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[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own. As proofreading is incomplete, you will find typos and other districting errors. I apologize in advance.]
Summary of
John Nolt
Logics
Part 5: Nonclassical Logics
Chapter 15: Mildly Nonclassical Logics
15.3 Supervaluations
15.3.1 [Basic technique of supervaluation]
Brief summary:
A three-valued semantics was one way to deal with a number of situations where bivalence was unsatisfactory. A third value, I or indeterminate, was used. But instead of that third value, we can keep just T or F, although certain formulas can be assigned neither of these two values. These cases of no value are called truth value gaps. To evaluate the truth value of formulas with component truth value gaps, we can use a technique invented by Bas van Fraassen called supervaluation. First we begin by making a truth table and putting in the values we know, but we place gaps where there is no value. This is called a partial valuation. Next, we fill those gaps with T or F such that every possible combination of values is given. These are called classical completions. Finally, we make a supervaluation of these classical completions in the following way. If all the classical completions compute the formula as true, then the supervaluation is true. If all the classical completions compute the formula as false, then the supervaluation is false. And if not all the classical completions are either entirely T or F, then the supervaluation does not assign a value to the formula. Here is the formal definition of supervaluation:
DEFINITION A supervaluational model of a formula or set of formulas consists of
1. a partial valuation S, which assigns to each sentence letter of that formula or set of formulas the value T, or the value F, or no value. We use the notation ‘S(Φ)’ to denote the value (if any) assigned to Φ by S.
2. A supervaluation VS of S that assigns truth values VS (Φ) to formulas Φ according to these rules:
VS (Φ) = T iff for all classical completions S′ of S, S′(Φ) = T;
VS (Φ) = F iff for all classical completions S′ of S, S′(Φ) = F;
VS assigns no truth value to Φ otherwise.
(Nolt 415-416, boldface his)
Summary
Nolt recalls that we can have certain motivations to reject simply two logical values, true and false, and to add some third one [see section 15.2]. But we have further motivations that would make us want to hold on to just the truth and falsity without adding a third or more. One solution would be to say that propositions can be either true, false, or neither true nor false. Sentences with this third option “are said to exhibit truth-value gaps” (Nolt 414 boldface his). [The idea here might be the following. We have reason to think that certain propositions do not fall neatly under the valuations of true or false. But we also have the intuition that truth is something that admits of no quantitative variation. It is absolute in some sense. So the notion of a third value might be mathematically modelable, but philosophically speaking it is too difficult to conceive and believe strongly in the reality of. So this solution keeps the idea that there are no other logical values than true and false. But it deals with the problematic cases by saying that they simply have neither of these values, rather than saying they have a third value.] Nolt will now examine how we can calculate truth values when a component proposition lacks a value (414).
[Recall Bochvar’s three-valued semantics from section 15.2 where any instance of a constituent indeterminate value would make the whole formulation indeterminate.] Nolt notes that one option “would be to declare all such formulas truth-valueless. The result would be just like Bochvar's three-valued semantics, except that in the truth tables where Bochvar’s logic has an ‘I’, this semantics would have a blank” (415). [There were other ways to arrange the truth values for three-valued semantics,] or instead we “could create truth tables like those for other forms of three-valued logic” (415). But, Nolt says, “this approach is hardly novel, differing in no essential respect from three-valued logics themselves” (415). [The problem with that might be the following. Suppose we simply mimicked the three-value truth tables, only replacing the ‘I’ value with a blank. We might then ask what really the difference would be between them other than how we philosophically interpret the meaning of the third option. In other worlds, this “gap” option would be little more than a different way to interpret the indeterminately three-valued semantics.]
Nolt says that instead we should consider Bas van Fraassen’s technique of supervaluation, which “unlike most multivalued logics, preserves the validity of all classical inference patterns” (Nolt 415). The first thing we note about supervaluations is partial value assignment, where we assign to sentences letters either T, F, or no value.
In propositional logic, a supervaluational semantics assigns to sentence letters the value T, or the value F, or no value at all. We shall call such an assignment a partial valuation. (Note that at one extreme some partial valuations assign truth values to all the sentence letters of a formula or set of formulas and, at the other, some assign no truth values at all.)
(Nolt 415, boldface his)
[In other cases, we compute the truth value of complex formulas on the basis of the truth values of their component formulas.] In supervaluations, we will calculate the values of complex formulas by means of classical completions of partial valuations, rather than by calculating by means of “truth tables directly from the truth values of their components” (415).
[The basic idea with this technique seems to be the following. Although we might have truth gaps for certain constituent parts of a formula, that may not in the end have any effect on its overall value, were we to give it every possible classical valuation. And thus for some cases, we can determinately say that it maintains classical truth or falsity, despite the gaps in its internal values. So we begin with a partial valuation of a complex formula, meaning that some of the sentence letters will not originally be assigned a T or F value. The second step is a classical completion, where we exhaust every possible arrangement for these blank values to receive either T or F. The next thing we do is we see if regardless of every possible substitution, does the whole formula compute as either T or F, in all cases? If so, we say that the formula is T or F, depending on what all the evaluations compute it as. If instead they are not all of one or the other value, then we say that it has no value. Giving it this final valuation based on the classical completions of its partial valuation is called its supervaluation.]
A more interesting method – and one that, unlike most multivalued logics, preserves the validity of all classical inference patterns-is the technique of supervaluations invented by Bas van Fraassen. In propositional logic, a supervaluational semantics assigns to sentence letters the value T, or the value F, or no value at all. We shall call such an assignment a partial valuation. (Note that at one extreme some partial valuations assign truth values to all the sentence letters of a formula or set of formulas and, at the other, some assign no truth values at all.) The truth values of complex formulas, however, are not calculated by truth tables directly from the truth values of their components. Rather, the calculation takes into account all of what are called the classical completions of a given partial valuation.
