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| Truthlikeness |
Truth is the aim of inquiry. Despite this, progress in an inquiry does not always consist in supplanting falsehoods with truths. The history of science is replete with cases of falsehoods supplanting other falsehoods.
If such transitions are to constitute epistemic progress, then it must be possible for one falsehood better to realize the aim of inquiry—be more truthlike, be closer to the truth, or have more verisimilitude—than another. The notion of “truthlikeness” is thus fundamental for any theory of knowledge that endeavors to take our epistemic limitations seriously without embracing epistemic pessimism.
Given that truthlikeness is not only a much-needed notion but rich and interesting, it is surprising that it has attracted less attention than the simpler notion of truth. The explanation is twofold. First, if knowledge requires truth, then falsehoods cannot constitute knowledge. The high value of knowledge has obscured other epistemic values such as the comparative value of acquiring more truthlike theories.
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| theory of probability |
Second, if knowledge requires justification, then the notion of probability often takes center stage. There has been a long and deep confusion between the notions of subjective probability (seemingly true) and the notion of truthlikeness. This, together with the high degree of development of the theory of probability, obscured the necessity for a theory of truthlikeness.
Sir Karl Popper was the first to notice the importance of the notion. Popper was long a lonely advocate of both scientific realism and fallibilism: that, although science aims at the truth,most theories have turned out to be false and current theories are also likely to be false.
This seems a bleak vision indeed and fails to do justice to the evident progress in science. Popper realized that the picture would be less bleak if a succession of false (and falsified) theories could nevertheless constitute steady progress toward the truth.
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| observational accuracy |
Further, even if actually refuted by some of the data, the general observational accuracy of a false theory might be good evidence for the theory’s approximate truth, or high degree of truthlikeness. That our theories, even if not true, are close to the truth, may be the best explanation available for the accuracy of their observable consequences.
Note that truthlikeness is no more an epistemic notion than is truth. How truthlike a theory is depends only on the theory’s content and the world, not on our knowledge. The problem of our epistemic access to the truthlikeness of theories is quite different from the logically prior problem of what truthlikeness consists in.
Popper proposed a bold and simple account of truthlikeness: that theory B is more truthlike than theory A if B entails all the truths that A entails, A entails all the falsehoods that B entails, and either B entails at least one more truth than A or A entails at least one more falsehood than B.
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| more truthlike |
This simple idea undoubtedly has virtues. Let the Truth be that theory that entails all and only truths (relative to some subject matter). On Popper’s account the Truth is more truthlike than any other theory, and that is as it should be.
The aim of an inquiry is not just some truth or other. Rather, it is the truth, the whole truth, and nothing but the truth about some matter—in short, the Truth—and the Truth realizes that aim better than any other theory.
The account also clearly separates truthlikeness and probability. The Truth generally has a very low degree of (subjective) probability, but it definitely has maximal truthlikeness. Furthermore, the account yields an interesting ranking of truths—the more a truth entails, the closer it is to the Truth.
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| theory of motion |
Popper’s account also has some defects. For example, it does not permit any falsehood to be closer to the Truth than any truth. (Compare Newton’s theory of motion with denial of Aristotle’s theory.) But its most serious defect is that it precludes any false theory being more truthlike than any other.
The flaw is simply demonstrated. Suppose theory A entails a falsehood, say f, and we attempt to improve on A by adding a new truth, say t. Then the extended theory entails both t and f and hence entails their conjunction: t&f. But t&f is a falsehood not entailed by A.
Similarly, suppose A is false and we attempt to improve it by removing one of its falsehoods, say f. Let g be any falsehood entailed by the reduced theory B. Then g f is a truth entailed by A but not B. (If B entailed both g and g f, it would entail f.) So truths cannot be added without adding falsehoods, nor falsehoods subtracted without subtracting truths.
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| quantitative measures |
Maybe this lack of commensurability could be overcome by switching to quantitative measures of true and false logical content. Indeed, Popper proposed such accounts, but the problem they face is characteristic of the content approach, the central idea of which is that truthlikeness is a simple function of two factors—truth-value and logical content/strength.
