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.

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.

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]

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.

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.

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

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

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.

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

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.

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.

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.

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.

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.

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

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.

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

${\displaystyle eEc:cCe}$

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.

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

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]

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.

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

${\displaystyle FCe:\lnot cCe}$

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.

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

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

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.

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

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.

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.

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.

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.

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.

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.

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.

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

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.

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

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

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

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

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

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

• Formal theory of causality

• 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).