Priest (21.6) An Introduction to Non-Classical Logic, ‘Existence and Quantification,’ summary

 

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

 

21.6

Existence and Quantification

 

 

 

 

Brief summary:

(21.6.1) We can add inner and outer quantifiers to our quantified 3-valued logics. Outer quantifiers behave as normal, but inner quantifiers are  problematic when existential statements take the value i, and so inner quantifiers are primitive. (21.6.2) We now wonder if it makes sense for the existential predicate to have  non-classical values. (21.6.3) According to a certain view, we can think of existence statements of the form ℭa taking the value i under the sense of neither true nor false. (21.6.4) One argument for truth-valueless existence statements could be that non-denoting ones are valueless. “But the claim about non-denotation is not very plausible as far as the existence predicate goes. Supposing that the name ‘Sherlock Holmes’ does not denote anything, it would seem that ‘Sherlock Holmes exists’ is false, not truth-valueless” (464). (21.6.5) Another possibility is to say that existence statements can be neither true nor false when they state the existence of something bound up with a future contingency. So, we might say, “‘The first Pope of the 25th century will exist (but does not yet)’ or ‘Hilary will exist’ – where ‘Hilary’ rigidly designates the first Pope of the 25th century – is neither true nor false. But this seems wrong. If there is such a Pope, this is true” (464). (21.6.6) There is a stronger argument for truth-valueless existence statements, namely, ones that call for verificationism. So if “one can verify neither ‘a exists’ nor its negation, for some suitable a, then this statement is neither true nor false. Thus, for example, ‘The author of the Dao De Ching in fact existed’, or ‘Laozi in fact existed’ might be of this kind” (464). (21.6.7) Another way that we can have valueless existence statements would be borderline ranges of vague predicates, as for example during the gradual process of death where during a certain period some but not all vital bodily functions have ceased and thus when there is “a grey area where it is vague as to whether or not someone exists”. (21.6.8) We can also think of borderline existence cases as involving the value i with the sense of both true and false. For, “What intuition tells us, after all, is that the statement in question seems to be as true as it is false, as false as it is true; and, as far as that goes, the symmetric positions, both and neither, would seem to be as good as each other. Hence, borderline cases of existence might deliver existence statements that are both true and false” (464). (21.6.9) There are existence statements involving paradoxical self-reference that can be considered both true and false. Priest gives the example of Berry’s paradox. “Consider all those (whole) numbers that can be specified in English by a (context-independent) description with less than, say, 100 words. There is a finite number of these, so there are many numbers that cannot be so specified. There must therefore be a least. But there cannot be such a number, since if it did exist it would be specified by the description ‘the least (whole) number that cannot be specified in English by a description with less than 100 words’. The least whole number that cannot be specified in English by a description with less than 100 words both does and does not, therefore, exist” (465).

 

 

 

 

 

Contents

 

21.6.1

[Outer and Inner Domain Quantifiers in 3-Valued Logics]

 

21.6.2

[Wondering About the Sense of the Existential Predicate Taking Non-Classical Values]

 

21.6.3

[Existence Statements as Gaps]

 

21.6.4

[Existence Gaps and Non-Denotation]

 

21.6.5

[Existence Gaps and Future Contingents]

 

21.6.6

[Existence Gaps and Verificationism]

 

21.6.7

[Dying as Involving a Vague, Valueless Existential Predication]

 

21.6.8

[Borderline Existence Statements as Both True and False]

 

21.6.9

[Paradoxes of Self-Reference Involving Existence Statements, Like Berry’s Paradox, as Both True and False]

 

 

 

 

 

Summary

 

21.6.1

[Outer and Inner Domain Quantifiers in 3-Valued Logics]

 

[We can add inner and outer quantifiers to our quantified 3-valued logics. Outer quantifiers behave as normal, but inner quantifiers are  problematic when existential statements take the value i, and so inner quantifiers are primitive.]

 

[We will now recall some matters from section 13.5  regarding the useful distinction and addition of inner and outer quantifiers:

(13.5.1) We might want a free logic where quantifiers range over all objects and not just existent ones. (13.5.2) Quantifiers ranging over the outer domain D are called the outer quantifiers, and they are written as ∃ and ∀. The quantifiers that range over the inner domain E are called inner quantifiers, and they are written as ∃^E and ∀^E. (13.5.3) We read ∀xA as ‘Every x is such that A’; ∀^ExA as ‘Every existent x is such that A’; ∃xA as ‘Some x is such that A’ or as ‘Something is A’; and ∃^ExA as ‘there exists an x such that A’ or as ‘there is an x such that A’. (13.5.4) We should not think that the existential quantifier of natural language necessarily implies existence. (13.5.5) There is an argument for reading the existential quantifier as “there exists”. The argument wants to avoid problems like the ontological argument, so it does not allow existence to be a predicate. Instead, it sees as the only other viable option for expressing existence as being the existential quantifier. Part of the thinking is that only things that are there can be predicated. But this is not a convincing argument, because there are many examples of predication of non-existing objects, like Zeus being worshipped. (13.5.6) If we wish, we can define inner quantifiers in terms of outer ones, which means that “in a free logic with outer quantifiers, we can dispense with inner quantifiers altogether,” namely, in the following way:

