Formal dictionary

The Template:W by Template:W is an inspiration for this resource.

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

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 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 ..., e_n) is a causal chain means: e₁ is a cause of e₂, e₂ is a cause of e₃ and so on until e_n-1 is a cause of e_n

$CC(e_1,e_2,e_3...,e_n):e_{1}C{e}_2\wedge e_{2}C{e}_3...\wedge e_{n-1}C{e}_n$

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.

$cCIe:\mathrm{\neg }cCe\wedge \mathrm{\neg }eCc$

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

Template:Formal dictionary/Formalize 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.

$cDCe:cCe\wedge \mathrm{\neg }\mathrm{\exists }d(cCd\wedge dCe)$

Effect

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

$eEc:cCe$

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.

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.

$FCe:\mathrm{\neg }cCe$

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.

$ϵFSCe:cCe\to c\in ϵ$

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.

$cICe:cCe\wedge \mathrm{\neg }cDCe$

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.

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.

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.

$\mathrm{\forall }x\mathrm{\forall }y(\mathrm{\forall }{X}_{\in B}(Xx\leftrightarrow Xy)\to \mathrm{\forall }{Y}_{\in A}(Yx\leftrightarrow Yy))$

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.

$\mathrm{\forall }x\mathrm{\forall }{X}_{\in A}(Xx\to \mathrm{\exists }{Y}_{\in B}(Yx\wedge \mathrm{\forall }y(Yy\to Xy)))$

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

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).