Sunday, September 14, 2008

A causality calculus

Let us design a first-order calculus including causality as non-truth-functional connective for which causality semantics is defined at the same level as the rest of the (truth-functional) connectives, according to the philosophical position stated in a previous entry. We assume we are given an infinite set of variables V and a set R of predicate symbols with associated arities.

Propositional constants: F (falsehood).

Connectives: → (only if), > (causal implication).

Quantifiers: V (for all).

Axioms and rules of inference: the usual ones for standard first-order calculus plus the following axiom schemas dealing with causality:

(C1) p > q, q > rp > r

(C2) (p > p) → F

(C3) p > q → (p and q)

(C4) p > r → (p or q) > r

(C5) p > (q and r) → p > q

These axioms deserve some comment: C1 states the usual transitivity of causation, while C2 says that no event causes itself; C3 allows us to speak about causation only among events actually taking place, which makes this a "realist" semantics. C4 and C5 translate some usual properties of material implication to causal implication.

Semantics: a valuation v specifies several mappings:

  • From V to some set U.
  • From each n-ary predicate in R to some n-ary relation in U.
  • From each formula to the set {F,T}.

and obeys the following rules:

  • v(F) = F.
  • v(φψ) = T iff v(φ) = F or v(ψ) = T.
  • v(R(t1,...,tn)) = T iff (v(t1),...,v(tn)) is in v(R).
  • v(Vx φ) = T iff v'(φ) = T for every valuation v' identical to v except possibly at x.
  • The relation on formulas of the calculus defined by v(φ > ψ) = T is consistent with axiom schemas C1,..., C5. In particular, this implies that the relation thus defined is a strict partial order.

As usual, a statement is valid if it is true for any valuation. It is easy to see that there are valuations with the properties described above: just take any valuation for standard first-order predicate calculus and augment it with v(φ > ψ) = F for all φ, ψ (we may call this a Humean valuation).

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