# 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.

## Definitions
### Accessibility relation

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

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.

### Actual property

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

Let x be an entity and w$_1$ and w$_2$ two possible worlds. Then x changes aristotelically from w$_1$ to w$_2$ means: there is at least one possible world w accessible from w$_1$ and with access to w$_2$ (or identical to w$_2$) such that P is a potential property of x in w$_1$ and an actual property in w.

### Causal chain

A sequence of events (e$_1$, e$_2$, e$_3$ ..., e$_n$) is a causal chain means: e$_1$ is a cause of e$_2$, e$_2$ is a cause of e$_3$ and so on until e$_{n-1}$ is a cause of e$_n$
$CC(e_{1},e_{2},e_{3}...,e_{n}):e_{1}Ce_{2}\land e_{2}Ce_{3}...\land e_{n-1}Ce_{n}$

### Causal independence

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.
$cCIe:\lnot cCe\land \lnot eCc$

### Cause

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)

Let x be an entity, and w$_1$ and w$_2$ two possible worlds. Then x changes from w$_1$ to w$_2$ means: there is at least one property P and at least one possible world w accessible from w$_1$ and with access to w$_2$ (or identical to w$_2$) such that x has P in w$_1$ and lacks it in w, or lacks it in w$_1$ and has it in w.

### Change (2)

Let x be an entity, and w$_1$ and w$_2$ two possible worlds. Then x changes from w$_1$ to w$_2$ means: there is at least one property P such that x has P in w$_1$ and lacks it in w$_2$, or lacks it in w$_1$ and has it in w$_2$.

### Determinism

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

### Direct cause

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.
$cDCe:cCe\land \lnot \exists d(cCd\land dCe)$

### Element

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

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

### Entity

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

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

### Essential property

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.

### Event

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

Let e be an event. Then e is a first cause means: there is no event c that is a cause of e.
$FCe:\lnot cCe$

### Full set of causes

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.
$\epsilon FSCe:cCe\to c\in \epsilon$

### Identity

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

### Indirect cause

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.
$cICe:cCe\land \lnot cDCe$

### Metaphysical probability

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

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

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

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

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.

### Set

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)

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.
$\forall x\forall y(\forall X_{\in B}(Xx\leftrightarrow Xy)\rightarrow \forall Y_{\in A}(Yx\leftrightarrow Yy))$

### Supervenience (2)

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.
$\forall x\forall X_{\in A}(Xx\rightarrow \exists Y_{\in B}(Yx\land \forall y(Yy\rightarrow Xy)))$

## Theorems
### Potential properties are not actual

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

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

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

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

### Essential properties do not change

If x changes a property P from w$_1$ to w$_2$, then P is not an essential property of x.
Suppose x changes a property P from w$_1$ to w$_2$. Then, by the definition of change, there's at least one possible world w accessible from w$_1$ and with access to w$_2$ (or identical to w$_2$) where x exists, and x has P in w$_1$ but lacks it in w, or lacks it in w$_1$ but has it in 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
## Theories

* Formal theory of causality

## Guidelines for contributors

* 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).