When Popper wrote Logik der Forschung the notion of truth was problematic and he avoided talking about it until Tarski developed the use of metalanguages to permit a formal statement of the correspondence theory of truth that was robust in terms of modern logic. A good place to find Popper’s take on this is in the 1962 Addendum to volume 2 of The Open Society and its Enemies. It is also explained clearly in one of his lectures on the philosophy of science that he delivered each year for a decade or so at the London School of Economics (not yet published).
He wanted to be able to say that scientists seek the truth (without ever knowing we have found it) and he also wanted to be able to form a critical preference for one theory that was better than another, especially nearer to the truth. He tried to go beyond the intuitive idea of getting nearer to the truth with a formal definition of ‘truthlikeness’ or‘verisimilitude’ that combined the ideas of both truth and content.
In Conjectures and Refutations (1963) he defined or formalized the content with reference to the logical consequences of theories. The idea was to compare the ‘truth-content’ of a theory, which is the class of true propositions which may be derived from it, with the ‘falsity-content’ which is the class of the theory’s false consequences.
I am not aware of anyone ever actually doing the sums for any live theories so it is difficult to know what it really means but the problem for Popper came a few years later in the the 1970s when his colleague Davaid Miller and others revealed fundamental defects in the formal definition.
The details of the problems are explained in the Stanford Encyclopedia. The problems are (possibly) significant because verisimilitude was largely important in Popper’s system because of its application to theories which are known to be false. In this connection, Popper had written:
Ultimately, the idea of verisimilitude is most important in cases where we know that we have to work with theories which are at best approximations—that is to say, theories of which we know that they cannot be true. (This is often the case in the social sciences). In these cases we can still speak of better or worse approximations to the truth (and we therefore do not need to interpret these cases in an instrumentalist sense). (Conjectures and Refutations, 235).
The Stanford author (Thornton) explained that the deficiencies discovered by the critics were seen by many as devastating, precisely because the most significant of these related to the levels of verisimilitude of false theories.
In 1974, Miller and Tichý, working independently of each other, demonstrated that the conditions specified by Popper in his accounts of both qualitative and quantitative verisimilitude for comparing the truth- and falsity-contents of theories can be satisfied only when the theories are true. In the crucially important case of false theories, however, Popper’s definitions are formally defective. For while Popper had believed that verisimilitude intersected positively with his account of corroboration, in the sense that he viewed an improbable theory which had withstood critical testing as one the truth-content of which is great relative to rival theories, while its falsity-content (if it exists) would be relatively low, Miller and Tichý proved, on the contrary, that in the case of a false theory t2 which has excess content over a rival theory false t1 both the truth-content and the falsity-content of t2 will exceed that of t1. With respect to theories which are false, therefore, Popper’s conditions for comparing levels of verisimilitude, whether in quantitative and qualitative terms, can never be met.
This had a strange result because people who had previously attached little importance to the theory of verisimilitude suddenly came to see it as central to his philosophy of science, and consequently liked to think that the whole Popperian edifice had been subverted. I can recall the day in 1974 when a friend in Philosophy, a gentle and easygoing soul, approached me with malicious glee to announce that Popper’s program was in ruins! This seemed strange, but who was I to argue with a professional philosopher?
In Realism and the Aim of Science Popper replied to criticism (pp xxxv to xxxvii) with the argument that the notion that successive theories may be better approximations to the truth is not damaged by the failure of a formal definition. Two examples were (1) the progress from the earth-centred solar system to the sun-centred system with circular orbits to the system with elliptical orbits, and (2) the progress from Darwin’s ideas about heredity, to Mendel, to the steps that resulted in uncovering the genetic code.
“These examples show, I believe, that a formal definition of verisimilitude is not needed for talking sensibly about it.” (xxxvi)
He went on to expand on his view about the use and abuse of definitions.
I have often argued against the need for definitions. They are never really needed, and rarely of any use, except in the following sort of situation: we may by introducing a definition show that not only are fewer basic assumptions needed for a good theory but that our theory can exlain more than without the definition. In other words, a new definition is only of interest if it strengthens a theory. I thought that I could do this with my theory of the aims of science…[this did not work]. But the widely held view that scrapping this definition weakens my theory is completely baseless.
The assertion that my authority is damaged by this incident is obviously true, but I never claimed or wished to have any authority. The assertion that my theory is damaged has been advanced without even attempting to give a reason, and seems to me just incompetent.