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 II:
Quantification and Identity
21
Many-valued Logics
21.2
Quantified Many-valued Logics
Brief summary:
(21.2.1) A “propositional many-valued logic is characterised by a structure ⟨V, D,{fc: c ∈ C}⟩, where V is the set of truth values, D ⊆ V is the set of designated values, and for each connective, c, fc is the truth function it denotes. An interpretation, v, assigns values to propositional parameters; | the values of all formulas can then be computed using the fcs; and a valid inference is one that preserves designated values in every interpretation” (456-457). (21.2.2) “A quantified many-valued logic is characterised by a structure of the form ⟨D, V, D,{fc: c ∈ C}, {fq: q ∈ Q}⟩. V, D, and {fc: c ∈ C} are as before. D is a non-empty domain of quantification, and if q is the set of quantifiers in the language, for every q ∈ Q, fq is a map from subsets of V into V. (In a free many-valued logic, there is an extra component, the inner domain, E, and E ⊆ D.)” (457). (21.2.3) An “evaluation, v, assigns every constant a member of D and every n-place predicate an n-place function from the domain into the truth values. (So if P is any predicate, v(P) is a function with inputs in D and an output in V.) Given an evaluation, every formula, A, is then assigned a value, v(A), in V recursively, as follows. If P is any n-place predicate: v(Pa1 . . . an) = v(P)(v(a1), . . . , v(an)) . For each n-place propositional connective, c: v(c(A1, . . . , An)) = fc(v(A1), . . . , v(An)) as in the propositional case. And for each quantifier, q: v(qxA) = fq({v(Ax(kd)): d ∈ D}) . (In a free many-valued logic, ‘D’ is replaced by ‘E’.)” (357). (21.2.4) An “inference is valid if it preserves designated values. Thus, Σ ⊨ A iff for every interpretation, whenever v(B) ∈ D, for all B ∈ Σ, v(A) ∈ D” (457).
[The Structure of Propositional (Non-Quantificational) Many-Valued Logics]
[The Structure of Quantified Many-Valued Logics]
[The Evaluation Function]
[Validity]
Summary
[The Structure of Propositional (Non-Quantificational) Many-Valued Logics]
[A “propositional many-valued logic is characterised by a structure ⟨V, D,{fc: c ∈ C}⟩, where V is the set of truth values, D ⊆ V is the set of designated values, and for each connective, c, fc is the truth function it denotes. An interpretation, v, assigns values to propositional parameters; | the values of all formulas can then be computed using the fcs; and a valid inference is one that preserves designated values in every interpretation” (456-457).]
[Priest begins by reviewing propositional many-valued logic from section 7.2.2. Recall that this structure also works for classical logic:
Let C be the class of connectives of classical propositional logic {∧,∨,¬, ⊃}. The classical propositional calculus can be thought of as defined by the structure ⟨V, D, {fc; c ∈ C}⟩. V is the set of truth values {1,0}. D is the set of designated values {1}; these are the values that are preserved in valid inferences. For every connective, c, fc is the truth function it denotes. Thus, f¬ is a one-place function such that f¬(0) = 1 and f¬(1) = 0; f∧ is a two-place function such that f∧(x, y) = 1 if x = y = 1, and f∧(x, y) = 0 otherwise; and so | on. These functions can be (and often are) depicted in the following ‘truth tables’.
| f¬ | |
| 1 | 0 |
| 0 | 1 |
| f∧ | 1 | 0 |
| 1 | 1 | 0 |
| 0 | o | o |
(pp.120-121, section 7.2.2)
And we applied it to a 3-valued logic in section 7.3 (the following is from the brief summary):
The structure of many-valued logics can be formulated as:
⟨V, D, {fc; c ∈ C}⟩
V is the set of assignable truth values. D is the set of designated values, which are those that are preserved in valid inferences (like 1 for classical bivalent logic). C is the set of connectives. c is some particular connective. And fc is the truth function corresponding to some connective, and it operates on the truth values of the formula in question. In classical logic: the assignable truth values of V are true and false, or 1 and 0; the designated values are just 1, and the connectives are: f¬, f∧, f∨, f⊃. [A ≡ B we are defining as (A ⊃ B) ∧ (B ⊃ A)] And finally, the connective functions operate on truth values in accordance with certain rules (displayed often as the truth tables for connectives that we are familiar with. [This was covered in the previous section.] K3 (strong Kleene three-valued logic) and Ł3 are two sorts of three-valued logics. Both keep D as {1}, and both extend V to {1, i, 0}. 1 is true , 0 is false, and i is neither true nor false. We have the same connectives (excluding for simplicity the biconditional), and the connective functions associated with them are defined in the following way. K3 in particular has these assignments for the connective functions:
| f¬ | |
| 1 | 0 |
| i | i |
| 0 | 1 |
| f∧ | 1 | i | 0 |
| 1 | 1 | i | 0 |
| i | i | i | 0 |
| 0 | 0 | 0 | 0 |
| f∨ | 1 | i | 0 |
| 1 | 1 | 1 | 1 |
| i | 1 | i | i |
| 0 | 1 | i | 0 |
| f⊃ | 1 | i | 0 |
| 1 | 1 | i | 0 |
| i | 1 | i | i |
| 0 | 1 | 1 | 1 |
One problem with K3 is that every formula can obtain the undesignated value i by assigning all of its propositional parameters the value of i. (Simply look at the tables above where the input values are i. You will see in all cases the output is i too.) This means there are no logical truths in K3. One remedy for making the law of identity a logical truth is by changing the value assignment for the conditional such that when both antecedent and consequent are i, the whole conditional is 1.
