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[The following is summary. All boldface in quotations are mine 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 16: Radically Nonclassical Logics
16.1 Infinite Valued and Fuzzy Logics
Brief summary:
There are vague predicates or concepts which can potentially lead to counter-intuitive inferences. Consider for example the vague predicate in this inference:
A global population of 1,000,000,000 is sustainable.
If a global population of 1,000,000,000 is sustainable, so is a global population of 1,000,000,001.
∴ A global population of 1,000,000,001 is sustainable.
(Nolt 420)
Suppose we reiterate the premises, each time building from the prior conclusion’s numerical value, and adding one more in the process. After a while, the population number will get very large, and the conclusion will no longer be true (it will not satisfy the predicate any more). An infinite-valued semantics can deal with these situations. It allows us to assign partial values to propositions, such that instead of true and false we have 0, 1, and all the decimal values between. In our example, each iteration would receive slightly less of a truth value, and so eventually the reiterations would arrive at 0, corresponding to our intuition that their truth value should decrease as the population number increases. A common valuation scheme is:
1. V(¬Φ) = 1 – V(Φ)
2. V(Φ & Ψ) = min(V(Φ), V(Ψ))
3. V(Φ ∨ Ψ) = max(V(Φ), V(Ψ))
4. V(Φ → Ψ) = 1 + min(V(Φ), V(Ψ)) - V(Φ)
5. V(Φ ↔ Ψ) = 1 + min(V(Φ), V(Ψ)) - max((V(Φ), V(Ψ))
Consider a predicate like “is red”. Suppose we add the following argument to get “fresh blood is red”. This is clearly true. But what about “a sunset is red”? This is partly true. That means the set of things that belong to the predicate “is red” has items whose membership is not entire. They have a certain degree of membership, and that degree matches the truth value of the proposition predicating them. In other words, if “a sunset is red” has the truth value 0.2, that means sunsets only have a membership degree of 0.2 in the set of red things. These are fuzzy sets (that is, sets whose membership is a matter of degree), and they were invented by Lofti Zadeh. He also applied fuzzy sets to the logical values that can be assigned, such that a range of values would be assigned and certain values in that range are themselves assigned a partial value for their degree of membership in that range of truth values. Such a semantics based on fuzzy truth values is called a fuzzy logic. While such fuzzy systems involve perhaps too much complexity and arbitrarity, they have proven useful for artificial intelligence programming, and they have also appealed to people who are wary of too much logical or conceptual precision.
Summary
Nolt first has us consider the following argument [quoting]:
A global population of 1,000,000,000 is sustainable.
If a global population of 1,000,000,000 is sustainable, so is a global population of 1,000,000,001.
∴ A global population of 1,000,000,001 is sustainable.
(Nolt 420)
This argument seems valid, because it simply uses modus ponens. And “the premises are true – or at least almost true” (420). But what if we reiterate this inference 999 billion times, in each instance beginning with the previously increased figure and then adding yet another 1 to it? We would then conclude that a global population of one trillion is sustainable. But surely it is not.
Nolt explains that the problem arises because we begin with “almost true” premises that take the form:
If a global population of n is sustainable, so is a global population of n +1.
(Nolt 421)
Nolt says that at the beginning of the iterations, the conclusions that we draw are “either wholly true or approximately true” (421). But with each further inference, “the conclusions become less and less true so that by the end of the sequence we arrive at a conclusion that is wholly false” (421).
Nolt notes that we cannot pinpoint one particular inference in the sequence and claim that it is the precise source where the chain enters into error. “Rather, there is a gradual progression from truth into error” (421). He identifies the problem in this example as being that we are using a vague predicate, “is sustainable” (421). While there are clear cases where the populations are so great that they are obviously unsustainable and also ones where they are no doubt sustainable, there are also
intermediate cases in which it is ‘sort of true’ but ‘sort of false’ that the population is sustainable. Our vague notion of sustainability defines no sharp boundary at which a population of n is sustainable, but a population of n + 1 is not. Rather, as the numbers increase it becomes less and less true that the population is sustainable.
