Formal dictionary

This formal dictionary is a collection of definitions that are sufficiently formal to be used in theorems and theories.

The Ethics by Baruch Spinoza is an inspiration for this resource.

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

Definitions edit

Accessibility relation edit

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 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 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 edit

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

 

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

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) edit

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) edit

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 edit

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

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 edit

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

 

Entity edit

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 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

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 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

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 edit

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 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

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.

Sequence edit

Set edit

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) 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 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

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. 1.0 1.1 Essential vs Accidental Properties in the Stanford Encyclopedia of Philosophy