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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3
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2answers
152 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 ...
1
vote
2answers
128 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 ...
2
votes
1answer
35 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 ...
1
vote
1answer
31 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 ...
3
votes
1answer
57 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 ...
5
votes
1answer
58 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 ...
1
vote
1answer
53 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 ...
2
votes
2answers
84 views

Logical consequence in all structures in Kripke semantics

I read* the following definition of logical consequence in all structures within Kripke semantics:$$X\models A\iff\text{ for every } (W,R),\text{ if }(W,R)\models X,\text{ then }(W,R)\models A$$ ...
5
votes
3answers
259 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 ...
4
votes
0answers
105 views

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

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 ...
3
votes
2answers
122 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 ...
11
votes
11answers
1k views

A proportionality puzzle

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.
5
votes
4answers
212 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 ...
5
votes
3answers
375 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 ...
4
votes
2answers
112 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 ...
3
votes
1answer
267 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 ...
2
votes
2answers
213 views

Is there a proper term and/or symbol for an “agnostic” conclusion?

My question stems from the material conditional: $p \rightarrow q\\p\\\therefore\space q$ However, if $\bar p$ then the conditional is silent. I would like a way to represent this fact using, if ...
76
votes
22answers
7k views

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 <something>, ... One of my students just rose ...
1
vote
0answers
19 views

Can we say that CSL has the same expressive power of PCTL?

In other words, a part from the fact that continuous stochastic logic (CSL) deals with continuous time models whereas probabilistic computation time logic (PCTL) deals with discrete time models, is ...
1
vote
1answer
100 views

How many distinct functions for a set containing four elements? [closed]

How many distinct unary and binary functions can be defined on a set containing four elements? Edit: How many distinct unary and binary operations can be defined on a set containing four elements?
3
votes
3answers
203 views

Existence of maximal boolean-algebra sublattice (preserving top and bottom) of finite distributive lattice

If I regard a modal logic as some sort of many-valued logic, a "modal operator" projecting into a classical propositional logic context could sometimes be useful. Such an operator would provide a ...
0
votes
1answer
60 views

Which One of These Logical Theses Does Not Hold for Relevant Logics With a Connective for Conjunction?

I write in Polish notation and have included fully infixed notation here also which indicates parsing order. For every relevant logic simplification fails: Simplifcation: CpCqp or ...
0
votes
2answers
153 views

what are the Rosser Turquette axioms of Lukasiewicz 3 valued propositional logic?

I am trying to get my head around the Rosser-Turquette axiomatisation of Lukasiewicz n-valued logics, but cannot really follow it. Maybe if somebody can give me the axioms for 3 and 4 valued logic ...
5
votes
3answers
325 views

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 ...
4
votes
4answers
756 views

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 ...
5
votes
0answers
891 views

which branch of maths studies Standard Logical Matrices

In classical logic you have truthtables like: & | T | F ---|---|--- T | T | F F | F | F In many valued logic you have tables like: (this one is ...
12
votes
8answers
3k views

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 ...
1
vote
2answers
327 views

Example of a logic where a proof by contradiction does not imply a direct proof.

It makes sense that for any logic that has axioms, an inference rule and a statement you want to prove P, you can take a direct proof of P and turn it into a proof by contradiction (you have already ...
4
votes
4answers
173 views

Which CSL rules hold in Łukasiewicz's 3-valued logic?

CSL is classical logic. So I'm talking about the basic introduction and elimination rules (conditional, biconditional, disjunction, conjunction and negation). I'm not talking about his ...
0
votes
2answers
227 views

Intuitionistic logic

In propositional logic, I have now the following formulas. $X\equiv A \implies (B \vee C)$, $Y\equiv (A \implies B) \vee (A \implies C)$. I have already proven that Y implies X. But does X imply Y? ...
3
votes
1answer
414 views

Reference request: preparation for learning a little smooth infinitesimal analysis?

I'm interested in learning a little smooth infinitesimal analysis. There is a free book by Kock: Smooth Differential Geometry, http://home.imf.au.dk/kock/ . As I dive into it, I feel that I'm not ...
4
votes
4answers
3k views

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 ...
6
votes
2answers
191 views

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 ...