# Refined Verisimilitude

• Sjoerd D. Zwart
Chapter
Part of the Synthese Library book series (SYLI, volume 307)

## Abstract

The difference between verisimilitude and truthlikeness definitions is presented in Chapter 1; I then expounded on this difference in Chapters 2–3. Together, the first three chapters constitute the expository part of this book. In Chapters 4–5, I discussed the epistemic problem of approach-to-the-truth, and Miller’s extensional substitution argument. Putting forward the solution to Miller’s puzzle, in the preceding chapter, I paved the way for my refined verisimilitude proposal, which is the subject of this chapter. In a way, Chapters 1–5 can be viewed as preparatory steps leading up to my new approach-to-the-truth proposal presented in Section 6.4. As a prelude I first introduce the refined Δ-verisimilitude definition that merges the Δ-distances on the Lindenbaum and constituent algebra of a language. Then we will see that the ≤+ ordering of the ℒ-propositions can be combined with the total preorder on the constituents of the language. I shall partition the Lindenbaum algebra in equivalence classes of propositions of the same distance to the truth, and I shall prove that the merger of the two orderings is compatible with this partition and orders the equivalence classes. I call the resulting ordering the refined verisimilitude proposal. A more detailed summary of this chapter can be found at the end of Section 6.1.

## Keywords

Equivalence Class Partial Order Boolean Algebra Transitive Closure Atomic Proposition
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## References

1. 1.
Interestingly, in private conversation, Niiniluoto contended that his two dimensional truthlikeness continuum does not subsume my refined verisimilitude proposal.Google Scholar
2. 3.
E.g. Chang and Keisler (1973), p.46–47.Google Scholar
3. 4.
Although T, yr E Sent(2)/H are equivalence classes of sentences in the sequel, I will use them as if they are a single sentence and write ç wGoogle Scholar
4. 5.
Clearly, as the constituent and Lindenbaum algebra both are Boolean algebras, the more abstract version of our proposals only uses set algebras; it disregards the identity of the atomic propositions.Google Scholar
5. 8.
See Davey and Priestley (1990), p. 25.Google Scholar
6. 9.
In case the language is finite and the truth T is complete, the truth ideal consists of the contradiction and T. The truth ideal is less trivial if T is incomplete or unaxiomatizable.Google Scholar
7. 10.
See e.g. Davey and Priestley (1990), p.145.Google Scholar
8. 11.
For more details see Davey and Priestley (1990), p.145).Google Scholar
9. 12.
See e.g. Davey and Priestley (1990), p.18–19.Google Scholar
10. 13.
See e.g. Stoll (1961), p.263.Google Scholar
11. 14.
Intuitively, the verisimilitude of a theory is the similarity between that theory and the true theory regarding empirical content. Consequently, it make no sense to contribute verisimilitude to the tautology and the contradiction, which lack empirical content.Google Scholar
12. 15.
Counterbalancing logical strength and false consequences is less counter intuitive if the truth is complete; it might even be helpful in language dynamics, where the choice of conceptual frameworks is at stake.Google Scholar
13. 16.
This might be the beginning of a compound definition if the truth is incomplete.Google Scholar
14. 19.
A relation R is a weak ordering of the set A iff R is transitive and stongly connected in A.“ Google Scholar
15. 20.
See e.g. Suppes (1960), p.84.Google Scholar
16. 21.