Formal dictionary

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The Ethics by Baruch Spinoza 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[edit | edit source]

Accessibility relation[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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

Direct cause[edit | edit source]

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 | edit source]

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 | edit source]

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

Entity[edit | edit source]

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 | edit source]

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

Essential property[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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

Indirect cause[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

Set[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

Potential properties are not actual[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

Theories[edit | edit source]

Guidelines for contributors[edit | edit source]

  • 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 | edit source]

  1. 1.0 1.1 Essential vs Accidental Properties in the Stanford Encyclopedia of Philosophy