by Corry Shores
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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part I:
Propositional Logic
11.
Fuzzy Logics
11.4
The Continuum-valued Logic Ł
Brief summary:
(11.4.1) One way to construct a fuzzy logic is as a many-valued logic with a continuum of values from 0 (completely false) to 1 (completely true), including all real number values between, such that 0.5 is half true, and so on. (11.4.1a) To formulate the semantics for the connectives, we for now will use the oldest and most philosophically interesting means to do so. (11.4.2) Priest next gives the semantics for the connectives in our continuum many-valued logic.
f¬ (x) = 1 − x
f∧(x, y) = Min(x, y)
f∨(x, y) = Max(x, y)
f→(x, y) = x ⊖ y
where Min means ‘the minimum (lesser) of’; Max means ‘the maximum (greater) of’; and x ⊖ y is a function defined as follows:
if x ≤ y, then x ⊖ y = 1
if x > y, then x ⊖ y = 1 − (x − y) (= 1 − x + y)
| Note that we could say ‘x ≥ y’ instead of ‘x > y’ in the second clause, since if x = y, 1 − (x − y) = 1. Note, also, that we could define x ⊖ y equivalently as Min(1, 1 − x + y).
(225-226)
(11.4.3) The formulations of the semantic evaluations for the connectives in fuzzy logic hold to the basic intuitions we have about how they should operate. (11.4.4) Priest next notes that: if x ≤ y, then y ⊖ z ≤ x ⊖ z ; and if x ≤ y, then z ⊖ x ≤ z ⊖ y . (11.4.5) Our continuum-valued fuzzy logic “is a generalisation of both classical propositional logic, and Łukasiewicz’ 3-valued logic;” for, if we use only 1 and 0, we get the outcomes for classical semantics, and if we use just 0, 0.5, and 1 (with 0.5 understood as i), we get the outcomes for Ł3. (11.4.6) The designated value is context dependent, and so “any context will determine a number, ε, somewhere between 0 and 1, such that the things that are acceptable are exactly those things with truth value x, where x ≥ ε” (226). (11.4.7) Validity is defined as: “Σ ⊨ε A iff for all interpretations, v, if v(B) ≥ ε for all B ∈ Σ, then v(A) ≥ ε” (226). (In other words, an inference is valid under the following condition: whenever the premises are at least as high as the ((context-determined designated fuzzy)) value ε, then so too is the conclusion at least as high as ε. (11.4.8) Our fuzzy logic is called Ł, and its context-independent definition of validity is: Σ ⊨ A iff for all ε, where 0 ≤ ε ≤ 1, Σ ⊨ε A . (11.4.9) A set of truth-values X can be listed in descending numerical order. Suppose it is in an infinite set following a pattern like {0.41, 0.401, 0.4001, 0.40001, . . .}. Even though there would be no least member, there is still however a number that would be the greatest possible figure that is still less than or equal to all the members, in this case being 0.4. And it is called the greatest lower bound of set X, abbreviated as Glb(X). (11.4.10) A simpler characterization of validity would be: Σ ⊨ A iff for all v, Glb(v[Σ]) ≤ v(A). (11.4.11) Given the semantic evaluation for conjunction and the conditional, we can formulate validity in the following way: {B1, . . . , Bn} ⊨ A iff for all v, v((B1 ∧ . . . ∧ Bn) → A) = 1. “Thus (for a finite number of premises), validity amounts to the logical truth of the appropriate conditional when the set of designated values is just {1}, that is, the logical truth of the conditional in ⊨1. The logic with just 1 as a designated value is usually written as Łℵ, and it is called Łukasiewicz’ continuum-valued logic” (227).
Contents
11.4.1
[Fuzzy Logic as a Continuum Many-Valued Logic]
11.4.1a
[Ways to Formulate the Semantics for Connectives]
11.4.2
[The Semantics for Connectives]
11.4.3
[The Thinking Behind the Semantic Formulations for Connectives]
11.4.4
[Some Conditional Formulations with a z Value]
11.4.5
[Continuum Semantics as a Generalization of Classical Semantics and Ł3]
11.4.6
[The Designated Value]
11.4.7
[Validity]
11.4.8
[The Context-Independent Formulation of Validity in Ł.]
