# Formal Dictionary

The **Formal Dictionary** is a growing collection of definitions that are sufficiently formal to be used in formal proofs.

You can transclude the terms in this dictionary and use them to prove theorems and build theories.

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

## Contents

- 1 Definitions
- 1.1 Accessibility relation
- 1.2 Accidental property
- 1.3 Actual property
- 1.4 Aristotelian change
- 1.5 Causal chain
- 1.6 Causal independence
- 1.7 Cause
- 1.8 Change (1)
- 1.9 Change (2)
- 1.10 Determinism
- 1.11 Direct cause
- 1.12 Element
- 1.13 Effect
- 1.14 Entity
- 1.15 Essence
- 1.16 Essential property
- 1.17 Event
- 1.18 First cause
- 1.19 Full set of causes
- 1.20 Identity
- 1.21 Indirect cause
- 1.22 Metaphysical probability
- 1.23 Object
- 1.24 Possible world
- 1.25 Potential property
- 1.26 Property
- 1.27 Sequence
- 1.28 Set
- 1.29 Supervenience (1)
- 1.30 Supervenience (2)

- 2 Theorems
- 3 Theories
- 4 Guidelines for contributors
- 5 Notes and references

## 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 Wikipedia article Accessibility relation.

### 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 Wikipedia article Cause.

### 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 Wikipedia article Element (mathematics).

### 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 Wikipedia article Entity.

### 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 Wikipedia article Event (philosophy).

### 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 Wikipedia article Object.

### 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 Wikipedia article Possible world.

### 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 Wikipedia article Property.

### 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 Wikipedia article Set (mathematics).

### 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