# Formal glossary of philosophy

 Subject classification: this is a philosophy resource.
 Educational level: this is a research resource.
The Ethics by Baruch Spinoza is an inspiration for this project.

The formal glossary of philosophy is a collection of definitions of philosophical terms that are sufficiently formal to allow proofs about them.

You can use the terms in this glossary to prove theorems and build philosophical theories. If you want to use the terms in this glossary, the recommended way is to transclude them with the Template:Formal glossary of philosophy.

## Guidelines

• When defining a term, link only the first appearance of each other term in to its definition in the dictionary.
• Try not to link outside of the dictionary (except Wikipedia articles when explaining primitive terms). 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.
• 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).

## Definitions

### Accessibility relation

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term here.

### 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.[1]

### 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 w1 and w2 two possible worlds. Then x changes aristotelically from w1 to w2 means: there is at least one possible world w accessible from w1 and with access to w2 (or identical to w2) such that P is a potential property of x in w1 and an actual property in w.

### Causal chain

A sequence of events (e1, e2, e3 ..., en) is a causal chain means: e1 is a cause of e2, e2 is a cause of e3 and so on until en-1 is a cause of en

${\displaystyle CC(e_{1},e_{2},e_{3}...,e_{n}):e_{1}Ce_{2}\land e_{2}Ce_{3}...\land e_{n-1}Ce_{n}}$

### 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.

${\displaystyle cCIe:\lnot cCe\land \lnot eCc}$

### Cause

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term here.

### Change (1)

Let x be an entity, and w1 and w2 two possible worlds. Then x changes from w1 to w2 means: there is at least one property P and at least one possible world w accessible from w1 and with access to w2 (or identical to w2) such that x has P in w1 and lacks it in w, or lacks it in w1 and has it in w.

### Change (2)

Let x be an entity, and w1 and w2 two possible worlds. Then x changes from w1 to w2 means: there is at least one property P such that x has P in w1 and lacks it in w2, or lacks it in w1 and has it in w2.

### 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.

${\displaystyle cDCe:cCe\land \lnot \exists d(cCd\land dCe)}$

### Element

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term here.

### Effect

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

${\displaystyle eEc:cCe}$

### Entity

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term here.

### 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.[1]

### Event

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term here.

### 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.

${\displaystyle FCe:\lnot cCe}$

### 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.

${\displaystyle \epsilon FSCe:cCe\to c\in \epsilon }$

### 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.

${\displaystyle cICe:cCe\land \lnot cDCe}$

### 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

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term here.

### Possible world

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term here.

### 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

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term here.

### Set

This is a primitive notion or term—an undefined term used to define others. You can get an intuitive grasp of the intended meaning of the term here.

### 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.

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

### 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.

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

## 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 w1 to w2, then P is not an essential property of x.

Suppose x changes a property P from w1 to w2. Then, by the definition of change, there's at least one possible world w accessible from w1 and with access to w2 (or identical to w2) where x exists, and x has P in w1 but lacks it in w, or lacks it in w1 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

## Notes and references

1. Essential vs Accidental Properties in the Stanford Encyclopedia of Philosophy