11.4.9
[The Greatest Lower Bound]
11.4.10
[Simplified Characterization of Validity]
11.4.11
[Validity in Terms of the Conditional in Łℵ]
Summary
11.4.1
[Fuzzy Logic as a Continuum Many-Valued Logic]
[One way to construct a fuzzy logic is as a many-valued logic with a continuum of values from 0 (completely false) to 1 (completely true), including all real number values between, such that 0.5 is half true, and so on.]
[In section 11.3.9, we noted that to deal with sorites situations where there are continuous changes but no apparent discrete truth-value breaks, that we could use fuzzy logics, which employ continuous gradations of truth value. Now Priest notes that we can construct a fuzzy logic as a many-valued logic with a continuum of truth values. We take 1 as completely true, and 0 as completely false. And truth values can take take all real numbers between 1 and 0 such that 0.5 would be half true, and so on.]
A natural way to construct a fuzzy logic is as a many-valued logic with a continuum of truth values. Let the truth values, V, be the set of real numbers (decimals) between 0 and 1, {x: 0 ≤ x ≤ 1}. This is often written as [0,1]. 1 is completely true; 0 is completely false; 0.5 is half true; etc.
(224)
[contents]
11.4.1a
[Ways to Formulate the Semantics for Connectives]
[To formulate the semantics for the connectives, we for now will use the oldest and most philosophically interesting means to do so.]
[We will now look at ways to formulate the semantics for connectives. Later in the chapter Priest will return to this matter with a discussion of something called a t-norm. But for now, we will use the oldest and most philosophically interesting way to formulate the connectives.]
What are the semantic functions that correspond to the connectives ∧, ∨, ¬ and →? There are various ways to answer this question, based on the general notion of something called a t-norm. Details can be found in the technical appendix to this chapter, 11.7a. For the rest of this chapter, we will concentrate on the oldest and, perhaps, most interesting answer for philosophical purposes.
(224)
[contents]
11.4.2
[The Semantics for Connectives]
[Priest next gives the semantics for the connectives in our continuum many-valued logic.]
[Priest next gives the semantics for the connectives. Let us consider them each on the basis of our intuition. Negation we often think of as giving the flip side or “other” of the value under negation. Here it is formulated: f¬ (x) = 1 − x. So suppose that we want the negation of something that is true. We would want 0. So 1 – 1 would give us 0. Suppose instead we want the negation of something false. We would want 1. And 1 – 0 is in fact 1. Suppose something is only a little bit true, like 0.1 true. It negation should be very true, in a proportional way. And in fact, 1 – 0.1 gives us 0.9, which fits our intuitions about how negation should function. All other cases of negation should match our intuitions for the same reasons. For the next two connectives, we need a notion of “minimum (lesser) of” and “maximum (greater) of”. These notions remind me a little bit of the diamond lattices from section 8.4.3. For conjunction, normally when at least one of the conjuncts is false, that “drags down” the whole conjunction’s value to false, regardless of the truth-value of the other conjunct. So with our continuum-valued logic, we will “drag” the whole conjunction’s value down to whatever the lowest of the two conjunct values are, ignoring the other value. (The formulation is: f∧(x, y) = Min(x, y) .) We may have a similar intuition for disjunction. Normally if one of the disjuncts is true, that “lifts up” the whole disjunction’s value to true. So for the continuum-valued logic, we take the higher of the two disjunct values, ignoring the other, thereby “lifting” the disjunction’s value up to it. (The formulation is: f∨(x, y) = Max(x, y) .) It gets more complicated of course for the conditional. Normally the cases where a conditional is true is when {1} the antecedent is 1 and the consequent 1 (so here they are equal), {2} when the antecedent is 0 and the consequent 0 (so here again they are equal) or {3} when the antecedent is 0 and the consequent 1 (so here the antecedent is less than the consequent). In all cases of the conditional being true, the antecedent is either less than or equal to the consequent. And, it is false if the the antecedent is 1 and the consequent is 0 (thus when the antecedent is more than the consequent). But we still need to puzzle through the formulation and consider our intuitions. The formulation will make it such that when the antecedent is less than or equal to the consequent, the whole conditional is 1 rather than some partial value, whereas when the antecedent is greater than the consequent, then it can take a partial value. More specifically, in that case we calculate it in the following way. We subtract the consequent’s value from antecedent, and that value we subtract from 1. Suppose we have the simple values that the antecedent is 1 and the consequent is 0. We would want the outcome to be 0. Here we have 1 – (1 – 0) which equals 0, as we expected. But suppose that the antecedent is simply true but consequent is only a little true, like 0.25 true. Then we have 1 – (1 – 0.25), or 0.25. Or suppose the antecedent is simply true and the consequent is mostly true, at 0.75. Then the conditional is 0.75 true. Now, suppose that the