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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
21.7
Neutral Free Logics
Brief summary:
(21.7.1) We will now examine neutral free logics, where applying a predicate to a non-existent object always results in the semantic value neither true nor false (i). (21.7.2) Free logics are rendered neutral free logics by the addition of the neutrality constraint: if, for some 1 ≤ j ≤ n, dj ∉ E, then v(P)(d1, . . . , dn) = i. (In other words, formulas that predicate at least one non-existent object will be valued i, here understood as neither true nor false.) (21.7.3) Neutral free logics can alternatively be defined by using only a domain E of existents and by using the v denotation function for names as a partial function that leaves some values undefined, and so:
if v(a1) = d1, . . . , v(an) = dn then v(Pa1 . . . an) = v(P)(d1 . . . dn)
if any of v(a1), …, v(an) is undefined, v(Pa1 . . . an) = i
(466)
(21.7.4) We can use this strategy also to give an alternative definition for negative free logics: “The denotation function for names is taken to be partial, and the truth conditions of atomic sentences are given as [in the section above], replacing ‘= i’ with ‘≠ 1’ ” (466). So (I presume, perhaps incorrectly):
if v(a1) = d1, . . . , v(an) = dn then v(Pa1 . . . an) = v(P)(d1 . . . dn)
if any of v(a1), …, v(an) is undefined, v(Pa1 . . . an) ≠ 1
(21.7.5) “The Neutrality Constraint gives rise to valid inferences that are not valid in a positive free logic. For example […], Pa1 . . . an ⊨ ℭa1 ∧ . . . ∧ ℭan and ¬ Pa1 . . . an ⊨ ℭa1 ∧ . . . ∧ ℭan. Negative free logics make the first of these valid, but not the second” (466). (21.7.6) In neutral free logics, we would say that statements with non-existent objects like “The greatest prime number is even” and “The King of France is bald” are valued i or neither true nor false. But we cannot say that all statements with non-existent objects are neither true nor false. “For it would seem that ‘The greatest prime number exists’ and ‘The King of France exists’ are both false, not neither true nor false” (466). But, that prevents there from being an obvious formal standard for determining which statements are exceptions. And so, by making one arbitrary exception for existence statements of non-existent objects, what stops us from making other exceptions, like saying that certain statements regarding non-existents are true, like “Homer worshipped Zeus” and “I am thinking about Sherlock Holmes”? (21.7.7) “Hence, though some sentences with non-denoting terms may be neither true nor false, not all would seem to be; the most appropriate free logic, even in a many-valued context, would appear to be a positive one” (467).
[Non-Existent Objects as Creating Gaps, in Neutral Free Logics]
[The Neutrality Constraint]
[An Alternative Definition of Neutral Free Logics Using a Partial v Function]
[Using This Strategy to Alternatively Define Negative Free Logics]
[Additional Valid Inferences in Neutral Free Logics]
[A Problem with Neutral Free Logics: Arbitrary Exceptions]
[Positive Free Logics as the Best for Many Values]
Summary
[Non-Existent Objects as Creating Gaps, in Neutral Free Logics]
[We will now examine neutral free logics, where applying a predicate to a non-existent object always results in the semantic value neither true nor false (i). ]
[In chapter 13 we dealt with free logics. They involve distinguishing a subset of the domain that is the set of existent things, with the remainder being non-existent ones (see section 13.2.2). In section 13.4 we distinguished positive, negative, and neutral free logics. And in section 13.4.1 we saw that this distinction between existent and non-existent domain members eliminated certain problematic inferences, like the inference that anything that can be predicated must exist (Ax(a) ⊨ ∃xA) and that it is impossible for nothing to exist (∃x(Px ∨ ¬Px).) But in section 13.4.2, we learned that:
Some might still want to use free logics to accommodate non-existing things, but they might think that non-existing things should not have positive properties. For, while existing things have such tangible, physical properties that allow them to be seen and be physically interactable, non-existing things do not. (So we might want to say that Sherlock Holmes is in our domain, but we might also want to say that as a non-existing object, he cannot actually live on Baker St. For, only physically real things can have spatial location.) To disallow non-existing objects from having positive properties, we could apply the negativity constraint: If ⟨d1, . . . , dn⟩ ∈ v(P) then d1 ∈ v(ℭ), and …and dn ∈ v(ℭ). (In other words, if something belongs to a predicate, it needs to be an existent thing.) Free logics with the negativity constraint are called negative free logics.
