## Saturday, 28 November 2009

### Towards a Teleological Logic (Part 3)

Thus far, I have tried to provide an intuitive feel for a basic teleological logic (henceforth, BTL). We may now introduce some additional regimentation by specifying the syntax of BTL. Let us assume that we have a simple propositional language, L. The alphabet of L consists of:
(i) a denumerable set Π of propositional variables p, q, r, p1 ,p2, . . .

(ii) the primitive logical connectives ⊤ (verum), ⊥ (falsum), ~ (negation), □ (teleological necessity), ◊ (teleological possibility), ∧ (conjunction), ∨ (disjunction), → (material implication), and ↔ (material equivalence).

(iii) the parentheses ( ).
The well formed formulas (wffs) of L consists of the smallest set Σ such that:
(a) every propositional variable in Π is in Σ,
(b) ⊤ and ⊥ are in Σ,
(c) If p is in Σ then so are ~p, □p and ◊p
(d) If p, q are in Σ, then so are (p ∧ q), (p ∨ q), (p → q) and (p ↔ q).
The sentences under (a) and (b) are the atomic sentences of L. ⊤ and ⊥ are 0-place connectives; ~, □ , ◊ are 1-place connectives; and all remaining connectives are 2-place. I propose the following axiom schemata for BTL:
BTL:
A1. All tautologous wffs of L
A2. □(p → q) → (□p → □q)
A3. □p → ~ □~p
R1. If ⊢ p and ⊢ p → q, then ⊢ q
R2. If ⊢ p then ⊢ ⊤ p
It should be clear to the observant reader that BTL is simply modal system D, with the relevant notation amended to express a teleological interpretation. A1 is standard in all normal modal systems. According to A2, if a material conditional holds in all T-worlds, and its antecedent holds in all T-worlds, then the consequent of the material conditional also holds in all T-worlds. This is the K axiom present in all normal modal logics, also known as the distribution axiom.

A3 follows from conditions imposed on the binary relation R, which restricts access to worlds that are teleologically ideal (i.e., possible worlds in which every telos is realised). A3 tells us that for any world Γ that is a member of some frame G, there is some world Δ in G such that R(Γ,Δ). A3 guarantees that there is always a possible world fitting the conditions of the accessibility relation; thus ensuring that there is always a T-world we may refer to when we need to formerly represent a teleological claim. In addition to A1-A3, BTL includes Modus Ponens, which is represented by R1. When A1 and R1 are combined, they yield the full inferential power of the propositional calculus. R2 tells us that if p is a theorem, then the claim that p obtains in all T-worlds is also a theorem.

Taking BTL as our starting point, and using our quotidian intuitions about purposiveness as a guide, I believe we may assess which axioms should and should not be included in a plausible teleological logic. For example, we know that if Δ stands in relation R to Γ, such that R(Γ,Δ), and some world Ω stands in relation R to Δ, such that R(Δ,Ω), then Ω must itself be a T-world. Since (by definition) all T-worlds stand in relation R to Γ, it follows that Ω stands in the relation R to Γ, such that R(Γ,Ω). This means that, under a teleological interpretation, the relation R is transitive. This is equivalent to the following axiom:
A4. □p → □□p (□ -4)
Earlier, it was noted that the actual world is not a member of the set of T-worlds. Given a teleological interpretation of the accessibility relation, it follows that the actual world is not accessible from itself. This entails the denial of Reflexivity; the frame condition on R, according to which R(Γ,Γ) for every Γ that is a member of G. Thus, under a teleological reading of R, the following axiom turns out to be false:
(*) □p → p (□ -M)
Moreover, since some non-T-world Γ (i.e., the actual world) may fail to stand in the relation R with respect to some given T-world Δ, such that R(Δ ,Γ) is false, even though Δ stands in the relation R with respect to Γ, such that R(Γ,Δ) is true, the following axiom also turns out to be false:
(**) p → □◊p (□ -B)
The upshot is that under a teleological interpretation, R is not Symmetric. The denial of (**) follows from the fact that the actual world is not a world in which all the purposes found in a given T-world are realised. Consequently, while all T-world stand in the relation R to the actual world, the actual world does not stand in the relation R to any T-world. In fact, only another T-world Ω can stand in the relation R to some other T-world Δ since (intuitively) it is only in some other T-world Ω that every telos found in Δ is realised. However, this still falls short of the claim that R is Euclidean; the frame condition that if R(Γ,Δ) and R(Γ,Ω), then R(Δ,Ω). All that has been asserted so far is that if Ω stands in the relation R to some world Δ, then Ω must be a T-world. This is consistent with the possibility that Ω fails to stand in relation R to Δ. Nevertheless, there seems to be some intuitive traction to the idea that every telos found in some T-world is realised in all other T-worlds. This suggests that all T-worlds stand in the relation R to each other. When this observation is combined with the fact that all T-worlds stand in relation R to the actual world, this yields the following axiom:
A5. ◊p →□◊p (□-5)
A5 tells us that R is Euclidean. Moreover, if all T-worlds stand in the relation R to all T-worlds, then all T-worlds stand in the relation R to themselves. Consider: if T-worlds are possible worlds in which every telos is realised, then every telos found in a given T-world must be realised in that T-world. It follows that for any given T-world, it stands in the relation R to itself. However, as was noted earlier, the actual world is not a T-world, so that the actual world fails to stand in the relation R to itself. This, we noted, entails the denial of Reflexivity. However, any world which occupies the second position in the two-place relation R(Γ,Δ) must (by definition) be a T-world, which means that it must stand in the R relation with itself. It follows that R is Shift Reflexive, such that if R(Γ,Δ) then R(Δ,Δ). This yields the following axiom:
A6. □(□ p → p) (□-□ M)
Moreover, we noted that since the actual world does not stand in the relation R to any T-world (even though all T-worlds stand in the relation R to the actual world), R is not Symmetric. Even so, if R(Γ,Δ) holds for some world Δ, and Ω is accessible from Δ, such that R(Δ,Ω), then Ω must be a T-world. But if Ω is a T-world, and given that all T-worlds are accessible from each other, then Δ must stand in relation R to Ω, such that R(Ω,Δ). This means that R is Shift Symmetric, such that if R(Γ,Δ) holds for some world Δ, then R(Δ,Ω) only if R(Ω,Δ). Thus, we arrive at the following axiom:
A7. □ (◊□p → p) (□ - □ B)
Euclidean modal systems are usually assumed to be Transitive, Reflexive and Symmetric, as with system S5. However, while R is Transitive (given a teleological interpretation), it is not Reflexive and Symmetric. Instead, R is Shift Reflexive and Shift Symmetric. When Transitivity, the Euclidean axiom, Shift Reflexivity and Shift Symmetry are added to BTL, we arrive at what may be referred to as Sophisticated Teleological Logic (henceforth STL):

STL:
A1. All tautologous wffs of L
A2. □(p → q) → (□p → □q)
A3. □p → ~ □~p
A4. □ p → □□p
A5. ◊p → □◊p
A6. □ (□p → p)
A7. □ (◊□ p → p)
R1. If ⊢ p and ⊢ p → q, then ⊢ q
R2. If ⊢ p then ⊢ ⊤ p
In my next post on this topic, I will consider a few objections to STL (especially the concept of a T-world limned thus far) which will motivate a multimodal teleological logic; one in which T-worlds are indexed to sets of teleological objects.