antecedent is mostly true at 0.75 and the consequent is only a little true, at 0.25. Then the conditional is 0.5. So we should consider two issues with regard to our intuitions about conditionals. The first question is, why would the first cases not admit of fuzzy values? I guess the idea would be that part of our intuition of the conditional is that there is a certain allowance that is absolute. But why is it ok for example for something 0.5 to follow conditionally from something 0.49? Why intuitively is that conditional a 1, being the same value as a conditional where 1 follows conditionally from 0? This is not so obvious to me. My best guess is that whenever the antecedent is less than the consequent, then we think that the conditions are not being met sufficiently to make it false, and so it is true with no further qualification to say it is less than fully true. Furthermore, it is already tricky to understand why in classical logic when the antecedent is false the whole conditional is thereby a true conditional. For the same intuition or reasoning that would go so far as to hold this might also be at work in saying that simply being a tiny fraction less true than the consequent is enough to make the whole conditional true (the example we consider later will illustrate this possible intuition). Now what is the intuition that says when the antecedent is more true than the consequent, the conditional should have fuzzy values that are proportional to their numerical difference? Here maybe (and probably not) the idea is that a conditional is all about something following conditionally from something else, and when the consequent is less than the antecedent, then the “following conditionally” has been tampered with. When the antecedent is less than or equal to the consequent, then the “following conditionally” is not being tampered with. Consider, “if they are an adult, then they are responsible.” And suppose they are 0.49 an adult but 0.5 responsible. We would think that if they are 0.49 an adult, then they would need to be at least 0.49 responsible for this to be true. But in fact, we went over that value by a little, and so for even more reason it would be true when they are 0.5 responsible but only 0.49 an adult. However, suppose instead they are 0.9 an adult but they are 0.1 responsible. The conditions would have called for them to be at least 0.9 responsible. Yet they are not. They are much less responsible than that. But they are still a little responsible, so the conditional is still a little true (being 0.2 true). Still, why do we not say that it is 0 true? My guess would be that it would go against our intuitions for the borderline cases. Suppose someone is 0.5 an adult but 0.49 responsible. It would seem to go too far to say that the conditional here is 100 percent false. It seems instead to have nearly met the conditions to be completely true, and so we should not evaluate the conditional as completely false. That is the best I have at the moment for making these notions intuitive. (In fact, Priest will give the actual reasoning later in section 11.4.3, but I wanted to work through it now when they are presented.)]
According to this:
f¬ (x) = 1 − x
f∧(x, y) = Min(x, y)
f∨(x, y) = Max(x, y)
f→(x, y) = x ⊖ y
where Min means ‘the minimum (lesser) of’; Max means ‘the maximum (greater) of’; and x ⊖ y is a function defined as follows:
if x ≤ y, then x ⊖ y = 1
if x > y, then x ⊖ y = 1 − (x − y) (= 1 − x + y)
| Note that we could say ‘x ≥ y’ instead of ‘x > y’ in the second clause, since if x = y, 1 − (x − y) = 1. Note, also, that we could define x ⊖ y equivalently as Min(1, 1 − x + y).
(225-226)
[contents]
11.4.3
[The Thinking Behind the Semantic Formulations for Connectives]
[The formulations of the semantic evaluations for the connectives in fuzzy logic hold to the basic intuitions we have about how they should operate.]
[Priest next discusses the thinking behind the connective formulations. As we went through similar thinking in 11.4.2, I will defer to the quotation below.]
The truth functions for negation, conjunction and disjunction are fairly natural. As the truth value of ‘Mary is a child’ goes down, the truth value of ‘Mary is not a child’ would seem to go up coordinately. A conjunction would seem to be just as good as its least true conjunct; and a disjunction would seem to be just as good as its most true. The truth function for → is anything but obvious. Here is its rationale. Consider A → B. If A is less true (or, better, no more true) than B, then the truth value of A → B is 1. That’s how it works, after all, with the standard 2-valued material conditional. If A is more true than B, then there is something faulty about the conditional: its truth value must be less than 1. How much less? The amount that the truth value falls in going from A to B. In particular, if it falls all the way from 1 to 0, then the value of A → B is 0. All this is exactly what ⊖ means.2
(225)
2. Fuzzy logic should not be confused with probability theory. Though fuzzy truth values and probability values are both real numbers in [0, 1], fuzzy truth values are truth functional – that is, the value of a compound is determined by the values of its components – whilst probabilities are not. Given a die, let A be ‘you roll 1, 2, or 3’, and B be ‘you roll 4, 5, or 6’. Then if P(A) is the probability of A, P(A ∧ A) = P(A) = 0.5, but P(A ∧ B) = 0, even though P(A) = P(B).