(421)
If we see the situation in these terms, then we are also thinking of truth as something that admits of degrees. “There are, it appears, not just two truth values, but a potential continuum of values, from wholly false to wholly true” (421). [We could then assign propositions any from an infinitely varying range of values.] “If we take this gradation of truth value seriously, the result is an infinite valued semantics” (421).
Since we are dealing with a range of quantitative variation between true and false, we will use 0 for false, 1 for true, and all degrees of variation between as decimals. Nolt explains that we now have many different options for how to revise our valuation rules. But he will introduce us to one of the simplest ones (421).
[In this semantics, a proposition’s negation seems to have the inverse value.]
If Φ is wholly true, then ¬Φ is wholly untrue and vice versa. Likewise, it seems reasonable to suppose that if Φ is three-quarters true, ¬Φ is only one-quarter true. Thus, as a general rule ¬Φ has all the truth that Φ lacks and vice versa. More formally,
1. V(¬Φ) = 1 – V(Φ)
(Nolt 421)
This also means that “the double negation of a formula has the same truth value as the formula itself” (421).
[In a conjunction, if even one conjuct is false, the whole conjunction is false, even if the other one is true. So we might think of the mechanics here that the lowest value drags the value of the whole down to it.]
A conjunction would seem to be true as the least true of its conjuncts. The truth conditionals for conjunctions are best expressed using the notation ‘min(x, y)’ to indicate the minimum or least of the two value x and y – or the value both ‘x’ and ‘y’ exress if x = y. Thus for example, min(0.25, 0.3) = 0.25 and min(1,1) = 1. The valuation rule for conjunctions, then, is
2. V(Φ & Ψ) = min(V(Φ), V(Ψ))
(Nolt 422)
[Now with a similar sort of thinking, note how normally in a disjunction, the whole disjunction will be true when at least one is true, even if the other is false. In other words, it is as if the highest value pulls the value of the whole disjunction up to it.]
Disjunctions are true as the most true of their disjuncts. This idea may be expressed by the notation ‘max(x, y)’, which indicates the maximum or greatest of the two values x and y, or the value both variables express if x = y:
3. V(Φ ∨ Ψ) = max(V(Φ), V(Ψ))
(Nolt 422)
[The reasoning behind the conditional valuation is a bit trickier for me to grasp, so I will guess. Recall how for conditionals that if the antecedent is false, then the whole conditional is true. Thus some degree of falsity in the antecedent is not a ‘problem’ (by ‘problem’ I mean something like a circumstance that would lead to falsity) and in fact does more to lend to the truth of the conditional. What is ‘problematic’ is not simply the antecedent being true but rather the antecedent’s being true while the consequent is false. As we will see, one idea in the infinite-valued evaluation for conditionals is that so long as the consequent is not more false than the antecedent, then the whole conditional is fully true. But if the consequent is less true than the antecedent, then the conditional is as false as the difference between the consequent and antecedent. (So if the consequent is 0.2 more false than the antecedent, then the whole conditional is 0.2 false and thus its overall truth value is 0.8, which is its quantity of being true). The reasoning here would seem to be that it is ‘problematic’ for the consequent to be less true than the antecedent, and the extent to which it is gives the extent to which the whole conditional is less than true. But I do not entirely understand why a partial truth for the antecedent and an at least or more partial truth of the consequent results in a full truth value rather than a partial one. I can only think that the partial falsity of the antecedent is not ‘problematic’ but only the truthfulness implying falsity is, which only begins to happen when the consequent is less true than the antecedent.]
