The 101st Philosopher's Carnival

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Towards a Teleological Logic (Part 4)

In this post, I wish to consider a pair of objections to the account of a T-world defended thus far. One putative difficulty with characterising T-worlds as possible worlds in which every telos is realised is that it seems to preclude compensatory and/or conflicting purposes. The notion of a compensatory purpose applies to teleological objects that have the telos of “filling in” for when some other teleological object fails to realise its telos. For example, we can imagine a system equipped with an emergency self-destruct sequence that only initiates if there is a failure in all other safety protocols. If we conceive of T-worlds as worlds in which every telos is realised, then such a self-destruct mechanism will never have the opportunity to realise its telos since there will never be the required failure elsewhere in the system. This suggests that in a given T-world, compensatory purposes are never realised. However, if compensatory purposes remain unrealised, then a T-world cannot really be a world in which every telos is realised. Thus, the possibility of compensatory purposes appears to threaten the concept of a T-world with incoherence.

One possible reply to the above objection, which will ultimately prove inadequate, is to distinguish between cases in which some X does not have the opportunity to fulfil the function for which it is designed and cases in which X is presented with such an opportunity, but fails to do so. Moreover, we may say that a teleological object only fails to realise its telos in the latter case. This suggestion appears to have some intuitive traction since few would regard a safety mechanism as somehow defective simply because it never had the opportunity to perform its function. On this view, some world Γ counts as a non-T-world just in case some X in Γ is presented with an opportunity to realise its telos and yet fails to do so. Thus, far from implying the failure of compensatory purposes, T-worlds virtually guarantee that such purposes are realised by removing the antecedent conditions for their failure. Consequently, the concept of a T-world remains coherent, even if there are teleological objects with compensatory purposes.

The above reply seems right, as far as it goes. But there are at least two reasons for thinking that it does not go very far. First, while it appears to avoid the problem posed by compensatory purposes in the case of artefacts, it is not clear that the reply generalises to biological systems. We can certainly imagine a biological system or process that has the function of “filling in” for some other biological system or process, should the latter fail to realise its telos. Moreover, we can imagine such a system evolving in the actual world since individuals within a biological population with such a system acting as “back-up” would display greater reproductive success than their conspecifics that lacked such “safety nets”. However, it is not clear that such a “back-up” system could ever evolve in a T-world. Since the original system will always realise its telos, the presence of a “back-up” system will never offer any evolutionary advantage to its possessor, and will therefore never become subject to selective pressures. The upshot is that there could never be a biological system with a compensatory purpose in a T-world.

Second, the definition of T-worlds as possible worlds in which every telos is realised seems to be at odds with the fact that there may be conflicting purposes. The notion of a conflicting purpose applies to teleological objects whose telos involves preventing some other teleological object from realising its telos. For example, we can imagine a type of antibiotic whose purpose it is to prevent the DNA found in a particular bacteria from performing its replicatory function. Insofar as we define a T-world as one in which every telos is realised, then both the bacterial DNA and the antibiotic cannot coexist in the same T-world. But since we have assumed a constant domain semantics, and since there are possible worlds in which both the bacterial DNA and the antibiotic exist (e.g., the actual world), this appears to throw the notion of a T-world into jeopardy.

I believe we may overcome the challenge posed by both compensatory and conflicting purposes via a multimodal teleological logic, in which the accessibility relation Ri is indexed to a particular biological system or human artefact, i. Instead of a single “common” accessibility relation R, there is a series Ri, Rj, Rk . . . , indexed to sets of teleological objects (e.g., the set of human eyes, the set of human ears, the set of hammers). The Kripke frame for the corresponding language, L, in which {□i|i ∈ I} represents the set of necessity operators of L, consists of a non-empty set of possible worlds G, and the binary relation Ri, for each i ∈ I. The satisfaction relation for □i is defined as follows:

(5.1) w ⊩ □i φ if and only if ∀u(Ri(w,u) → u ⊩ φ).

According to (5.1), φ is true in all T-worlds relative to some biological system or artefact i just in case φ holds in any world that stands in the relation R to some other world. Instead of speaking of T-worlds in which every telos is realised, we now speak of T-worlds indexed to some biological system or artefact, i, such that i always realises the telos for which it was selected or designed in the actual world.

On this view, the possible worlds in which the antibiotic performs its function constitutes the set of T-worlds indexed to that antibiotic, while the possible worlds in which the DNA of a particular bacterium realises its telos represents the set of T-worlds indexed to that bacterial DNA. Since each teleological object is now indexed to its own set of T-worlds, there can be no conflicting purposes within T-worlds. Thus, the coherence of the concept of a T-world is preserved. The multimodal account also avoids the problem posed by biological systems with compensatory purposes. Since the set of T-worlds indexed to some biological system i includes worlds in which some other biological system j fails to realise its telos, this allows i to increase the reproductive fitness of its host when i serves as a “back-up” system in the eventuality of j failing to realise its telos. The upshot is that on the multimodal account, there is no difficulty posed by cases of compensatory or conflicting purposes.

The 100th Philosopher's Carnival

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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.

UW Graduate Student Conference

THE 5TH BIENNIAL UNIVERSITY OF WASHINGTON
GRADUATE STUDENT CONFERENCE IN PHILOSOPHY
November 13 & 14, 2009
Theme: Moral Psychology

CONFERENCE SCHEDULE:

Friday

3:30 PM KEYNOTE ADDRESS

“Responsibility and Mental Agency”
Pamela Heironymi (UCLA)
Savery Hall, Room 264

5:30 PM RECEPTION (Savery Hall Third Floor Philosophy Department Table)


Saturday (all sessions in Savery Hall Room 264)

9:00 – 9:30 LIGHT BREAKFAST PROVIDED (Savery Hall Room 264)

9:30 – 10:20 SESSION 1: “Responsibility and Affective Skills in the Psychopath”
Garrett Pendergraft (University of California, Riverside)
COMMENTS: Janice Moskalik (University of Washington)

10:30 – 11:20 SESSION 2: “Irresistible Motivation”
Todd Beattie (Princeton University)
COMMENTS: Jason Benchimol (University of Washington)

11:30 – 12:20 SESSION 3: “Hard Feelings and Forgiveness”
Grant Rozeboom (Stanford University)
COMMENTS: Patrick Smith (University of Washington)

12:20 – 1:30 LUNCH

1:30 – 2:20 SESSION 4: “Evaluation without Hyper-intellectualisation”
Avery Archer (Columbia University)
COMMENTS: Rachel Fredericks (University of Washington)

2:30 – 3:20 SESSION 5: “Liberal Universalism and How We Understand the Past”
George Tsai (University of California, Berkeley)
COMMENTS: Amy Reed (University of Washington)

3:30 – 4:20 SESSION 6: “Is Self-Binding Morally Wrong?”
Jeff Sebo (New York University)
COMMENTS: Fareed Awan (University of Washington)

4:30 – 5:20 SESSION 7: “Why So Serious? An Inquiry On Racist Jokes”
Luvell Anderson (Rutgers University)
COMMENTS: Elizabeth Scarbrough (University of Washington) &
Jonathan Rosenberg (University of Washington)

5:30 – 7:00 BANQUET (Savery Hall Third Floor Philosophy Department Table)

8:00 – ? PARTY (at the “Philosophy House”)