(225)
[contents]
11.4.4
[Some Conditional Formulations with a z Value]
[Priest next notes that: if x ≤ y, then y ⊖ z ≤ x ⊖ z ; and if x ≤ y, then z ⊖ x ≤ z ⊖ y .]
[Priest next notes some other numerical relations for the conditional that are formulated with a z value, but I am not sure yet what they are for. See them below.]
Note that:
if x ≤ y, then y ⊖ z ≤ x ⊖ z
if x ≤ y, then z ⊖ x ≤ z ⊖ y
For the first of these, suppose that x ≤ y (and so, that −y ≤ −x): if x ≤ z, then x ⊖ z = 1, so the result follows. If z < x ≤ y, then y ⊖ z = 1 − y + z ≤ 1 − x + z = x ⊖ z. The second conditional is left as an exercise.
(225)
[contents]
11.4.5
[Continuum Semantics as a Generalization of Classical Semantics and Ł3]
[Our continuum-valued fuzzy logic “is a generalisation of both classical propositional logic, and Łukasiewicz’ 3-valued logic;” for, if we use only 1 and 0, we get the outcomes for classical semantics, and if we use just 0, 0.5, and 1 (with 0.5 understood as i), we get the outcomes for Ł3.]
[Priest then brings to our awareness that if we stick to the classical 1 and 0 values, then the formulations will give us the classical outcome values. (We saw this in our bracketed comments in section 11.4.2). Also, if we think of 0.5 as i, then our continuum-valued semantics are the same as Łukasiewicz’ 3-valued logic Ł3 (see section 7.3.2 and section 7.3.8).]
Notice that if we restrict ourselves to just the values 1 and 0, then the truth functions of 11.4.2 are exactly the same as those of classical truth tables. It is less obvious, but is easy to check, that if we restrict ourselves to just the values 1, 0.5 and 0, then the truth functions are exactly the same as those of Ł3 (7.3.2 and 7.3.8), thinking of → as ⊃, and 0.5 as i. In this sense, the logic is a generalisation of both classical propositional logic, and Łukasiewicz’ 3-valued logic.
(225)
[contents]
11.4.6
[The Designated Value]
[The designated value is context dependent, and so “any context will determine a number, ε, somewhere between 0 and 1, such that the things that are acceptable are exactly those things with truth value x, where x ≥ ε” (226).]
[Recall the notion of the designated value from section 7.2. Designated values are “the values that are preserved in valid inferences” (p.120, section 7.2.2). In classical logic, it is just the 1 value that is preserved in valid inferences. Now we wonder, what will be the designated value in our fuzzy logic? Priest says that context will decide how true something needs to be: “If you buy a new car, you expect it not to have been driven at all. (So ‘this is a new car’ needs to have truth value 1.) But you would still describe it as a new car to a friend, even if you had bought it and driven it around for a few weeks. (So in this context, ‘this is a new car’ need have truth value only 0.95, say.)” (226). Thus: “if A is acceptable as true, and B is truer than A, then B is acceptable as true as well. What all this means is that any context will determine a number, ε, somewhere between 0 and 1, such that the things that are acceptable are exactly those things with truth value x, where x ≥ ε” (226).]
What of the designated values of the logic? In general, things do not have to be completely true to be acceptable. If I ask for a red apple, and you give me one with a very small patch of green (so that ‘this is red’ is, say, 0.95 true), that’s probably good enough. How true something has to be to be acceptable will depend on the context. If you buy a new car, you expect it not to have been driven at all. (So ‘this is a new car’ needs to have truth value 1.) But you would still describe it as a new car to a friend, even if you had bought it and driven it around for a few weeks. (So in this context, ‘this is a new car’ need have truth value only 0.95, say.) But at any rate, if A is acceptable as true, and B is truer than A, then B is acceptable as true as well. What all this means is that any context will determine a number, ε, somewhere between 0 and 1, such that the things that are acceptable are exactly those things with truth value x, where x ≥ ε.