| f⊃ | 1 | i | 0 |
| 1 | 1 | i | 0 |
| i | 1 | 1 | i |
| 0 | 1 | 1 | 1 |
This new system, where everything else is identical to K3except for the above alternate valuation for the conditional, is called Ł3 (Łukasiewicz’ three-valued logic).
(From the brief summary to section 7.3)
]
As we saw in 7.2.2, a propositional many-valued logic is characterised by a structure ⟨V, D,{fc: c ∈ C}⟩, where V is the set of truth values, D ⊆ V is the set of designated values, and for each connective, c, fc is the truth function it denotes. An interpretation, v, assigns values to propositional parameters; | the values of all formulas can then be computed using the fcs; and a valid inference is one that preserves designated values in every interpretation.
(456-457)
[The Structure of Quantified Many-Valued Logics]
[“A quantified many-valued logic is characterised by a structure of the form ⟨D, V, D,{fc: c ∈ C}, {fq: q ∈ Q}⟩. V, D, and {fc: c ∈ C} are as before. D is a non-empty domain of quantification, and if q is the set of quantifiers in the language, for every q ∈ Q, fq is a map from subsets of V into V. (In a free many-valued logic, there is an extra component, the inner domain, E, and E ⊆ D.)” (457).]
[Let us now go part by part through the addition of quantification into this many-valued system. First recall from section 21.2.1 above that the structure without the quantification element is:
⟨V, D,{fc: c ∈ C}⟩
Now Priest writes:
A quantified many-valued logic is characterised by a structure of the form ⟨D, V, D,{fc: c ∈ C}, {fq: q ∈ Q}⟩. V, D, and {fc: c ∈ C} are as before.
(457)
By saying “V, D, and {fc: c ∈ C} are as before”, Priest is pointing out what has remained from the propositional structuration. And that leaves us with:
D is a non-empty domain of quantification
We saw this in section 12.3.1:
An interpretation of the language is a pair, ℑ = ⟨D, v⟩. D is a non-empty set (the domain of quantification); v is a function such that:
• if c is a constant, v(c) is a member of D
• if P is an n-place predicate, v(P) is a subset of Dn
(Dn is the set of all n-tuples of members of D, {⟨d1, ..., dn⟩: d1, ..., dn ∈ D}. By convention, ⟨d⟩ is just d, and so D1 is D.)
(Priest 264, section 12.3.1. Note, the first formulation should look like:
)
Next Priest writes:
and 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 recall from section 21.2.1 above that V is the set of truth values. And recall from section 12.2.3 that the quantifiers are ∀ and ∃:
If A is any formula, and x is any variable, then ∀xA, ∃xA are formulas. I will omit outermost brackets in formulas.
(p.264, section 12.2.3)
And in section 12.3.2, we saw how formulas are evaluated in quantification logic, including the quantifiers:
Given an interpretation, truth values are assigned to all closed formulas. To state the truth conditions, we extend the language to ensure that every member of the domain has a name. For all d ∈ D, we add a | constant to the language, kd, such that v(kd) = d. The extended language is the language of ℑ, and written L(ℑ). The truth conditions for (closed) atomic sentences are:
v(Pa1 ... an) = 1 iff ⟨v(a1), ..., v(an)⟩ ∈ v(P) (otherwise it is 0)
The truth conditions for the connectives are as in the propositional case (1.3.2). For the quantifiers:
v(∀xA) = 1 iff for all d ∈ D, v(Ax(kd)) = 1 (otherwise it is 0)
v(∃xA) = 1 iff for some d ∈ D, v(Ax(kd)) = 1 (otherwise it is 0)
(264-265)
So again, Priest is writing now:
if q is the set of quantifiers in the language, for every q ∈ Q, fq is a map from subsets of V into V.
My best guess is that here, for the quantifiers ∀ and ∃, there are functions that take as their input values the truth-values for the formulas they quantify and output new values for the whole quantified formula, but I am not sure if that is so, and if it is so, I am not sure yet how that works. (There is more elaboration in section 21.2.3 below). Lastly Priest writes:
In a free many-valued logic, there is an extra component, the inner domain, E, and E ⊆ D.