There are many ways of dealing with conditionals; again we shall choose one of the simplest. We shall assume that a conditional is wholly true if its consequent is at least as true as its antecedent, but that if the consequent is less true than the antecedent by some amount x, then the conditional is less than wholly true by that amount. If, for example, V(Φ) = 0.3 and V(Ψ) = 0.4, then V(Φ → Ψ) = 1, since the degree of truth of the consequent exceeds that of the antecedent. If, however, the values are reversed, so that V(Φ) = 0.4 and V(Ψ) = 0.3, then, since the antecedent’s degree of truth exceeds the consequent’s by 0.1, the conditional is that much less than wholly true; in other words, V(Φ → Ψ) = 1 - 0.1 = 0.9. These truth conditions can be expressed by the equation
4. V(Φ → Ψ) = 1 + min(V(Φ), V(Ψ)) - V(Φ)
If Ψ is at least as true as Φ – that is, V(Φ) ≤ V(Ψ) – then min(V(Φ), V(Ψ)) - V(Φ) = 0 and so V(Φ → Ψ) = 1. If V(Φ) > V(Ψ), then min(V(Φ), V(Ψ)) - V(Φ) = V(Ψ) - V(Φ) so that V(Φ → Ψ) = 1 + V(Ψ) - V( Φ) = 1 - (V( Φ) - V(Ψ)); that is, the conditional is less than wholly true by the amount V(Φ) - V(Ψ).
Notice that Φ → Ψ is wholly false only if V(Φ) = 1 and V(Ψ) = 0. Otherwise, it has some degree of truth-that is, V(Φ → Ψ) > 0.
(422-423)
[For biconditionals, recall how normally a biconditional can be true even if both sides are false. What is important here is that both sides have the same value. So in our infinite-valued semantics, if both have the same partial value, then the whole biconditional will be true. And if there is a discrepancy, then the biconditional will be as false as the difference between them. Thus if there is 0.2 difference between them, its value is 0.8, because it is 0.8 true.]
For the biconditional, we shall assume that its truth value is 1 iff the truth values of its components are equal and that otherwise it is less than wholly true by the amount of their difference. Thus, if V(Φ) = V(Ψ), then V(Φ → Ψ) = 1. But if, say, V(Φ) = 0.3 and V(Ψ) = 0.7 so the difference between V(Φ) and V(Ψ) is 0.4, then V(Φ → Ψ) = 1 - 0.4 = 0.6. These truth conditions are expressible by the equation
5. V(Φ ↔ Ψ) = 1 + min(V(Φ), V(Ψ)) - max((V(Φ), V(Ψ))
(432)
By limiting our values we can obtain semantics that we have seen before. For example, we obtain classical semantics by limiting our values to 0 and 1, and we obtain Łukasiewicz’ three-valued semantics by having a third value 1/2 that we regard as the I value (see section 15.2). [Let us examine that last claim, because it is something Graham Priest also says. The negation of I is I, which in these quantities would be 0.5. That calculation results from V(¬Φ) = 1 – V(Φ) also. The conjunction of a true conjunct with an indeterminate one is I, or 0.5 in our restricted three quantity system. This quantity also results from V(Φ & Ψ) = min(V(Φ), V(Ψ)).]
We can test for validity using these semantics, and we find it often gives the results we would expect. For example, we would think that P ∨ ~P is invalid (not true on all valuations), and in fact it is not true for the value 0.5. (Nolt 423-424)
Nolt then notes something interesting. [Consider formulas of the form Φ ∨ ~Φ. Suppose Φ is 1. That means ~Φ is 0. The higher value is 1, and thus the disjunction is 1. Suppose Φ is 0. That means ~Φ is 1. The higher value is 1, and thus the disjunction is 1. Now suppose Φ has some intermediate value like 0.5. That means ~Φ is 0.5. The higher value is 0.5, and thus the disjunction is 0.5. If Φ is 0.3, then ~Φ is 0.7, and the whole disjunction is 0.7. If Φ is 0.7, then ~Φ is 0.3, and the whole disjunction is 0.7.] Formulas taking the form Φ ∨ ~Φ never take a value below 1/2, so they are always at least partly true. However, they are not valid, because they are not true under all valuations.
