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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
49 views

Proof within Łukasiewicz's infinite-valued logic $Ł_{א}$

Using the axiom system for Łukasiewicz's infinite-valued logic $Ł_{א}$, I need to construct a proof of the following: ⊢ (A → B) ∨ (B → A) ⊢ (A → (B → C)) → (B → (A → C)) A → B ⊢ (A ∧ C) → B A → B ⊢ ¬B ...
Amilio's user avatar
  • 11
1 vote
0 answers
53 views

Books on co-Heyting algebras (with a view to their logics).

I would like to know more about co-Heyting algebras, particularly from the perspective of their logics (like paraconsistent logics). What books are available out there on the topic? It might be that ...
Shaun's user avatar
  • 45.1k
2 votes
1 answer
67 views

Does Bi-Intuitionistic Logic turn Classical in sufficiently strong first-order theories?

Bi-Intuitionistic Logic adds to Intuitionistic Logic a binary connective $←$ known as co-implication or subtraction. A weak negation $\sim A$ is defined for Bi-Intuitionistic Logic as $\top ← A$; Bi-...
PW_246's user avatar
  • 1,258
-2 votes
1 answer
70 views

Is there a way to define classical implication in this logic?

I’m asking this question so that I may provide my own answer to it and share what I’ve discovered. I’ve already posted about a logic that results from modifying the Gödel-McKinsey-Tarski translation ...
PW_246's user avatar
  • 1,258
-2 votes
2 answers
184 views

Does this outwit Rosser's trick? [closed]

Reading about Rosser's trick made me instantly curious about the following question: Can we devise a logic (which semantically encapsulates useful maths - define that as you will) in which for any ...
it's a hire car baby's user avatar
2 votes
1 answer
62 views

Examples of finitely-valued logics that aren't algebraizable

I was reading this question which asks about the equivalent of Boolean algebras for relevant logic. In the comments, Noah Schweber mentions that the relevant logic E is not algebraizable. In the book ...
Greg Nisbet's user avatar
  • 11.7k
1 vote
0 answers
166 views

Boolean algebra is to classical logic like what is to relevant logic?

The Question: Boolean algebra is to classical logic like what is to relevant logic? Context: I guess this is a terminology question, so there's not much I can add, except that I've been interested ...
Shaun's user avatar
  • 45.1k
1 vote
1 answer
258 views

What exactly are capture and release?

Motivation: I'm interested in how different people resolve the Liar paradox and other, related phenomena, like the revenge Liar paradoxes, and so on. I have a copy of "Formal Theories of Truth,&...
Shaun's user avatar
  • 45.1k
1 vote
0 answers
133 views

Has this logic already been studied?

I have been spending the better part of a year thinking about the subtleties involved in balancing natural language intuitions for logic with the power and efficacy that Classical Logic and ...
PW_246's user avatar
  • 1,258
3 votes
1 answer
144 views

Question about Hilbert's axiom for the postive fragment of classical logic

In his axiomization of the positive fragment of Classical Logic, Hilbert included the axiom $(B \rightarrow A) \rightarrow [(C \rightarrow A) \rightarrow (((B \rightarrow C) \rightarrow A) \rightarrow ...
Michael Lee Finney's user avatar
3 votes
1 answer
90 views

How to prove a formula in Visser’s FPL

I’m currently reading through Visser’s “A Propositional Logic with Explicit Fixed Points“ (1980) and am having trouble proving a seemingly basic result. The system is the analogue to GL modal logic on ...
PW_246's user avatar
  • 1,258
4 votes
1 answer
415 views

doubly negated intuitionistic formulas

What is an example of some doubly negated statement in intuitionistic logic that is not equivalent to its classical version? Some background: for every statement $P$ we can find a classically ...
Keplerto's user avatar
  • 343
0 votes
0 answers
16 views

How does relevance postulate in Belief Revision guarantee minimal change?

I've just started studying belief bases and I am struggling to understand how the relevance postulate for contraction guarantees minimality. (Relevance) If q ∈ A and q ∉ A ÷ p, then there is a set A′ ...
Jonas's user avatar
  • 307
1 vote
1 answer
81 views

Is three-valued relevant intuitionistic logic a thing?