(From the brief summary of section 13.4.2)
And finally, in section 13.4.7 we noted that:
An alternative to negative free logics would be neutral free logics, which say that sentences containing names that do not refer to existent objects would be neither true nor false. We deal with this in ch.21.
(From the brief summary of section 13.4.7)
Priest notes now that “In positive free logics, applying a predicate to a non-existent object can result in any semantic value. In negative logics, it always results in the value false (0). In a neutral logic it is always neither true nor false (i);” and our particular focus now is on neutral logics now, as we covered the other two types before and postponed the neutral ones for this section.]
In 13.4 we noted that free logics can be classified as positive, negative, or neutral. In positive free logics, applying a predicate to a non-existent object can result in any semantic value. In negative logics, it always results in the value false (0). In a neutral logic it is always neither true nor false (i). We looked at positive and negative free logics in chapter 13. We are now in a position to see what a neutral free logic is like.
(465)
[The Neutrality Constraint]
[Free logics are rendered neutral free logics by the addition of the neutrality constraint: if, for some 1 ≤ j ≤ n, dj ∉ E, then v(P)(d1, . . . , dn) = i. (In other words, formulas that predicate at least one non-existent object will be valued i, here understood as neither true nor false.)]
[We next examine the neutrality constraint, which when added to a free logic will generate a neutral free logic. I am not certain about how this formulation works, so please see the quotation below. It says specifically:
if, for some 1 ≤ j ≤ n, dj ∉ E, then v(P)(d1, . . . , dn) = i
I will guess at the meaning. Overall, it seems to be saying that formulas (which in quantified logic contain predicates) that predicate at least one non-existent object will be valued i, which is presumably here neither true nor false.]
A neutral free logic is a logic with a value which may be thought of as neither true nor false, such as i in K3 or Ł3 (or the value n in FDE – see the next chapter), which satisfies the condition that for any n-place predicate:
if, for some 1 ≤ j ≤ n, dj ∉ E, then v(P)(d1, . . . , dn) = i.
Call this the Neutrality Constraint. (Depending on the context, the converse condition might also be plausible: if v(P)(d1, . . . , dn) = i then, for some 1 ≤ j ≤ n, dj ∉ E. Only non-existent objects give rise to truth value gaps.) Note that the Negativity Constraint can be added just as much to a many-valued logic as it can be to a two-valued logic, giving rise to a many-valued negative free logic.
(465)
[An Alternative Definition of Neutral Free Logics Using a Partial v Function]
[Neutral free logics can alternatively be defined by using only a domain E of existents and by using the v denotation function for names as a partial function that leaves some values undefined, and so: “if v(a1) = d1, . . . , v(an) = dn then v(Pa1 . . . an) = v(P)(d1 . . . dn) ; if any of v(a1), …, v(an) is undefined, v(Pa1 . . . an) = i” (466).]
[(ditto)]
Neutral free logics can be formulated in a different, but equivalent, way. We may dispense with the ‘outer domain’ altogether. The only domain we need is E. Instead of taking the denotation function for names, v, to be a total function, we let it be partial. That is, for some inputs the output may just not be defined – just as division is not defined if the divisor is zero. (Division is, in fact, a partial function.) The appropriate truth conditions for atomic sentences are then:
if v(a1) = d1, . . . , v(an) = dn then v(Pa1 . . . an) = v(P)(d1 . . . dn)
if any of v(a1), …, v(an) is undefined, v(Pa1 . . . an) = i.