Recall from section 13.2.1 and section 13.2.2 that in free logics we have the existence predicate ℭ, whose denotation is the set of existents in the inner domain E, which is a subset of the outer domain D, with the remainder in the domain being non-existent things.]
A quantified many-valued logic is characterised by a structure of the form ⟨D, V, D,{fc: c ∈ C}, {fq: q ∈ Q}⟩. V, D, and {fc: c ∈ C} are as before. D is a non-empty domain of quantification, and if q is the set of quantifiers in the language, for every q ∈ Q, fq is a map from subsets of V into V. (In a free many-valued logic, there is an extra component, the inner domain, E, and E ⊆ D.)
(457)
[The Evaluation Function]
[An “evaluation, v, assigns every constant a member of D and every n-place predicate an n-place function from the domain into the truth values. (So if P is any predicate, v(P) is a function with inputs in D and an output in V.) Given an evaluation, every formula, A, is then assigned a value, v(A), in V recursively, as follows. If P is any n-place predicate: v(Pa1 . . . an) = v(P)(v(a1), . . . , v(an)) . For each n-place propositional connective, c: v(c(A1, . . . , An)) = fc(v(A1), . . . , v(An)) as in the propositional case. And for each quantifier, q: v(qxA) = fq({v(Ax(kd)): d ∈ D}) . (In a free many-valued logic, ‘D’ is replaced by ‘E’.)” (357)]
[We now will examine how we evaluate constants and predicates in our quantified many-valued logic. We do so by an evaluation function v. For every constant it assigns a member of the domain D. (At this point it gets a bit complicated, and I may have this wrong. So please see the quotation below.) v also assigns “every n-place predicate an n-place function from the domain into the truth values. (So if P is any predicate, v(P) is a function with inputs in D and an output in V.)” (It is not obvious to me how this works, so let us look first at a simpler system. Recall the following regarding truth-conditions in quantified logic, from section 12.3.2:
Given an interpretation, truth values are assigned to all closed formulas. To state the truth conditions, we extend the language to ensure that every member of the domain has a name. For all d ∈ D, we add a | constant to the language, kd, such that v(kd) = d. The extended language is the language of ℑ, and written L(ℑ). The truth conditions for (closed) atomic sentences are:
v(Pa1 ... an) = 1 iff ⟨v(a1), ..., v(an)⟩ ∈ v(P) (otherwise it is 0)
The truth conditions for the connectives are as in the propositional case (1.3.2). For the quantifiers:
v(∀xA) = 1 iff for all d ∈ D, v(Ax(kd)) = 1 (otherwise it is 0)
v(∃xA) = 1 iff for some d ∈ D, v(Ax(kd)) = 1 (otherwise it is 0)
(pp.264-265, section 12.3.2)
As we can see, there the v both assigned constants to domain members, and it also assigned truth-values to predicates on the basis of those constant assignments. So if the constants in a predicate are assigned to domain members such that they form n-tuples that are assigned to the predicate, then the predicate is assigned as true. But here in our current section, I am not sure if that is still the case. It merely says, as we quoted above, that v also assigns “every n-place predicate an n-place function from the domain into the truth values. (So if P is any predicate, v(P) is a function with inputs in D and an output in V.)” I do not quite get that yet. I will wildly and vaguely guess that the idea is the following. We have a predicate P. Suppose it is a one-place predicate like, “is green.” We have members in our domain. Let us say that they are “grass” and “the sun.” Let v assign g to “grass” and s to “the sun”. Now, under the older formulations above, we would say that v assigns to P the 1-tuple <grass> (or g maybe). But now we are saying that it assigns a function which takes the input values from D and a truth-value output. So for example, perhaps, it assigns true for “grass” and false for “the sun”. That is the best I can gather from the wording right now. The connectives and the quantifiers seem to work in a similar way, where there is a function that will assign the truth-values in accordance with what we would want the outputs to be for that operator.]
Given this structure, an evaluation, v, assigns every constant a member of D and every n-place predicate an n-place function from the domain into the truth values. (So if P is any predicate, v(P) is a function with inputs in D and an output in V.) Given an evaluation, every formula, A, is then assigned a value, v(A), in V recursively, as follows. If P is any n-place predicate:
v(Pa1 . . . an) = v(P)(v(a1), . . . , v(an))
For each n-place propositional connective, c:
v(c(A1, . . . , An)) = fc(v(A1), . . . , v(An))
as in the propositional case. And 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}). Thus, the value of qxA is determined by the set of the values of substitution instances of A formed using the names of all members of the domain of quantification.
(457)
[Validity]
[An “inference is valid if it preserves designated values. Thus, Σ ⊨ A iff for every interpretation, whenever v(B) ∈ D, for all B ∈ Σ, v(A) ∈ D” (457).]
[(ditto)]
As in the propositional case, an inference is valid if it preserves designated values. Thus, Σ ⊨ A iff for every interpretation, whenever v(B) ∈ D, for all B ∈ Σ, v(A) ∈ D.
(457)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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