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.

"[...] if philosophers always agreed about the meaning of words, that would imply the disappearance of almost all their controversies."

- Descartes

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 ..., 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

$$CC(e_1,e_2,e_3...,e_n) : e_1Ce_2 \land e_2Ce_3 ... \land e_{n-1}Ce_n$$

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.

$$cCIe : \lnot cCe \land \lnot eCc$$

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
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.

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

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.

$$FCe : \lnot cCe$$

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.

$$\epsilon FSCe : cCe \to c \in \epsilon$$

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.

$$cICe : cCe \land \lnot cDCe$$

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.

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.

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

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.

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

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).