(226)
[contents]
11.4.7
[Validity]
[Validity is defined as: “Σ ⊨ε A iff for all interpretations, v, if v(B) ≥ ε for all B ∈ Σ, then v(A) ≥ ε” (226). (In other words, an inference is valid under the following condition: whenever the premises are at least as high as the ((context-determined designated fuzzy)) value ε, then so too is the conclusion at least as high as ε.]
[In section 11.4.6 above, Priest introduced the ε value for giving the designated values in our fuzzy logic. It is a value between 0 and 1 whose exact quantity is determined by the context. But once established, it means that an inference is valid whenever the premises are at least as high as ε, then so too is the conclusion at least as high as ε.]
Correspondingly, for every such ε, taking the set of designated values, Dε, to be {x: x ≥ ε}, will define a notion of validity. Thus Σ ⊨ε A iff for all interpretations, v, if v(B) ≥ ε for all B ∈ Σ, then v(A) ≥ ε.
(226)
[contents]
11.4.8
[The Context-Independent Formulation of Validity in Ł.]
[Our fuzzy logic is called Ł, and its context-independent definition of validity is: Σ ⊨ A iff for all ε, where 0 ≤ ε ≤ 1, Σ ⊨ε A .]
[(ditto)]
Each logic defined in this way is a perfectly good many-valued logic. But in logic, it makes sense to abstract from context and consider a notion of validity that is context-independent. Hence, it is natural to define the central notion of logical consequence as follows:
Σ ⊨ A iff for all ε, where 0 ≤ ε ≤ 1, Σ ⊨ε A
We will call this logic Ł.
(226)
[contents]
11.4.9
[The Greatest Lower Bound]
[A set of truth-values X can be listed in descending numerical order. Suppose it is in an infinite set following a pattern like {0.41, 0.401, 0.4001, 0.40001, . . .}. Even though there would be no least member, there is still however a number that would be the greatest possible figure that is still less than or equal to all the members, in this case being 0.4. And it is called the greatest lower bound of set X, abbreviated as Glb(X).]
[The next ideas get more mathematical, and I will likely missummarize them, so please consult the quotation below. We have a set of truth values, called X. And this set can be either infinite or finite. The members of the set are listed in descending numerical order. Suppose it is simply the finite set: {0.41, 0.401, 0.4001}. We now think of a value, either in or not in the set, that is less than or equal to all the members in the set. But the idea is that this number should be as high as possible. In our case here, that number would be 0.4001. Now suppose the set is infinite, and it follows the same pattern, but without terminating in some particular figure: {0.41, 0.401, 0.4001, 0.40001, . . .}. As such, it has no least member. However, we can still say that there is the highest possible number that is less than or equal to all the members in this infinite set, being namely 0.4. (I do not know much math, but intuitively the idea seems to be the following. The numbers seem to tend to a limit at 0.4. Even if they never attain it exactly in a single terminal member, 0.4 would still be the highest we can say that is less than or equal to all the members of the set.) It is called, the greatest lower bound of set X, abbreviated as Glb.]
A set of truth values, X, may have no least member. (Consider, for example, {0.41, 0.401, 0.4001, 0.40001, . . .}.) But there will always be a greatest number that is less than or equal to every number in the set. (In this case, the number is 0.4.) This is called the greatest lower bound of X (Glb(X)). If the set is finite, then the Glb of the set is, of course, its least member. Notice that, by definition, if x ∈ X, x ≥ Glb(X); and if for all x ∈ X, x ≥ y, then Glb(X) ≥ y.
(226)
[contents]
11.4.10
[Simplified Characterization of Validity]
[A simpler characterization of validity would be: Σ ⊨ A iff for all v, Glb(v[Σ]) ≤ v(A).]
[Priest next gives a simplified formulation of valid inference. (See the quotation for details).]