If truthlikeness were such a function, then among false theories truthlikeness would vary with logical strength alone. There are only two well-behaved options here: Truthlikeness either increases monotonically with logical strength, or else it decreases. But strengthening a false theory does not itself guarantee either an increase or a decrease in truthlikeness.
If it is hot, rainy, and windy (h&r&w), then both of the following are logical strengthenings of the false claim that it is cold (~h): It is cold, rainy, and windy (~h&r&w); it is cold, dry, and still (~h&~r&~w). The former involves an increase, and the latter a decrease, in truthlikeness.
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| actual world |
A quite different approach takes the likeness in truthlikeness seriously. An inquiry involves a collection of possibilities, or possible worlds, one of which is actual. Each theory selects a range of possibilities from this collection—that theory’s candidates for actuality. A proposition is true if it includes the actual world in its range. Each complete proposition includes just one such candidate.
The Truth, the target of the inquiry, is the complete true proposition—that proposition that selects the actual world alone. If worlds vary in their degree of likeness to each other, then a complete proposition is the more truthlike the more like actuality is the world it selects. This is a promising start, but we need to extend it to incomplete propositions.
The worlds in the range of an incomplete proposition typically vary in their degree of likeness to actuality, and the degree of truthlikeness of the proposition should be some kind of function thereof: average likeness is a simple suggestion that yields intuitively pleasing results. The framework can also be used in the analysis of related notions such as approximate truth or closeness to being true.
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| cutting the complexity |
There are two related problems with this program. The first concerns the measure of likeness between worlds. It would be a pity if this simply had to be postulated. The second concerns the size and complexity of worlds and the number of worlds that propositions typically select. Fortunately, there is available a handy logical tool for cutting the complexity down to a finite, manageable size. We can work with kinds of worlds rather than whole words.
The kinds at issue are specified by the constituents of first-order logic, a special case of which are the maximal conjunctions of propositional logic (like h&r&w, ~h&r&w, ~h&~r&~w). Constituents have two nice features. First, each depicts in its surface structure the underlying structure of a kind of world.
And, second, like the propositional constituents, they are highly regular in their surface structure, enabling degree of likeness between constituents to be extracted. (The world in which it is cold, rainy, and windy [~h&r&w] is more like the world in which it is hot, rainy, and windy [h&r&w] than it is like the world in which it is cold, dry, and still [~h&~r&~w]. In the propositional case, just add up the surface differences.)
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| The possibilities |
Since every statement is logically equivalent to a disjunction of constituents, we have here the elements of a quite general account of truthlikeness, one that can be extended well beyond standard first-order logic.
Not just any features count in a judgment of overall likeness. Such judgments clearly presuppose a class of respects of comparison. The possibilities specified by h&r&w and ~h&r&w differ in one weather respect and agree on two, whereas those specified by h&r&w and ~h&~r&~w differ in all three.
But now consider the following two states (where ∫ is the material biconditional): hot ∫ rainy, and hot ∫ windy. The possibility specified by h&r&w can equally be specified by h&(h ∫ r)&(h ∫ w); ~h&r&w by ~h&~(h ∫ r)&~(h ∫ w); and ~h&~r&~w by ~h&(h ∫ r)&(h ∫ w).
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| intuitive judgments |
Counting differences in terms of these new features does not line up with our intuitive judgments of likeness. Unless there is some objective reason for counting the hot-rainy-windy respects rather than the hot-(hot ∫ rainy)-(hot ∫ windy) respects, truthlikeness (unlike truth) seems robbed of objectivity.
This is the main objection to the likeness program. If sound, however, it would reach far indeed, for perfectly analogous arguments would establish a similar shortcoming in a host of important notions—similarity in general, structure, confirmation, disconfirmation, fit of theory to data, accuracy, and change.
The advocate of the objectivity of such notions simply has to grasp the nettle and maintain that some properties, relations, and magnitudes are more basic or fundamental than others. Realists, of course, should not find the sting too sharp to bear.
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| simply has to grasp |