by Corry Shores
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[Priest, Introduction to Non-Classical Logic, entry directory]
[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 II:
Quantification and Identity
21
Many-valued Logics
21.3
∀ and ∃
Brief summary:
(21.3.1) We now wonder, how does quantification in many-valued quantified logic behave (or more precisely, how do the functions for the quantifiers, f∀ and f∃, behave)? (21.3.2) “In classical logic, the universal quantifier acts essentially like a conjunction over all the members of the domain. So ∀xA is something like Ax(kd1) ∧ Ax(kd2) ∧ . . . , where d1, d2, . . . are all the members of the domain.” “Dually, the particular quantifier is something like a disjunction over all members of the domain: ∃xA is Ax(kd1) ∧ Ax(kd2) ∨ . . .” (458). We would expect that the universal and particular quantifiers behave the same way many-valued logics. (21.3.3) We can use greatest lower bound (Glb) of conjuncts (that is, the greatest truth-value that is less than or equal to the values assigned to the conjuncts) and least upper bound (Lub) of disjuncts (that is, the least truth-value greater than or equal to the value assigned to either disjunct) in order to evaluate universal and particular quantification, respectively: we define “f∀(X) as Glb(X), so that v(∀xA) is the greatest lower bound of {v(Ax(kd)): d ∈ D}” and we “define f∃(X) as Lub(X), and v(∃xA) is the least upper bound of {v(Ax(kd)): d ∈ D}” (458).
[Turning to the Behavior of f∀ and f∃]
[Universal and Particular Quantifiers in Classical Logics]
[Using Greatest Lower Bound and Least Upper Bound to Evaluate Universal and Particular Quantification]
Summary
[Turning to the Behavior of f∀ and f∃]
[We now wonder, how does quantification in many-valued quantified logic behave (or more precisely, how do the functions for the quantifiers, f∀ and f∃, behave)?]
[Recall from section 21.2.2 that the structure of our quantified many-valued logic is:
⟨D, V, D,{fc: c ∈ C}, {fq: q ∈ Q}⟩
where “if q is the set of quantifiers in the language, for every q ∈ Q, fq is a map from subsets of V into V” (457). So we evaluate the quantifiers by means of functions that take truth-values as inputs and assign to them somehow truth-values as outputs. Then in section 21.2.3 we learn that:
for each quantifier, q:
v(qxA) = fq({v(Ax(kd)): d ∈ D})
(In a free many-valued logic, ‘D’ is replaced by ‘E’.) For example, v(∀xA) = f∀({v(Ax(kd)): d ∈ D}).
(p.457, section 21.2.3)
This seems to suggest that the evaluation of the quantifiers involves substitutions and assigning truth-values by means of them. While it may not have been so obvious how exactly this works, it seems that in this chapter we will learn more about this.]
Of course, the main quantifiers in which we are interested (in this book, anyway) are the universal and particular quantifiers. So, given a many-valued logic, how would one expect f∀ and f∃ to behave?
(457)
[Universal and Particular Quantifiers in Classical Logics]
[“In classical logic, the universal quantifier acts essentially like a conjunction over all the members of the domain. So ∀xA is something like Ax(kd1) ∧ Ax(kd2) ∧ . . . , where d1, d2, . . . are all the members of the domain.” “Dually, the particular quantifier is something like a disjunction over all members of the domain: ∃xA is Ax(kd1) ∧ Ax(kd2) ∨ . . .” (458). We would expect that the universal and particular quantifiers behave the same way many-valued logics. ]
[Priest next explains how universal and particular quantifiers behave in classical logic. We saw something similar in Priest’s “Multiple Denotation, Ambiguity, and the Strange Case of the Missing Amoeba”, section 1.1. Here the symbol ⊗ means conjunction, and ⊕ disjunction, and the quantifiers were evaluated in the following way:
For the quantifiers, if we represent the natural generalisations of ⊕ and ⊗ by the same signs (so that ⊕X=1 iff 1 ∈ X, etc.), then the truth conditions of the quantifiers are as follows:
vs(∀vα) = ⊗{vs(v/a)(α); a ∈ D}
vs(∃vα) = ⊕{vs(v/a)(α); a ∈ D}
where s(v/a) is that evaluation of the variables that is the same as s, except that its value at v is a.
