Some Remarks on Popper’s Qualitative Criterion of Verisimilitude

Abstract

The paper sets up a general framework for defining the notion of verisimilitude. Popper’s own account of verisimilitude is then located within this framework; and his account is defended on the grounds that it can be seen to provide a reasonable structural or Pareto criterion, rather than a substantive criterion, of verisimilitude. Some other criteria of verisimilitude that may be located within the framework are also considered and their relative merits compared.

Appendix

The present appendix is not self-contained but presupposes some of the definitions already given in the main body of the text.

We first establish the sufficiency of Popper’s criterion given principles P1–P5 from Sect. 3. Given P1–P5, the following facts will obtain:

• (1) True(P) & P ≥ Q ⇒ P ≥_t Q

Proof

Suppose P ≥ Q and that P (and hence Q) is true. By P1, @ ≥_t Q. So by P4, P ∨ @ ≥_t P ∨ Q. But, given that P is true, P ∨ @ = P and, given that P ≥ Q, P ∨ Q = Q. Hence P ≥_t Q.□

• (2) Q⁺ ≥_t Q.

Proof

By P1, @ ≥_t Q. By P4, Q ∨ @ ≥_t Q ∨ Q. But Q ∨ @ = Q⁺ and Q ∨ Q = Q. Hence Q⁺ ≥_t Q.□

• (3) True(P) & P ≥ Q⁺ ⇒ P ≥_t Q.

Proof

Suppose P is true and P ≥ Q⁺. P ≥_t Q⁺ by (1). But Q⁺ ≥_t Q by (2); and so P ≥_t Q by P5.□

• (4) True(P) ⇒ Q ≥_f P.

Proof

By P2 and P4, P ∨ Q ≥_f P ∨ @. By P3 and P4, ⊥ ∨ Q ≥_f P ∨ Q; and so by P5, ⊥ ∨ Q ≥_f P ∨ @. But ⊥ ∨ Q = Q and, given that P is true, P ∨ @ = P. Hence Q ≥_f P.□

From these facts, we can establish the sufficiency of Popper’s criterion:

Theorem 1

Given P1–P5 and the right-to-left direction of (Df_v),

• P ≥_v Q according to Popper’s criterion ⇒ P ≥_v Q.

Proof

We easily confirm that it suffices to show:

• True(P) & P ≥ Q⁺ ⇒ P ≥_v Q

since this covers both the TT and the TF case.

So suppose P is true and P ≥ Q⁺. By (3) above, P ≥_t Q; by (4), Q ≥_f P; and so by the right-to-left direction of (Df_v), P ≥_v Q.□

We turn to the characterization of Hilpinen’s criterion (Sect. 4) in terms of truth- and falsity-content.

Lemma 2

• (i) P ${ \succcurlyeq }$_t Q iff P↑ ⊆ Q↑

• (ii) P ${ \succcurlyeq }$_f Q iff P↓ ⊆ Q↓

Proof

(i) Suppose P ${ \succcurlyeq }$_t Q and pick a wʹ ∈ P↑. Then wʹ ${ \succcurlyeq }$ w for some w ∈ P. So there is a v ∈ Q such that w ${ \succcurlyeq }$ v. By ${ \succcurlyeq }$ transitive, wʹ ${ \succcurlyeq }$ v; and so wʹ ∈ Q↑.

Now suppose P↑ ⊆ Q↑ and pick a w ∈ P. Since P ⊆ P↑, w ∈ Q↑. So for some v ∈ Q, w ${ \succcurlyeq }$ v, as required.

(ii) Similar.□

We now consider the relationship between Popper’s and Hilpinen’s criterion for verisimilitude:

Lemma 3

(Soundness) If P ≥_t,f,v Q in the world space S then P ≥_t,f,v Q in any ranked space $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$.

Proof

The case for v will follow from the cases for t and f. For if P ≥_v Q in S then P ≥_t Q and P ≥_f Q in S; so P ≥_t Q and P ≥_f Q in $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$; and so P ≥_v Q in $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$.

Let us first consider the case in which P ≥_t Q. We distinguish two subcases.

Subcase 1

P false or Q true. Suppose P ≥_t Q in S. Then P ⊆ Q. So P↑ ⊆ Q↑; and hence, by lemma 2(i), P ≥_t Q in $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$.

Subcase 2

P true, Q false. Suppose P ≥_t Q in S. Then P ⊆ Q⁺. So P↑ ⊆ Q⁺↑. But Q⁺↑ = (Q ∪ {@})↑ = Q↑ since @ ∈ Q↑; and so P ≥_t Q in $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$.

Let us now consider the case in which Q ≥_f P. We distinguish three sub-cases:

Subcase 1

P true. Suppose Q ≥_f P. Since this holds automatically, we must show Q ≥_f P whenever P is true. Take any v ∈ Q. Since P is true, @ ∈ P. But @ ${ \succcurlyeq }$ v by Weak Centering.

Subcase 2

P false, Q true. This case is trivial since it is not one in which Q ≥_f P can hold.

Subcase 3

P, Q false. Suppose Q ≥_f P in S. Then Q ⊆ P. So P↓ ⊆ Q↓; and hence P ≥_f Q in $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$ by lemma 2(ii).□

Lemma 4

(Completeness) For any world-space S = (W, @):

if P ≥_t,f,v Q in every linear ranked space $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$ then P ≥_t,f,v Q in S.

Proof

As before, it suffices to prove the result for the alethic values t and f.

Case A

Suppose that P ≱_t Q in a world-space S = (W, @). Again, we distinguish sub-cases according to the truth-values of P and Q.