I have explored non-classical logics recently, and I was wondering if three-valued relevant intuitionistic was/could be a thing. It seems to be an interesting combination to me since it is closer to ...
user avatar
-1 votes
1 answer
210 views

What are proofs in constructivist logic?

The difference in syntax between classical and constructivist mathematics is, as far as I've understood, not because constructivists think a well-formed proposition may be untrue and unfalse at the ...
user110391's user avatar
  • 1,079
3 votes
1 answer
96 views

Proving chain order from Peirce's Law

I am trying to prove any one of the following statements $((p \to r) \lor (q \to s)) \to ((p \to s) \lor (q \to r))$ $(p \to q) \lor (q \to r)$ $((p \to q) \to q) \to (p \lor q)$ $(p \to q) \lor (q \...
Michael Lee Finney's user avatar
2 votes
1 answer
69 views

Examples of propositional logics with conditionals based on conditional probability?

Are there any propositional logics with conditionals whose semantics are based on conditional probability? I can see two design challenges that come with defining a conditional-probability-flavored ...
Greg Nisbet's user avatar
  • 11.7k
4 votes
2 answers
190 views

Is there a list of logics?

There are a lots of logics. Some of them are: Propositional logic Predicate logic Second order logic $n$ order logic Fuzzy logic Modal logic Multivalued logic etc So I`d like to know whether there ...
Gauss 'n Roses's user avatar
0 votes
1 answer
37 views

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{...
Martín S's user avatar
  • 159
1 vote
0 answers
55 views

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 ...
Greg Nisbet's user avatar
  • 11.7k
1 vote
0 answers
48 views

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-...
Greg Nisbet's user avatar
  • 11.7k
1 vote
0 answers
79 views

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 (...
Greg Nisbet's user avatar
  • 11.7k
7 votes
3 answers
884 views

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 ...
Nat Kuhn's user avatar
  • 295
0 votes
0 answers
46 views

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 =...
Eddie Chau's user avatar
0 votes
0 answers
57 views

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(\...
tox123's user avatar
  • 1,594
2 votes
0 answers
71 views

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 ...
Greg Nisbet's user avatar
  • 11.7k
6 votes
3 answers
239 views

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 ...
the gods from engineering's user avatar
0 votes
1 answer
121 views

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: ...
user avatar
1 vote
2 answers
228 views

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 ...
Luiz Martins's user avatar
3 votes
1 answer
298 views

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 ...
Luiz Martins's user avatar
5 votes
1 answer
140 views

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 ...
Floridus Floridi's user avatar
5 votes
1 answer
379 views

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 ...
user56834's user avatar
  • 13k
2 votes
1 answer
82 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 ...
Constantly confused's user avatar
3 votes
0 answers
89 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 ...
Eddie Chau's user avatar
2 votes
1 answer
106 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 ...
ferdinand's user avatar
  • 632
3 votes
1 answer
219 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 ...
Dannyu NDos's user avatar
  • 2,029
4 votes
1 answer
311 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 $71$ ...
Shaun's user avatar
  • 45.1k
1 vote
2 answers
172 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, ...
whatsgoingon's user avatar
2 votes
1 answer
540 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 ...
Shaun's user avatar
  • 45.1k
1 vote
0 answers
49 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 ...
TomR's user avatar
  • 1,313
3 votes
1 answer
231 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 ...
Daniil Kozhemiachenko's user avatar
3 votes
0 answers
88 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 ...
lfba's user avatar
  • 451
5 votes
1 answer
634 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 ...
Shaun's user avatar
  • 45.1k
0 votes
1 answer
170 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\...
Shaun's user avatar
  • 45.1k
5 votes
3 answers
88 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 "...
Threnody's user avatar
  • 898
2 votes
1 answer
88 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 $\bot$ ...
Pteromys's user avatar
  • 7,220
1 vote
1 answer
251 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 ...
user avatar
3 votes
2 answers
214 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 ...
Daniil Kozhemiachenko's user avatar
2 votes
1 answer
64 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 $\mathcal{L}_0$...
Kevin's user avatar
  • 311
1 vote
2 answers
673 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 ...
user192972's user avatar