  The Correspondence between Propositional Modal Logic with Axiom $\Box\varphi \leftrightarrow \Diamond \varphi$ and the Propositional Logic - Intelligent Information Processing VII (IIP 2014) Access content directly
Conference Papers Year : 2014

## The Correspondence between Propositional Modal Logic with Axiom $\Box\varphi \leftrightarrow \Diamond \varphi$ and the Propositional Logic

Meiying Sun
• Function : Author
• PersonId : 990784
Yuefei Sui
• Function : Author

#### Abstract

The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom $\Box\varphi \leftrightarrow\Diamond\varphi$ to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □k(k ≥ 0) only occurs before an atomic formula p, and $\lnot$ only occurs before a pseudo-atomic formula of form □k p. Maximally consistent sets of K +  have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □k pi (k,i ≥ 0) is corresponding to a propositional variable qki, each formula in K +  then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K +  and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K +  follow directly from those of P + .

#### Domains

Computer Science [cs]

### Dates and versions

hal-01383327 , version 1 (18-10-2016)

### Licence  Attribution

### Identifiers

• HAL Id : hal-01383327 , version 1
• DOI :

### Cite

Meiying Sun, Shaobo Deng, Yuefei Sui. The Correspondence between Propositional Modal Logic with Axiom $\Box\varphi \leftrightarrow \Diamond \varphi$ and the Propositional Logic. 8th International Conference on Intelligent Information Processing (IIP), Oct 2014, Hangzhou, China. pp.141-151, ⟨10.1007/978-3-662-44980-6_16⟩. ⟨hal-01383327⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

179 View