Other formulas are valid even with there being an infinity of valuations, like Φ → Φ. [see p.424 for the metatheorem and proof.]
Nolt then explains, “In general, formulas which are classically valid may take any truth value on the infinite-valued semantics, though those of particular forms may be confined to a particular range of values or even to one value” (424).
Nolt then discusses the fact that an infinite-valued semantics allows for a variety of ways to define validity [see pp.424-425 for more details on the options.]
Nolt then returns to the original example. He says that in our infinite-valued semantics, each additional iteration of the proposition taking the form
A global population of n + 1 is sustainable
takes a slightly lesser truth value (425). Note that the reasoning is based on the conditional:
If a global population of n is sustainable, so is a global population of n + 1.
[Here in the first case where the population is specified as six billion, our antecedent has the value 1, but each iteration has a value less than that. We subtract them to get the value of the whole conditional. Thus with each iteration, this conditional gets less and less than 1.]
“A long | sequence of such inferences, then, may lead us from near truth to absolute falsehood” (426).
We see that infinite-valued semantics can be potentially useful for certain cases of vagueness. But, Nolt explains, it is not “wholly satisfactory” (426). He notes that while we may know that “A global population of 6,000,000,001 is sustainable” is slightly less true than “A global population of 6,000,000,000 is sustainable,” we still have no obvious way to calculate precisely how much less of a truth value it should receive (426).
Nolt says that one reason we cannot make this determination is that we simply do not understand the ecosystem enough to know the degree to which additional population is unsustainable. The other reason is that the predicate “is sustainable” is vague. It is qualitative and vague and thus hard to precisely quantify (426).
[The next idea seems to be that Lofti Zadeh tried to improve upon infinite-valued semantics by saying that even the truth values can be fuzzy in the sense that they take on a range of values, and values as members in that range have a certain degree of membership. I am just guessing here, but the idea might be the following. We might say that the global population sentence from above is not simply 0.8 for instance but somewhere between 0.7-0.9, with the values near 0.7 being less “likely” to hold in the sense that 0.7 does not take a very strong degree of membership in the set of values, but it could still be in it. Let me quote:]
In the mid-1960s, Lofti Zadeh set out to improve upon infinite-valued semantics by making the truth values themselves imprecise. That is, instead of assigning to a statement like ‘A global population of 6,000,000,000 is sustainable’ an arbitrarily precise numerical truth value, Zadeh proposed that we assign it an imprecise range of values. By this he meant not merely an interval of values (say, the interval between .4 and .5, which itself is a precisely defined entity), but a fuzzy interval of values. A fuzzy interval is a kind of fuzzy set. And a fuzzy set is a set for which membership is a matter of degree.
(Nolt 426, italics and boldface his, underlining mine)
[Zedah claimed that most of our concepts (and maybe by this we are to understand ‘predicates’) actually are matters of fuzzy sets rather than classical sets. In other words, many things taking a predicate do not take it to the fullest degree, and thus there are members of the set of things taking the predicate which have less strong of a membership. The next idea in this paragraph might be something like what we said above in brackets, where numerical truth values within a certain range are assigned their own degree of membership in that group of values. Let me quote so you can see:]
Most concepts, Zadeh argued, define not classical sets, but fuzzy sets. Take the concept of redness. Some things are wholly and genuinely red. But others are almost red, somewhat red, only a little bit red, and so on. So, whereas fresh blood or a red traffic light might be fully a member of the set of all red things, the setting sun might be, say, halfway a member and a peach only slightly a member. Now in fuzzy-set theory, membership is assigned strict numerical values from 0 to 1, like the truth values in infinite-valued semantics. But in defining truth values, Zadeh compounds the fuzziness. He might, for example, define a truth value AT (almost true), which is a fuzzy set of numerical values in which, say, numbers no greater than 0.5 have membership 0, 0.6 has membership 0.3, 0.7 has membership 0.5, 0.9 has membership 0.8, and 0.99 has membership 0.95. Such a fuzzy set of numerical values is for Zadeh a truth value. A logic whose semantics is based on such fuzzy truth values is called a fuzzy logic.
