# Formal dictionary

This **formal dictionary** is a collection of definitions that are sufficiently formal to be used in theorems and theories.

If you want to contribute, check out the guidelines first.

“ | [...] if philosophers always agreed about the meaning of words, that would imply the disappearance of almost all their controversies. | ” |

— Descartes, Rules for the Direction of the Mind, Rule XIII |

## Definitions edit

### Accessibility relation edit

This is a primitive term. You can help the formal dictionary by finding a good definition for it. |

**Accessibility relation** is a primitive term, an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term by reading the article Accessibility relation at Wikipedia.

### Accidental property edit

Let *P* be a property, *x* an entity and *w* a possible world. Then *P* is an **accidental property** of *x* in *w* means: *x* has *P* in *w*, but in at least one possible world, *x* exists without *P*.^{[1]}

### Actual property edit

Let *P* be a property, *x* an entity and *w* a possible world. Then *P* is an **actual property** of *x* in *w* means: *P* is a property of *x* in *w*.

### Aristotelian change edit

Let *x* be an entity and *w _{1}* and

*w*two possible worlds. Then

_{2}*x*

**changes aristotelically**from

*w*to

_{1}*w*means: there is at least one possible world

_{2}*w*accessible from

*w*and with access to

_{1}*w*(or identical to

_{2}*w*) such that

_{2}*P*is a potential property of

*x*in

*w*and an actual property in

_{1}*w*.

### Causal chain edit

A sequence of events (*e _{1}*,

*e*,

_{2}*e*...,

_{3}*e*) is a

_{n}**causal chain**means:

*e*is a cause of

_{1}*e*,

_{2}*e*is a cause of

_{2}*e*and so on until

_{3}*e*is a cause of

_{n-1}*e*

_{n}

### Causal independence edit

Let *c* and *e* be events. Then *c* is **causally independent** of *e* means: *c* is not a cause of *e* and *e* is not a cause of *c*.

### Cause edit

This is a primitive term. You can help the formal dictionary by finding a good definition for it. |

**Cause** is a primitive term, an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term by reading the article Cause at Wikipedia.

### Change (1) edit

Let *x* be an entity, and *w _{1}* and

*w*two possible worlds. Then

_{2}*x*

**changes**from

*w*to

_{1}*w*means: there is at least one property

_{2}*P*and at least one possible world

*w*accessible from

*w*and with access to

_{1}*w*(or identical to

_{2}*w*) such that

_{2}*x*has

*P*in

*w*and lacks it in

_{1}*w*, or lacks it in

*w*and has it in

_{1}*w*.

### Change (2) edit

Let *x* be an entity, and *w _{1}* and

*w*two possible worlds. Then

_{2}*x*

**changes**from

*w*to

_{1}*w*means: there is at least one property

_{2}*P*such that

*x*has

*P*in

*w*and lacks it in

_{1}*w*, or lacks it in

_{2}*w*and has it in

_{1}*w*.

_{2}### Determinism edit

This is a semi-formal definition. You can help the formal dictionary by formalizing it further. |

**Determinism** means: every possible world has direct access to exactly one possible world.

### Direct cause edit

Let *c*, *d* and *e* be events. Then *c* is a **direct cause** of *e* means: *c* is a cause of *e* and there is no *d* such that *c* is a cause of *d* and *d* is a cause of *e*.

### Element edit

This is a primitive term. You can help the formal dictionary by finding a good definition for it. |

**Element** is a primitive term, an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term by reading the article Element (mathematics) at Wikipedia.

### Effect edit

Let *c* and *e* be events. Then *e* is an **effect** of *c* means: *c* is a cause of *e*.

### Entity edit

This is a primitive term. You can help the formal dictionary by finding a good definition for it. |

**Entity** is a primitive term, an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term by reading the article Entity at Wikipedia.

### Essence edit

Let *x* be an entity. Then the **essence** of *x* is the set of all its essential properties.

### Essential property edit

Let *P* be a property and *x* be an entity. Then *P* is an **essential property** of *x* means: in every possible world where *x* exists, *x* has *P*.^{[1]}

### Event edit

This is a primitive term. You can help the formal dictionary by finding a good definition for it. |

**Event** is a primitive term, an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term by reading the article Event (philosophy) at Wikipedia.

### First cause edit

Let *e* be an event. Then *e* is a **first cause** means: there is no event *c* that is a cause of *e*.