It is not difficult to see that the truth value of any sentence comes out the same under this policy. (The truth conditions make this clear for atomic sentences. For other formulas, this follows by a simple induction.)
(466)
[Using This Strategy to Alternatively Define Negative Free Logics]
[We can use this strategy also to give an alternative definition for negative free logics: “The denotation function for names is taken to be partial, and the truth conditions of atomic sentences are given as [in the section above], replacing ‘= i’ with ‘≠ 1’ ” (466). So: if v(a1) = d1, . . . , v(an) = dn then v(Pa1 . . . an) = v(P)(d1 . . . dn) ; if any of v(a1), …, v(an) is undefined, v(Pa1 . . . an) ≠ 1 ]
[(ditto)]
Note that we can follow the same strategy with respect to negative free logics as well. The denotation function for names is taken to be partial, and the truth conditions of atomic sentences are given as in 21.7.3, replacing ‘= i’ with ‘≠ 1’.2
(466)
2. An even stronger constraint replaces ‘= i’ with ‘= 0’. But this constraint, equivalent in a classical context, is less natural in a many-valued context. The intuition behind the Negativity Constraint is simply that atomic sentences containing names that do not refer to (existent) objects cannot be true.
(466)
[Additional Valid Inferences in Neutral Free Logics]
[“The Neutrality Constraint gives rise to valid inferences that are not valid in a positive free logic. For example […], Pa1 . . . an ⊨ ℭa1 ∧ . . . ∧ ℭan and ¬ Pa1 . . . an ⊨ ℭa1 ∧ . . . ∧ ℭan. Negative free logics make the first of these valid, but not the second” (466).]
[(ditto)]
The Neutrality Constraint gives rise to valid inferences that are not valid in a positive free logic. For example, as is easy to check, Pa1 . . . an ⊨ ℭa1 ∧ . . . ∧ ℭan and ¬ Pa1 . . . an ⊨ ℭa1 ∧ . . . ∧ ℭan. Negative free logics make the first of these valid, but not the second.
(466)
[A Problem with Neutral Free Logics: Arbitrary Exceptions]
[In neutral free logics, we would say that statements with non-existent objects like “The greatest prime number is even” and “The King of France is bald” are valued i or neither true nor false. But we cannot say that all statements with non-existent objects are neither true nor false. “For it would seem that ‘The greatest prime number exists’ and ‘The King of France exists’ are both false, not neither true nor false” (466). But, that prevents there from being an obvious formal standard for determining which statements are exceptions. And so, by making one arbitrary exception for existence statements of non-existent objects, what stops us from making other exceptions, like saying that certain statements regarding non-existents are true, like “Homer worshipped Zeus” and “I am thinking about Sherlock Holmes”?]
[(ditto) (Note: I am not entirely sure I grasp the problem with exceptions. Is it simply that we have no way to formally determine which exceptions there should be? Is it that by having these arbitrary exceptions, we somehow make it needless to have the neutrality constraint to begin with?)]
Neutral free logics are usually motivated by examples such as ‘The greatest prime number is even’ and ‘The King of France is bald’. But note that one would seem to have to make exceptions for the existence predicate itself. For it would seem that ‘The greatest prime number exists’ and ‘The King of France exists’ are both false, not neither true nor false. And once one has made an exception for one predicate, it seems somewhat arbitrary not to admit other exceptions, such as those we noted in connection with negative free logics in 13.4.6.
(466)
[Positive Free Logics as the Best for Many Values]
[“Hence, though some sentences with non-denoting terms may be neither true nor false, not all would seem to be; the most appropriate free logic, even in a many-valued context, would appear to be a positive one” (467).]
[(ditto)]
Hence, though some sentences with non-denoting terms may be neither true nor false, not all would seem to be; the most appropriate free logic, even in a many-valued context, would appear to be a positive one.
(467)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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