<p>Any logic defined semantically in a similar way is trivially consistent, as it has a model.</p>

<p>Any finite-valued logic has a faithful translation into classical logic, hence its set of valid formulas, or even the consequence relation, are recursively enumerable.</p>

<p>Completeness is a relationship between semantics and a proof system. You only gave the semantics, not any proof system, hence the question is meaningless. There <em>exist</em> complete recursively axiomatized proof systems for your kind of logic, but this is just another way to state that the logic is recursively enumerable.</p>

CC BY-SA 3.0
2014-08-10T22:04:16.740Z
Thanks Emil for your clarification!
Could you say something about the consistency? 
Is there an example proof of completeness for a similar thing, that you can point me to?
Daniel
CC BY-SA 3.0
2014-08-11T12:29:36.657Z
For example, complete calculi for several propositional 3-valued logics are presented in http://www.cs.tau.ac.il/~aa/articles/3valued_jsl.pdf , and completeness proofs for Hilbert and sequent calculi for a particular first-order 3-valued logic is given in section 6.2.1 of http://math.cas.cz/~jerabek/papers/phd.pdf . It would help a lot if you learn at least the basic notions of the theory of multi-valued logics, such as in http://plato.stanford.edu/entries/consequence-algebraic (propositional) or http://www.carlesnoguera.cat/files/ch2.pdf (both propositional and first-order). ...
Emil Jeřábek
CC BY-SA 3.0
2014-08-11T12:32:44.220Z
... (The theory is geared towards a different class of logics; 3-valued logics are often not algebraizable, e.g. Kleene’s logic. However, basic concepts of consequence relations, logical matrices, and algebraic semantics apply to them just fine.)
Emil Jeřábek
CC BY-SA 3.0
2014-08-10T10:35:03.760Z
2014-08-10T10:35:03.760Z
Emil Jeřábek
<p>Based on the explanations here [1] I know that 3-valued <em>first order</em> logic has many different versions, which differ in the definition of their operations (e.g. implication). All of these (as far as I know) are complete, i.e. any probable statement could be proven. Also these are (just as in the 2-valued case are Semi-decidable, i.e. the correctness of statement given axioms, could be proved, also wrong statements could not be disproved.) Also these are consistent, i.e. they don't contain contradiction. </p>

<p>Suppose I define my arbitrary 3-valued first order logic (by arbitrary, I mean, arbitrary $\Rightarrow$, AND, OR). Is it true that, it is </p>

<ul>
<li><p>COMPLETE, </p></li>
<li><p>SEMI-DECIDABLE </p></li>
<li><p>CONSISTENT</p></li>
</ul>

<p>for any $\Rightarrow$, AND, OR operations?</p>

<p>[1] <a href="http://en.wikipedia.org/wiki/Three-valued_logic" rel="nofollow">http://en.wikipedia.org/wiki/Three-valued_logic</a> </p>

<lo.logic><computability><decidability><logic-programming>
Consistency and completeness of any arbitrary 3-valued logic?
Daniel
# Consistency and completeness of any arbitrary 3-valued logic?

(Asked by: username_0 on 2014-08-10)

Based on the explanations here [1] I know that 3-valued *first order* logic has many different versions, which differ in the definition of their operations (e.g. implication). All of these (as far as I know) are complete, i.e. any probable statement could be proven. Also these are (just as in the 2-valued case are Semi-decidable, i.e. the correctness of statement given axioms, could be proved, also wrong statements could not be disproved.) Also these are consistent, i.e. they don't contain contradiction.

Suppose I define my arbitrary 3-valued first order logic (by arbitrary, I mean, arbitrary $\Rightarrow$, AND, OR). Is it true that, it is

* COMPLETE,
* SEMI-DECIDABLE
* CONSISTENT

for any $\Rightarrow$, AND, OR operations?

[1] <http://en.wikipedia.org/wiki/Three-valued_logic>

**Tags:** lo.logic, computability, decidability, logic-programming


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## 1 answers

### Answer by username_1 on 2014-08-10. Score: 6 | Accepted answer

Any logic defined semantically in a similar way is trivially consistent, as it has a model.

Any finite-valued logic has a faithful translation into classical logic, hence its set of valid formulas, or even the consequence relation, are recursively enumerable.

Completeness is a relationship between semantics and a proof system. You only gave the semantics, not any proof system, hence the question is meaningless. There *exist* complete recursively axiomatized proof systems for your kind of logic, but this is just another way to state that the logic is recursively enumerable.

> **Comments:**
> **username_0**: Thanks Emil for your clarification! Could you say something about the consistency?  Is there an example proof of completeness for a similar thing, that you can point me to?  
> **username_1**: For example, complete calculi for several propositional 3-valued logics are presented in http://www.cs.tau.ac.il/~aa/articles/3valued_jsl.pdf , and completeness proofs for Hilbert and sequent calculi for a particular first-order 3-valued logic is given in section 6.2.1 of http://math.cas.cz/~jerabek/papers/phd.pdf . It would help a lot if you learn at least the basic notions of the theory of multi-valued logics, such as in http://plato.stanford.edu/entries/consequence-algebraic (propositional) or http://www.carlesnoguera.cat/files/ch2.pdf (both propositional and first-order). ...  
> **username_1**: ... (The theory is geared towards a different class of logics; 3-valued logics are often not algebraizable, e.g. Kleene’s logic. However, basic concepts of consequence relations, logical matrices, and algebraic semantics apply to them just fine.)  

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