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
21.5
Their Free Versions
Brief summary:
(21.5.1) Quantified many-valued logics still have the problematic inferences Pa ⊨ ∃xPx and ∀xPx ⊨ Pa, and they can be solved using free logics. (21.5.2) Our quantified 3-valued logics are structured in the following way: “We take the language to contain an existence predicate, ℭ. An interpretation is a triple ⟨D, E, v⟩. D is the domain of all objects, and E ⊆ D contains those that are thought of as existent. For every constant, c, v(c) ∈ D. For every n-place predicate, P, v(P) is a function such that if d1, . . . , dn ∈ D, v(P)(d1, . . . , dn) ∈ V. v(ℭ) is such that: v(ℭ)(d) ∈ D iff d ∈ E . Truth conditions are as in the non-free case, except that for the quantifiers v(∀xA) = Min({Ax(kd): d ∈ E}) (not D), and v(∃xA) = Max({Ax(kd): d ∈ E})” (461-462). (21.5.3) Using these semantics, counter-models can be constructed for the above problematic inferences. (21.5.4) In our free version of many-valued logics, we establish validity and invalidity in the same way as the non-free versions. When D = E, then “anything valid in any many-valued free logic is valid in the corresponding non-free logic.” “Conversely, suppose that the inference with premises Σ and conclusion A is valid in one of our 3-valued logics. Let C be the set of constants that occur in A and all members of Σ, and let Π = {ℭc: c ∈ C} ∪ {∃x ℭx}. (The quantified sentence is redundant if C ≠ φ.) Then Π∪Σ ⊨ A in the corresponding free logic (where quantifiers are inner)” (462).
[Using Free Logics to Solve Problematic Inferences in Many-Valued Quantified Logics]
[The Structure and Truth-Conditions of Our 3-Valued Logics]
[The Invalidity of the Problematic Inferences Showable Using Counter-Models]
[Validity and Invalidity in Free Many-Valued Logics. Comparisons with Non-Free Versions.]
Summary
[Using Free Logics to Solve Problematic Inferences in Many-Valued Quantified Logics]
[Quantified many-valued logics still have the problematic inferences Pa ⊨ ∃xPx and ∀xPx ⊨ Pa, and they can be solved using free logics.]
[Recall from section 12.6.3 a problem in quantified logic where anything predicated must exist (thus Pegasus, which is predicated as a mythological figure, as having wings, etc., must exist), on account of the validity of Pa ⊨ ∃xPx. Priest says that this is valid in 3-valued logics too. Priest also notes that ∀xPx ⊨ Pa is also valid and is problematic, but I am not sure where we have seen it before (if it is in section 12.6, I have not found it yet. I came across something similar in section 15.2.3. So maybe the problem with it comes about when a is non-existent. Maybe furthermore this is problematic if P is taken to be the existence predicate. But I am just guessing very poorly here, sorry.) Now recall from from chapter 13 how free logics deal with these problems by using free logics. Regarding free logics, recall now from the brief summary of section 13.2 that in
(13.2.1) […] free logics we have the one-place existence predicate ℭ. We can think of ℭa as meaning ‘a exists’. (13.2.2) In free logics, we have our main domain of all objects, D, and we have the “inner domain” E. It is a a subset of D that we think of as being the set of all existent objects. (“An interpretation for the language is a triple ⟨D, E, v⟩, where D is a non-empty set, and E (the ‘inner domain’) is a (possibly empty) subset of D. One can think of D as the set of all objects, and E as the set of all existent objects” (290).) So suppose D contains Sherlock Holmes, the Pegasus, and Julius Caesar. Here, although all of them are in D, only Caesar is in E. (13.2.3) “As in classical logic, v assigns every constant in the language a member of D, and every n-place predicate a subset of Dn. In any interpretation, v(ℭ) = E” (290).
(from the brief summary of section 13.2, quotation is Priest’s)
In section 13.4 and section 13.5 we examined these solutions more closely. The important point now is that we can use free logic in a similar way to solve these problems in many-valued quantified logics.]
It is not difficult to check that in all the 3-valued logics in our compass
Pa ⊨ ∃xPx
∀xPx ⊨ Pa
Thus, for the first, if Pa is designated in an interpretation then v(P)(v(a)) ∈ D, in which case v(∃xPx) ∈ D. But one might well have reservations about these inferences, as we have already observed in 12.6. And just as one can formulate a free version of classical logic, as we did in chapter 13, one can formulate free versions of many-valued logics.
(461)
[The Structure and Truth-Conditions of Our 3-Valued Logics]
[Our quantified 3-valued logics are structured in the following way: “We take the language to contain an existence predicate, ℭ. An interpretation is a triple ⟨D, E, v⟩. D is the domain of all objects, and E ⊆ D contains those that are thought of as existent. For every constant, c, v(c) ∈ D. For every n-place predicate, P, v(P) is a function such that if d1, . . . , dn ∈ D, v(P)(d1, . . . , dn) ∈ V. v(ℭ) is such that: v(ℭ)(d) ∈ D iff d ∈ E . Truth conditions are as in the non-free case, except that for the quantifiers v(∀xA) = Min({Ax(kd): d ∈ E}) (not D), and v(∃xA) = Max({Ax(kd): d ∈ E})” (461-462).]