(426, boldface his, underlining mine)
[The next ideas I will probably misrepresent. Let me try the following. Consider the predicate, “is a number”. For this, there is the set of numbers, which when named as the argument taking this predicate, make the proposition true. Otherwise it is false. We are thinking here of classical semantics. Now, suppose we are using infinite valued logic. This means “x is a number” could potentially have a value between 1 and 0. But this example does not work, because it is not easy to think of such partial-valued cases. But for “x is red”, we have already seen cases where something is not entirely red, like a sunset. Nolt’s point seems to be that so long as we are implementing partial values for sentences with predicates taking at least one argument, then that means we are referring to objects that have only a degree of membership in the set of things corresponding to that predicate, and thus we are dealing with fuzzy sets. The next point I do not get, but let us consider a possibility. What we just noted is infinite-valued semantics. A fuzzy logic would have a semantics where not only do we say that a sunset has a 0.5 degree membership, but also that this somehow is a fuzzy value, meaning perhaps that there is a range of values from 0.4 to 0.6 indicating its fuzzy membership in this set. That is a guess, so please consult the text. The final point might be that by complicating the system in this way, and also by admitting of many potentially arbitrary determinations for how to assign the fuzzy values, we make our basic concepts become perhaps too complex for practical purposes.]
If infinite-valued semantics presents a bewildering array of choices of truth conditions and semantic concepts, fuzzy logic compounds the complication. Already in infinite-valued predicate logic, the extensions assigned to predicates must be fuzzy sets; for if they were classical sets, atomic formulas containing n-place predicates (n > 0) would always be either true or false. So, for example, the predicate ‘is red’ has as its extension the fuzzy set of red things described above. Consequently, infinite-valued semantics has the following truth clause for one-place predicates Φ and names α: |
V(Φα) = x iff V(α) is a member of the fuzzy set V(Φ) to degree x.
But on Zadeh’s fuzzy semantics the extensions of predicates are structures still more complex than fuzzy sets, structures which, when applied to the extensions of names, yield fuzzy truth values. The valuation rules for the operators, and the semantic concepts of validity, consistency, and so on, must all be redefined once again to accommodate these fuzzy values. In the process these concepts “splinter” even more wildly than concept of validity does in infinite-valued logic.
(426-427)
In fact, Zadeh’s attempts to minimize the arbitrarity by limiting the values and correlating them with natural language expressions does not really solve the problem.
Yet despite the complication upon complication, arbitrariness remains. Zadeh suggests that a fuzzy logic should not employ all possible fuzzy truth values (a very large set of values indeed!), but only a small finite range of them, and that it should correlate them with such natural language expressions as ‘very true’, ‘more or less true’, and so on. But which fuzzy set of numbers corresponds to the English expression ‘more or less true’? And why should we suppose that precisely that set is what we mean when we say that a particular sentence is more or less true? The choice of any particular fuzzy value is just as arbitrary as the assignment of a precise numerical truth value to a vague statement. The arbitrariness does not go away; it is merely concealed in the complexity.
(427)
Nolt concludes by noting certain successes of fuzzy logic. It has proved useful in artificial intelligence programming. It has proved attractive to those who do not like too much conceptual precision. Its popularity has even led to some ambiguous uses for the term.
Arbitrariness notwithstanding, fuzzy logic has found useful application in artificial intelligence devices. But it has also acquired a certain unwarranted mystique. Many people are attracted to the idea of a (warm and ?) fuzzy logic because it sounds as if it might offer relief from overtaxing precision. As a result of this popularity, the term ‘fuzzy logic’ is often used loosely. In popular science publications it may mean no more than an infinite-valued logic – or even statistical or just plain muddle-headed reasoning.
(427)
From:
Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.
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