539 views

What are some practical applications of mathematical/formal logic to science and humanities? [closed]

I am studying a bit of this and so far it seems that, apart from math and computer science, the discipline of Logic is very self facing, with logicians proving things for other logicians. It left me ...