(Priest, “Multiple Denotation, Ambiguity, and the Strange Case of the Missing Amoeba”, section 1.1.2, p.362)
And the following comes from the brief summaries of this article for section 1.1.1 and section 1.1.2:
Our interpretation function v assigns to each sentence either 1 or 0. These two values are then calculated for complex formulas using the truth-functions for logical operators: ⊖ for f¬ (negation); ⊗ for f∧ (conjunction); and ⊕ for f∨ (disjunction). Validity is truth preservation: an inference is valid only if when all its premises are true, so too is the conclusion.
We extend this semantics (for determining truth values and validity) to a first-order language (that is, one with quantified variables ranging over objects). Such an interpretation has the structure ⟨D, d⟩. D is the domain of objects, and d is a function that assigns objects to constants, and n-tuples (of domain members) to n-place predicates. We also have function s, which assigns to variables objects in the domain. The vs function assigns a truth-value to formulas containing constants or variables by determining whether or not the d function (for constants) or the s function (for variables) yields objects that lie in the interpretation for the predicate. Quantified formulas are evaluated by determining the truth values for the variable substitutions in the sentences. A universal quantification is assigned the value 1 if the truth valuation for all the substitutions yields 1 in each case, and it is 0 otherwise. (An existential quantification is 1 if the truth valuation of one or more substitutions yields 1, and it is o otherwise.) Validity is truth preservation in all interpretations and variable-evaluations.
(Brief summaries for (Priest, “Multiple Denotation, Ambiguity, and the Strange Case of the Missing Amoeba”, section 1.1.1 and section 1.1.2)
So let us look again at the evaluation of the quantifiers from that article:
vs(∀vα) = ⊗{vs(v/a)(α); a ∈ D}
vs(∃vα) = ⊕{vs(v/a)(α); a ∈ D}
(Priest, “Multiple Denotation, Ambiguity, and the Strange Case of the Missing Amoeba”, section 1.1.2, p.362)
As we can see, to evaluate the universal quantifier, we make the substitutions for the variable with items in the domain, and if every substitution makes the formula true, then the universally quantified formula is true. The evaluation of the conjunction operator makes it such that even one false substitution will make the whole conjunction false and thus the universally quantified formula false. For the particular quantifier, we do something similar, only now the operator on the substitutions is disjunction. This means that if at least one substitution is true, the particularly quantified formula is true. Priest is using different notation now in our current section, but it seems to be the same structure:
In classical logic, the universal quantifier acts essentially like a conjunction over all the members of the domain. So ∀xA is something like Ax(kd1) ∧ Ax(kd2) ∧ . . . , where d1, d2, . . . are all the members of the domain.
(458)
As far as I can understand, this formula is saying that we substitute each member of the domain in for the variable of the formula, and if they are all true, then the universally quantified formula is true. The particular quantifier also seems to work like how we noted above with disjunction:
Dually, the particular quantifier is something like a disjunction over all members of the domain: ∃xA is Ax(kd1) ∧ Ax(kd2) ∨ . . . .
(458)
Priest’s final point here is that “It is natural to suppose that the two quantifiers should work the same way in a many-valued logic” (458).]
In classical logic, the universal quantifier acts essentially like a conjunction over all the members of the domain. So ∀xA is something like Ax(kd1) ∧ Ax(kd2) ∧ . . . , where d1, d2, . . . are all the members of the domain. Of course, if the domain is infinite, the conjunction is infinite, so one cannot actually express this in the language. (Though there are formal languages that permit infinite conjunctions and disjunctions.) But the sense is intuitively clear enough. Dually, the particular quantifier is something like a disjunction over all members of the domain: ∃xA is Ax(kd1) ∧ Ax(kd2) ∨ . . . . It is natural to suppose that the two quantifiers should work the same way in a many-valued logic.