Let S be any partial valuation. Then a classical completion of S is a classical valuation (one that assigns each sentence letter in the relevant formula or set of formulas one and only one of the values T or F) that fills in all the truth-value gaps left by S. In other words, a classical completion of S does not change any truth-value assignment to a sentence letter that has already been made by S, but merely supplements assignments made by S, giving each sentence letter of the given formula or set of formulas a truth value. Since each truth-value gap can be filled in by the assignment either of T or of F, each partial valuation, unless it is classical to begin with, has more than one classical completion. Consider, for example, the formula ‘(P∨Q)&(R∨S)’, and let S be the partial valuation of this formula that assigns the value T to both ‘P’ and ‘R’ but no value to ‘Q’ or ‘S’.
(Nolt 415, emphases his)
[Let us first try to give the partial valuation for this situation in a table format.
As we can see, we have only entered the data that we are sure of. The rest is left out, and thus this is a partial valuation. In the next step, we will fill in every possible combination of values for Q and S, and then we evaluate them on the basis of normal classical valuation rules, in this case for disjunction and finally for conjunction. The result will be that all possible classical completions of this partial valuation will make it true, which means that the gaps would not interfere with classical evaluation of this formula, as it will not matter what values the gaps might have.]
Then S has with respect to this formula four classical completions, corresponding to the four ways of assigning truth value to the sentence letters ‘Q’ and ‘R’. Each classical completion of S is represented by a horizontal line in the truth table below:
The columns under ‘P’ and ‘R’ list only T’s, because these are the values assigned by the partial valuation S, and they are retained in each classical completion.
To determine the truth value of the compound formula ‘(P∨Q)&(R∨S)’, we expand S into a new nonclassical valuation VS called the supervaluation of S. | This is done by calculating the truth value of ‘(P∨Q)&(R∨S)’ on each of the classical completions of S, using the valuation rules of classical logic, as in the table above. If ‘(P∨Q)&(R∨S)’ is true on each of these classical completions (as the table shows that it is), then ‘(P∨Q)&(R∨S)’ is true on the supervaluation VS. If it had been true on none of them, then it would have been false on VS. And if it had been true on some but not others, then it would have been assigned no truth value on VS.
A supervaluation, then, is constructed in two stages. First, we define a partial valuation S, which assigns to each sentence letter the value T, or the value F, or no value. Next, at the second stage, we construct all the classical completions of S and use the classical valuation rules to calculate the truth values of complex formulas on each of these classical completions. A formula Φ (whether atomic or complex) is then assigned a truth value by the supervaluation VS if and only if all the classical completions of S agree on that truth value; if not, VS assigns no value to Φ. More formally:
DEFINITION A supervaluational model of a formula or set of formulas consists of
1. a partial valuation S, which assigns to each sentence letter of that formula or set of formulas the value T, or the value F, or no value. We use the notation ‘S(Φ)’ to denote the value (if any) assigned to Φ by S.
2. A supervaluation VS of S that assigns truth values VS (Φ) to formulas Φ according to these rules:
VS (Φ) = T iff for all classical completions S′ of S, S′(Φ) = T;
VS (Φ) = F iff for all classical completions S′ of S, S′(Φ) = F;
VS assigns no truth value to Φ otherwise.
(Nolt 415-416, boldface his)
As we can see, supervaluation may leave truth-value gaps in complex formulas. For example, if we have a disjunction, and one term is assigned false but the other a gap, then the classical completion will not all have the same value for the disjunction (416).
[For the next point, recall Kleene’s three-valued semantics from section 15.2.
(Nolt 412)
] We might be tempted to say that supervaluations follow the pattern of Kleene’s semantics, where we can simply take the value ‘I’ to be equivalent to a gap. So consider disjunction in the Kleene system, where both disjuncts are indeterminate. That makes the whole disjunction indeterminate. Similarly, the classical completion for a partial evaluation where both disjuncts are valueless would one giving false (where both disjuncts are false) and the rest giving true, and thus the whole would be valueless, just as in Kleene the whole is indeterminate. However, this does not hold in all cases, as for example with the formula ‘P∨~P’, where ‘P’ is either indeterminate or has no truth-value. [As we see in the Kleene system, when a formula is I, then its negation is I. Thus both P and ~P would be I in our example. And when we have a disjunction where both disjuncts are I, as in our case here, then the whole disjunction is I. However, for supervaluations, the negation of a formula with no value also has no value. For, supposing it is true, then the negation is false. And supposing it is false, then the negation is true. There is not a consistent outcome, so P has no value and ~P has no value. Then, when we make a classical completion of ‘P∨~P’, we see that it is always true, no matter what we assign it (given that both P and ~P are gaps).
Thus supervaluations are not equivalent to Kleene’s three-valued semantics.]
Nolt then defines validity and consistency for supervaluations. “a formula is valid iff it is true on all supervaluations; and a sequent is valid iff there is no supervaluation on which its premises are true and its conclusion is untrue. (And, of course, there are two ways of being untrue: being false and being truth-valueless.)” (Nolt 417).
Nolt then shows how “the logic that results from these stipulations is just classical propositional logic – even though, as we have seen, its semantics differs significantly from that of classical logic. Thus, for example, an inference is valid on supervaluational semantics if and only if it is valid on classical bivalent semantics, despite the fact that the former, but not the latter, permits truth-value gaps” (Nolt 417).
[Nolt next proves this with a complicated metatheorem and proof (417-418). After that he extends supervaluation semantics to predicate logic (418-419). These parts are left for another summary.]
From:
Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.
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