Subcase A1

P is false or Q true. Then P ⊆ Q. So for some w₀, w₀ ∈ P but w₀ ∉ Q. Since P is false or Q true, w₀ ≠ @. Choose ${ \succcurlyeq }$ so that w₀ ≺ w and w ≈ @ whenever w ≠ w₀. Thus the worlds are partitioned into two blocks, with w₀ the most distant world and all others the near worlds, as depicted below:

Then for no v ∈ Q is w₀ ${ \succcurlyeq }$ v; and consequently, P ≱_t Q in $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$ as so defined.

Subcase A2

P true, Q false. Then not P ⊆ Q⁺. So for some w₀, w₀ ∈ P but w₀ ∉ Q⁺. But since @ ∈ Q⁺, w₀ ≠ @. We may then proceed in the same manner as Case (1).

Case B

Suppose that Q ≱_f P in the world-space S = (W, @). We distinguish three sub-cases:

Subcase B1

P true. This case cannot arise.

Subcase B2

P false, Q true. Since never Q ≥_f P, we must show that Q ≥_f P always fails in some ranked space $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$. Since Q is true, @ ∈ Q and, since P is false, @ ∉ P. Choose ${ \succcurlyeq }$ so that w ≺ @ whenever w ≠ @ and w ≈_v whenever w, v ≠ @. Thus the worlds are again partitioned into two blocks, but this time with all worlds other than @ the distant worlds and @ the near world:

Then for no w ∈ P is w ${ \succcurlyeq }$ @ and consequently Q ≱_f P in $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$ as so defined.

Subcase B3

P, Q false. Then Q ⊆ P and so, for some world w₀, w₀ ∈ Q but w₀ ∉ P. Choose ${ \succcurlyeq }$ so that w₀ ≈ @, w ≺ @ whenever w ≠ w₀, @ and w ≈ v whenever w, v ≠ w₀, @, as depicted below:

Then for no w ∈ P is w ${ \succcurlyeq }$ w₀ and, consequently, Q ≱_f P in $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$ as so defined.□

Theorem 5

(Soundness and completeness) For any propositions P and Q in a world space S:

• P ≥_t/f/v Q in S iff P ≥_t/f/v Q in every $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$.

Proof

From Lemmas 3 and 4.□

Note that the proof only uses linearly ranked spaces with two blocks. If we insist upon Strong Centering, then the number goes up to three. Indeed, we may provide an exact characterization of the classes of Hilpinen rankings which give rise in this way to the Popperian ordering. Let X be a class of linear rankings on a world-space S. We say that such a class X is t/f/v-adequate if P ≥_t/f/v Q in S whenever P ≥_t/f/v Q in every $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$ for which ${ \succcurlyeq }$ ∈ X (by Soundness, the converses will also hold). Given a linear ranking ${ \succcurlyeq }$ on S, say that the world w is worst if w ≺_v for any world v ≠ w, i.e. w is most distant from actuality, and say that w is best if w $\succ$ v for any non-actual world v ≠ w, i.e. w is closer to actuality than any non-actual world.

Theorem 6

Let X be a class of linear rankings on a given world space S . Then:

• 1. X is t-adequate iff each non-actual world is worst in some ranking ${ \succcurlyeq }$ of X.

• 2. X is f-adequate iff each world is best in some ranking ${ \succcurlyeq }$ of X.

• 3. X is v-adequate iff each non-actual world is worst in some ranking ${ \succcurlyeq }$ of X and each world is best in some ranking ${ \succcurlyeq }$ of X.

Proof

(i) Suppose each non-actual world is worst in some ranking ${ \succcurlyeq }$ of X. We may then follow the method of proof in Subcases (A1) and (A2) of the proof of theorem 4 to show that P ≥_t Q fails in $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$.

Suppose that some non-actual world w₀ is worst in none of the rankings ${ \succcurlyeq }$ of X. Let P = {w₀} and Q = W − {w₀}. Then P ≥_t Q in every $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$ for which ${ \succcurlyeq }$ ∈ X. However, P ≥_t Q fails in S.

(ii) Suppose any world is best in some ranking of ${ \succcurlyeq }$ of X. To take care of subcase (B2), we let that world be @ and, to take care of (B3), we let it be w₀.

Suppose that some world w is best in none of the rankings ${ \succcurlyeq }$ of X. We distinguish two cases. (a) w = @. Let P = {@} and Q = W − {@}. Then P ≥_f Q in every $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$ for which ${ \succcurlyeq }$ ∈ X while P ≥_f Q fails in S. (b) w ≠ @. Let P = {w} and Q = W − {w, @}. Then again P ≥_f Q in every $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$ for which ${ \succcurlyeq }$ ∈ X while P ≥_f Q fails in S.

(iii) From (i) and (ii) since X is v-adequate iff it is t-adequate and f-adequate.□

Corollary 7

For no linear rank on a world space S with at least three worlds is P ≥_v Q in S identical to P ≥_v Q in $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$.

Proof

We let the X in Theorem 6 be {${ \succcurlyeq }$}. The result then follows from the fact that no non-actual world in S can be both worst and best with respect to ${ \succcurlyeq }$.□

Let the discrete ordering ${ \succcurlyeq }$ on a world space be {(w, w): w ∈ W} ∪ {(@, w): w ∈ W} and let the discrete ranked space $\varvec{S}^{{\mathbf{ \succcurlyeq }}}$ be the one in which ${ \succcurlyeq }$ is the discrete ordering on S.

Theorem 8

The Popper relations ≥_t/f/v are the same as the Hilpinen relations ≥_t/f/v over a discrete ranked space.

Proof

It suffices to show that in a discrete ranked space,

• P↑ = P = P^t when P is true

• P↓ = ⊤ = P^f when P is true

• P↑ = P⁺ = P^t when P is false

• P↓ = P = P^f when P is false.

But these facts are readily verified.□