⊨ has, in fact, a very simple characterisation. If Σ is a set of formulas, let v[Σ] be {v(B): B ∈ Σ}. Then:
Σ ⊨ A iff for all v, Glb(v[Σ]) ≤ v(A)
| Proof: Suppose that Σ ⊭ A. Then there is some ε, such that Σ ⊨ε A. That is, for some v, and for all B ∈ Σ, v(B) ≥ ε, and v(A) < ε. But if every member of v[Σ] is ≥ ε, Glb(v[Σ]) ≥ ε. Hence, for this v, it is not the case that Glb(v[Σ]) ≤ v(A). Conversely, suppose that for some v, Glb(v[Σ]) > v(A). Let ε = Glb(v[Σ]). Then for all B ∈ Σ, v(B) ≥ ε, but v(A) < ε. That is, Σ ⊭ε A. Hence, Σ ⊭ A.
(226-227)
[contents]
11.4.11
[Validity in Terms of the Conditional in Łℵ]
[Given the semantic evaluation for conjunction and the conditional, we can formulate validity in the following way: {B1, . . . , Bn} ⊨ A iff for all v, v((B1 ∧ . . . ∧ Bn) → A) = 1. “Thus (for a finite number of premises), validity amounts to the logical truth of the appropriate conditional when the set of designated values is just {1}, that is, the logical truth of the conditional in ⊨1. The logic with just 1 as a designated value is usually written as Łℵ, and it is called Łukasiewicz’ continuum-valued logic” (227).]
[This final notion is quite complex, and I will missummarize it. So please consult the quotation below. The first notion is one we encountered above in section 11.4.9, namely, that for a finite set of truth-values, the greatest lower bound will be its lowest value. Next recall the semantic evaluation for conjunction from section 11.4.2 above.
f∧(x, y) = Min(x, y)
(p.225, section 11.4.2)
So given what we have said in the prior sections about validity, the inference is valid when the least of the premises is less than or equal to the conclusion. With what we say about conjunction, this is the same as: v(B1 ∧ . . . ∧ Bn) ≤ v(A), (when premises are B and conclusion is A). Next we need to recall the semantic evaluation for the conditional again from section 11.4.2 above.
f→(x, y) = x ⊖ y
if x ≤ y, then x ⊖ y = 1
if x > y, then x ⊖ y = 1 − (x − y) (= 1 − x + y)
(p.225, section 11.4.2)
Priest will now formulate validity under the form of a conditional, but again it is best to skip to his rendition. We notice in the semantic evaluation that the only way the conditional can be exactly 1 is if the antecedent is less than or equal to the consequent. And keep in mind that an inference is valid when all the premises are less than or equal to the conclusion. (I get confused with this, but maybe we need to keep in mind also the designated value here.) So combine what was said about the semantic evaluations of conjunction and the conditional, then whenever we have:
v((B1 ∧ . . . ∧ Bn) → A) = 1
that means the antecedent terms are all less than the consequent value. Now consider the B formulas as premises and the A as the conclusion. Given that under the above evaluation they would fulfill the requirement for validity, we also have:
{B1, . . . , Bn} ⊨ A
Priest then concludes: “Thus (for a finite number of premises), validity amounts to the logical truth of the appropriate conditional when the set of designated values is just {1}, that is, the logical truth of the conditional in ⊨1.” We call such a continuum logic with just 1 as the designated value Łℵ, and we call it: Łukasiewicz’ continuum-valued logic. (But I am not following so well the notion of infinity here. I also was not able to put all of these ideas together. So please read the quotation.)]
For a finite set, the Glb is its minimum. So if Σ = {B1, . . . , Bn}, then Σ ⊨ A iff for all v, Min(v(B1), . . . , v(Bn)) ≤ v(A) iff v(B1 ∧ . . . ∧ Bn) ≤ v(A).3 A little thought concerning ⊖ suffices to show that v(C) ≤ v(A) iff v(C → A) = 1. Hence:
{B1, . . . , Bn} ⊨ A iff for all v, v((B1 ∧ . . . ∧ Bn) → A) = 1
Thus (for a finite number of premises), validity amounts to the logical truth of the appropriate conditional when the set of designated values is just {1}, that is, the logical truth of the conditional in ⊨1. The logic with just 1 as a designated value is usually written as Łℵ, and called Łukasiewicz’ continuum-valued logic. Hence, to investigate Ł further, we may investigate Łℵ.4
(227)
3. Strictly speaking, the conjuncts should be bracketed in some way, since conjunction is a binary connective. But, however one inserts brackets, the value of the iterated conjunction is the same: the minimum of the values of the conjuncts. It therefore does no harm to omit the brackets.
4. ℵ is the Hebrew letter aleph, and, following Cantor, is used by logicians to denote a size of infinity.
(227)
[contents]
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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