(458)
[Using Greatest Lower Bound and Least Upper Bound to Evaluate Universal and Particular Quantification]
[We can use greatest lower bound (Glb) of conjuncts (that is, the greatest truth-value that is less than or equal to the values assigned to the conjuncts) and least upper bound (Lub) of disjuncts (that is, the least truth-value greater than or equal to the value assigned to either disjunct) in order to evaluate universal and particular quantification, respectively: we define “f∀(X) as Glb(X), so that v(∀xA) is the greatest lower bound of {v(Ax(kd)): d ∈ D}” and we “define f∃(X) as Lub(X), and v(∃xA) is the least upper bound of {v(Ax(kd)): d ∈ D}” (458). ]
[Priest next will explain how formulas built up by conjunction and disjunction, and thereby universal and particular quantification, can be evaluated in many-valued logics. For details, see section 8.4, section 11.4.9, and section 11a.4.2. We arrange the truth-values of V into some kind of ranking order from low to high. Then for conjunction, we value it at the greatest lower bound (Glb) value of the conjuncts, “that is, the greatest value that is less than or equal to” the values of the conjuncts. Thus we define “f∀(X) as Glb(X), so that v(∀xA) is the greatest lower bound of {v(Ax(kd)): d ∈ D}.” Disjunction is valued at the least upper bound (Lub), which is “the least value greater than or equal to” the value of either disjunct. Thus we “define f∃(X) as Lub(X), and v(∃xA) is the least upper bound of {v(Ax(kd)): d ∈ D}.”]
Taking this idea as our guide: in most many-valued logics, the truth values, V, are ordered in a certain way; when this is the case, v(A ∧ B) is naturally taken to be the greatest lower bound (Glb) of v(A) and v(B), that is, the greatest value that is less than or equal to v(A) and v(B) (see 11.4.9). If one of v(A) and v(b) is less than the other, then this is just the lesser of the two. But if neither is less than the other (which may happen if the order is not a linear one), then the Glb will be distinct from both of them. Thus, as we saw in 8.4, First Degree Entailment may be formulated as a four-valued logic, where the values are not linearly ordered. In FDE, if v(A) = n and v(b) = b, then v(A ∧ B) = 0. Generalising this to the infinite case, it is natural to define f∀(X) as Glb(X), so that v(∀xA) is the greatest lower bound of {v(Ax(kd)): d ∈ D}. Dually, in most logics with an ordering, v(A ∨ B) is naturally taken to be the least upper bound (Lub) of v(A) and v(B), that is, the least value greater than or equal to v(A) and v(B). So we may define f∃(X) as Lub(X), and v(∃xA) is the least upper bound of {v(Ax(kd)): d ∈ D}.
(458)
[A Possible Problem with This Bound Technique]
[We cannot use this greatest lower bound and least upper bound evaluation technique when the given set of truth values have none. But none of the logics we consider here are like that anyway.]
[(ditto)]
There is a rub. In some orderings, some sets may have no Glb or Lub. Thus, consider the integers ordered in the usual way: . . . , −2, −1, 0, 1, 2, . . . Any finite set of these has a Glb and a Lub, the least and the greatest member of the set, respectively. But the set of positive numbers has no upper bound at all, and a fortiori, no least upper bound. And the set of negative numbers has no lower bound, and a fortiori, no greatest lower bound. In cases where sets of semantic values may not have a Glb or a Lub, then, we cannot proceed in the way suggested. Fortunately, for the logics of concern in the present book, this is not something we will have to worry about.1
(459)
1. Interactions between the ordering and the set of designated values can also produce odd consequences. For example, if, in the ordering, there are undesignated values higher | than designated values, then it is possible for v(∀xA) to be designated whilst v(Ax(a)) is not. In this case, universal instantiation will fail to be valid. Consequences of this kind will also not feature in any of the particular many-valued logics with which we will be concerned in this book.
(492-493)
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From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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