### Full set of causes edit

Let *e* be an event and *ε* be a set of events. Then *ε* is a **full set of causes** of *e* means: for every event *c* that is a cause of *e*, *c* is an element of *ε*.

### Identity edit

Let *x* and *y* be two entities. Then *x* and *y* are **identical** means: they have the same properties.

### Indirect cause edit

Let *c* and *e* be events. Then *c* is an **indirect cause** of *e* means: *c* is a cause of *e*, but *c* is not a direct cause of *e*.

### Metaphysical probability edit

Let *p* be a proposition, *w* a possible world and *n* a real number between 0 and 1. Then the **metaphysical probability** of *p* in *w* is *n* means: the number of possible worlds accessible from *w* where *p* is true divided by the total number of possible worlds accessible from *w* equals *n*.

Note: we assume that the total number of possible worlds accessible from *w* is a finite number, else all metaphysical probabilities collapse to zero.

### Object edit

This is a primitive term. You can help the formal dictionary by finding a good definition for it. |

**Object** is a primitive term, an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term by reading the article Object at Wikipedia.

### Possible world edit

This is a primitive term. You can help the formal dictionary by finding a good definition for it. |

**Possible world** is a primitive term, an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term by reading the article Possible world at Wikipedia.

### Potential property edit

Let *P* be a property, *x* an event and *w* a possible world. Then *P* is a **potential property** of *x* in *w* means: *x* exists without *P* in *w*, but in at least one accessible possible world, *x* has *P*.

### Property edit

This is a primitive term. You can help the formal dictionary by finding a good definition for it. |

**Property** is a primitive term, an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term by reading the article Property at Wikipedia.

### Sequence edit

### Set edit

This is a primitive term. You can help the formal dictionary by finding a good definition for it. |

**Set** is a primitive term, an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term by reading the article Set (mathematics) at Wikipedia.

### Supervenience (1) edit

Let *A* and *B* be two sets of properties. Then *A*-properties **supervene** on *B*-properties means: all entities that are *B*-indiscernible are *A*-indiscernible.

### Supervenience (2) edit

Let *A* and *B* be two sets of properties. Then *A*-properties **supervene** on *B*-properties means: anything that has an *A*-property has some *B*-property such that anything that has that *B*-property also has that *A*-property.

## Theorems edit

### Potential properties are not actual edit

If *P* is a potential property of *x* in *w*, then *P* is not an actual property of *x* in *w*.

### Actual properties are not potential edit

If *P* is an actual property of *x* in *w*, then *P* is not a potential property of *x* in *w*.

### Essential properties are actual edit

If *P* is an essential property of *x*, and *x* exists in *w*, then *P* is an actual property of *x* in *w*.

### Potential properties are not essential edit

If *P* is a potential property of *x* in *w*, then *P* is not an essential property of *x*.

### Essential properties do not change edit

If *x* changes a property *P* from *w _{1}* to

*w*, then

_{2}*P*is not an essential property of

*x*.

Suppose *x* changes a property *P* from *w _{1}* to

*w*. Then, by the definition of change, there's at least one possible world

_{2}*w*accessible from

*w*and with access to

_{1}*w*(or identical to

_{2}*w*) where

_{2}*x*exists, and

*x*has

*P*in

*w*but lacks it in

_{1}*w*, or lacks it in

*w*but has it in

_{1}*w*. In either case, there's at least one possible world where

*x*exists without

*P*, so by the definition of essential property,

*P*is not an essential property of

*x*. QED

### Some changes are not Aristotelian edit

## Theories edit

## Guidelines for contributors edit

- When adding a primitive term, use the Template:Formal dictionary/Primitive.
- Do not link your definitions outside of the dictionary, for example to Wikipedia. One of the goals of the dictionary is to be able to track the definitions back to the primitives. Linking out of the dictionary defeats this purpose. If you want to link to a term that hasn't been defined yet, just create a section for it and leave its definition for later, or mark it as a primitive term.
- When defining a term, link only the first appearance of each other term to its definition in the dictionary.
- If you want to add a different definition for an already existing term, distinguish them with numbers between parenthesis, like in Change (1) and Change (2).

## Notes and references edit

- ↑
^{1.0}^{1.1}Essential vs Accidental Properties in the Stanford Encyclopedia of Philosophy