## On Quantified Modal Logic (1999)

Venue: | Fundamenta Informatica |

Citations: | 5 - 0 self |

### BibTeX

@ARTICLE{Fitting99onquantified,

author = {Melvin Fitting},

title = {On Quantified Modal Logic},

journal = {Fundamenta Informatica},

year = {1999},

volume = {39},

pages = {13--211}

}

### OpenURL

### Abstract

Propositional modal logic is a standard tool in many disciplines, but first-order modal logic is not. There are several reasons for this, including multiplicity of versions and inadequate syntax. In this paper we sketch a syntax and semantics for a natural, well-behaved version of first-order modal logic, and show it copes easily with several familiar di#culties. And we provide tableau proof rules to go with the semantics, rules that are, at least in principle, automatable. 1 Introduction Propositional modal logic is a well-known tool, since possible worlds can represent computational states or moments of time or ways an agent thinks the world is. The addition of quantifiers, however, opens the door to a labyrinth full of twists and problems (see [7], for instance) and comparatively few have been willing to enter. Should quantificational domains be constant domain or varying or varying with restrictions, and what does "should" mean in this context anyway? Should constant symbol...

### Citations

818 | Dynamic logic
- Harel, Kozen, et al.
(Show Context)
Citation Context ...say that the full force of Herbrand’s theorem can be proven, even in a modal setting. 4.5 Dynamic Logic One of the interesting applications of multi-modal logic is dynamic logic, a logic of programs=-=, [8]-=-. In addition to the usual machinery of modal logic, a class of actions is introduced, with the class of actions closed under various operations, such as sequencing, repetition, and so on. For each ac... |

215 | First-Order Modal Logic
- Fitting
- 1998
(Show Context)
Citation Context ... formulas: 1. 〈λx.♦P(x)〉(c) 2. ♦〈λx.P(x)〉(c) What we now do is set out a formal modal semantics incorporating this idea, which we refer to as predicate abstraction. I have sketched these=-= ideas before [1, 2]-=- but these were essentially preliminary versions. The definitive treatment is in [6] and what follows is a partial sketch. I have already said what a modal frame was above, but the present notion of m... |

96 | Quantification in modal logic - Garson - 1977 |

11 | Modality and reference
- Thomason, Stalnaker
- 1968
(Show Context)
Citation Context ...behind many of the well-known “paradoxes” of modal logic. (A solution to this problem exists, though it is still not as well-known as it deserves to be. It was explicitly introduced to modal logic=-= in [10, 11]-=-, though as we will see, the underlying ideas are much earlier.) It is always interesting to find that problems in disparate areas have a common solution. The difficulty mentioned above with non-rigid... |

9 | Modal Logic Should Say More Than It Does
- Fitting
- 1991
(Show Context)
Citation Context ... formulas: 1. 〈λx.♦P(x)〉(c) 2. ♦〈λx.P(x)〉(c) What we now do is set out a formal modal semantics incorporating this idea, which we refer to as predicate abstraction. I have sketched these=-= ideas before [1, 2]-=- but these were essentially preliminary versions. The definitive treatment is in [6] and what follows is a partial sketch. I have already said what a modal frame was above, but the present notion of m... |

9 |
On Denoting. Mind 14:479–493
- Russell
- 1905
(Show Context)
Citation Context ...int out that in classical mathematical logic, all contexts are truth-functional, and the problem fades into the background.sOn Quantified Modal Logic 3 2.2 Russell Bertrand Russell, in a famous paper =-=[9], ga-=-ve a formal theory of definite descriptions like “the positive square root of 3” in a formal language. This was to be of special significance later in Principia Mathematica since it allowed classe... |

9 |
Abstraction in first-order modal logic
- Stalnaker, Thomason
- 1968
(Show Context)
Citation Context ...behind many of the well-known “paradoxes” of modal logic. (A solution to this problem exists, though it is still not as well-known as it deserves to be. It was explicitly introduced to modal logic=-= in [10, 11]-=-, though as we will see, the underlying ideas are much earlier.) It is always interesting to find that problems in disparate areas have a common solution. The difficulty mentioned above with non-rigid... |

7 | A modal Herbrand theorem
- Fitting
- 1996
(Show Context)
Citation Context ...abstraction allows us to prove a reasonable version for first-order modal logics. Since the full statement of the resulting theorem is somewhat complex, I only raise a few of the issues, and refer to =-=[3] f-=-or a fuller treatment. Classically, the first step of formula processing in the Herbrand method involves the introduction of Skolem functions. To cite the simplest case, the formula (∃x)P(x) is repl... |

2 |
Herbrand’s theorem, and the assignment statement
- Russell
- 1998
(Show Context)
Citation Context .... While our book is primarily aimed at philosophers, it contains much discussion of formal semantics and tableau-based proof procedures. A very brief summary of the material from the book appeared in =-=[5]-=-; this is an expanded version of that paper and presents a rough outline of our approach. Also, many of the basic ideas are more general than first-order, and a full treatment of higher-order modal lo... |

1 |
Barcan both ways. Forthcoming
- Fitting
- 1997
(Show Context)
Citation Context ...approach can simulate the others. The drawback is that both the converse Barcan formula and the Barcan formula are axiom schemes, with infinitely many instances. But, as we showed in [6], and also in =-=[4], these infini-=-tely many instances can be replaced by single formulas! In this section we briefly sketch how. We begin with the converse Barcan formula—the schema �(∀x)ϕ ⊃ (∀x)�ϕ. Requiring validity of... |