[We will now structure the quantified many-valued free logic similarly to what we saw above in section 21.5.1 in reference to section 13.2. We will have the existence predicate ℭ, whose denotation is the set of existent things in the domain, which is the subset E of the domain D, with the remainder being the set of non-existent things in the domain. Every constant is assigned a member in the domain. I am not entirely sure I understand the predicate evaluation, but it seems to be that the v function assigns a truth-value to predicates when its assigned constituents are in the domain (see the quotation below). I also may not follow the evaluation of the existence predicate, but it seems to be saying that the value of an existence predicate is a designated value only if the predicated thing is in the set of existent things. But I am not sure why we are using the designated values here. Finally, the truth conditions are the same as in the non-free cases, except for how the quantifiers are truth-evaluated. In section 21.2.3, Priest discusses the evaluation of formulas:
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.
(p.457, section 21.2.3)
I am not exactly sure how the connectives are evaluated, but I will guess it is in accordance with the tables given in section 7.3 and section 7.4. The quantifiers are evaluated as:
v(∀xA) = Min({Ax(kd): d ∈ E}) (not D)
v(∃xA) = Max({Ax(kd): d ∈ E}).
]
We take the language to contain an existence predicate, ℭ. An interpretation is a triple ⟨D, E, v⟩. D is the domain of all objects, and E ⊆ D contains those that are thought of as existent. For every constant, c, v(c) ∈ D. For every n-place predicate, P, v(P) is a function such that if d1, . . . , dn ∈ D, v(P)(d1, . . . , dn) ∈ V. v(ℭ) is such that:
v(ℭ)(d) ∈ D iff d ∈ E
| Truth conditions are as in the non-free case, except that for the quantifiers v(∀xA) = Min({Ax(kd): d ∈ E}) (not D), and v(∃xA) = Max({Ax(kd): d ∈ E}).
(461-462)
[The Invalidity of the Problematic Inferences Showable Using Counter-Models]
[Using these semantics, counter-models can be constructed for the above problematic inferences.]
[(ditto). (Note: I have not yet carried out these exercises, but I would imagine they would be done using the trial and error method shown in section 21.4.6.)]
It is now not difficult to construct counter-models to the inferences of 21.5.1. Details are left as an exercise.
(462)
[Validity and Invalidity in Free Many-Valued Logics. Comparisons with Non-Free Versions.]
[In our free version of many-valued logics, we establish validity and invalidity in the same way as the non-free versions. When D = E, then “anything valid in any many-valued free logic is valid in the corresponding non-free logic.” “Conversely, suppose that the inference with premises Σ and conclusion A is valid in one of our 3-valued logics. Let C be the set of constants that occur in A and all members of Σ, and let Π = {ℭc: c ∈ C} ∪ {∃x ℭx}. (The quantified sentence is redundant if C ≠ φ.) Then Π∪Σ ⊨ A in the corresponding free logic (where quantifiers are inner)” (462)]
[(This last part is a bit complex, so see the quotation below. But I will stumble through it a bit up to then.) We establish validity and invalidity in the same manner as with the non-free versions of these logics. As we saw in section 21.4, we use argumentation techniques like reductio (section 21.4.4) and contraposition (section 21.4.5) to show validity. To make counter-models to show invalidity, we use a trial and error method (section 21.4.6). (I cannot say much more on this right now, as I did not try to learn those techniques yet.) Priest now has us note “the special case of a free interpretation where D = E is a non-free interpretation. Hence, anything valid in any many-valued free logic is valid in the corresponding non-free logic” (462). I do not quite grasp what that means yet. I suppose we are talking about cases of free logic interpretations where the set of existents is identical to the whole domain. And so, it would seem to me, to be structured at its basis as a non-free logic, with the E set being redundant to the domain. The final point in this paragraph I am not following, but I will stumble through it a bit. We think of an argument I think in a non-free 3-valued logic with a set of premises that we call Σ and the conclusion called A. In the premises and conclusion there may be constants. Now we form another set. First recall from section 0.1.8 that a union is defined in the following way:
The union of two sets, X, Y, is the set containing just those things that are in X or Y (or both). This is written as X ∪ Y. So a ∈ X ∪ Y if and only if a ∈ X or a ∈ Y.
(page xxviii, section 0.1.8)
So this other set, called Π, is defined as
{ℭc: c ∈ C} ∪ {∃x ℭx}
But I may not get that well. It seems to be the union of two sets, with the first set being the set of all the existing constants in the premises and conclusion and another set which I cannot grasp, but is maybe the set of existentially quantifiable members of the domain. I probably have that wrong, but even if it is correct, it would seem to be redundant with the first set. In fact, the next sentence reads, “(The quantified sentence is redundant if C ≠ φ.)” But I am not sure really how all this works. Next Priest writes, “Then Π∪Σ ⊨ A in the corresponding free logic (where quantifiers are inner)” (462). I do not grasp this either, but it seems to be saying that on the basis of the union of the set of existing things mentioned in the premises and conclusion of the argument along with those premises themselves, we can infer the conclusion. So I am really guessing here, but maybe that is like saying we start with a non-free logic, and we take the premises, and then we affirm the existence of items they are predicating as well as the existence of the conclusion’s items, and we can infer the conclusion itself in the free logic version of the 3-valued logic. Please see the quotation, as I am guessing poorly here.]
To establish the validity or invalidity of inferences in the free version of a many-valued logic, we may proceed as in the non-free case. But note the special case of a free interpretation where D = E is a non-free interpretation. Hence, anything valid in any many-valued free logic is valid in the corresponding non-free logic. Conversely, suppose that the inference with premises Σ and conclusion A is valid in one of our 3-valued logics. Let C be the set of constants that occur in A and all members of Σ, and let Π = {ℭc: c ∈ C} ∪ {∃x ℭx}. (The quantified sentence is redundant if C ≠ φ.) Then Π∪Σ ⊨ A in the corresponding free logic (where quantifiers are inner). (This is true even when the language contains the identity predicate, and is proved in 21.11.6.)
(462)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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