∃^ExA     ∃x(ℭx ∧ A)

∀^ExA     ∀x(ℭx ⊃ A) 

(p297). However, “There is no way of defining outer quantifiers in terms of inner quantifiers” (297). (13.5.7) These new semantics make one problematic inference no longer problematic, namely, A_x(a) ⊨ ∃xA, now meaning that if something can be predicated, it is either an existent or non-existent object (previously it implied that any predicable thing must be existent). But it may not make the logical truth ∃x(A ∨ ¬A) unproblematic (it implies now that there must be at least a non-existent object, while before it implied there must be at least an existent object.)

(From the brief summary of section 13.5)

Priest notes now that for our 3-valued logics, we can have outer-quantifiers ranging over the whole of the domain D. But, for certain technical reasons (see below, as I have not yet tried to grasp and summarize them yet), we cannot define the inner quantifiers by means of outer-ones in 3-valued logics where existence predicates take non-classical values (like i), and so “inner quantifiers will have to be taken as primitive.” (I am not sure yet how that works, but it seems in the next section we should figure that out as an exercise.) ]

As with the two-valued case, in the free 3-valued logics we have been talking about, one can have outer quantifiers, ranging over the whole of D. The definability of the inner (existentially loaded) quantifiers in terms of the outer quantifiers and the existence predicate is, however, more problematic. If, as in 13.5.3, we write the outer quantifiers as ∀ and ∃, and use a superscript E to indicate the existentially loaded quantifiers, what we require is:

1. v(∃^ExA) = v(∃x(ℭx ∧ A))

2. v(∀^ExA) = v(∀x(ℭx ⊃ A))

We know that v(ℭk_d) ∈ D iff d ∈ E. If ℭ is a classical predicate, in the sense that for all d ∈ D, v(ℭkd) = 1 or v(ℭkd) = 0, these equations hold. The details are straightforward, and left as an exercise. (Check that if the lefthand side is 1, so is the righthand side. Then check the opposite direction. Do the same thing for 0. The case for i then follows.) If, however, existential statements may take the value i, things may go wrong. Consider an interpretation with | two members, d and e, as follows:

 

       D

+---------------+
|               |
|     E         |
|  +-----+      |
|  |     |      |
|  |  e  |   d  |
|  |     |      |
|  +-----+      |
+---------------+

If v is as follows:

 

	v(ℭ)	v(P)
d	i	1
e	1	0

 

this is a K₃ and Ł₃ interpretation. It is not difficult to check that v(∃^ExPx) = 0, but v(ℭk_d ∧ Pk_d) = i = v(∃x(ℭx ∧ Px)).

 

If v is as follows:

 

	v(ℭ)	v(P)
d	0	0
e	i	1

 

this is an LP and RM₃ interpretation. It is not difficult to check that v(∃^ExPx) = 1, but v(ℭk_e ∧ Pk_e) = i = v(∃x(ℭx ∧ Px)). Hence, if the existence predicate is allowed to take non-classical values, inner quantifiers will have to be taken as primitive.

(463)

[contents]

 

 

 

 

 

 

21.6.2

[Wondering About the Sense of the Existential Predicate Taking Non-Classical Values]

 

[We now wonder if it makes sense for the existential predicate to have  non-classical values.]

 

[(ditto)]

Arranging for this is a simple matter, and left as an exercise. However, it does raise the question of whether it makes sense for the existence predicate to have a non-classical value, the answer to which is not so obvious.

(463)

[contents]

 

 

 

 

 

 

21.6.3

[Existence Statements as Gaps]

 

[According to a certain view, we can think of existence statements of the form ℭa taking the value i under the sense of neither true nor false.]

 

[(ditto)]

Suppose that we are in a logic where i is interpreted as neither true nor false. Could a sentence of the form ℭa take this value? The answer depends on what sorts of thing one takes to be neither true nor false; but on certain views about this, the answer could be ‘yes’.

(463)

[contents]

 

 

 

 

 

 

21.6.4

[Existence Gaps and Non-Denotation]

 

[One argument for truth-valueless existence statements could be that non-denoting ones are valueless. “But the claim about non-denotation is not very plausible as far as the existence predicate goes. Supposing that the name ‘Sherlock Holmes’ does not denote anything, it would seem that ‘Sherlock Holmes exists’ is false, not truth-valueless” (464).]

 

[(ditto)]

Some have argued that a sentence containing a non-denoting name has no truth value (see 7.8). If this is the case, and a does not denote anything, | ℭa has no truth value. But the claim about non-denotation is not very plausible as far as the existence predicate goes. Supposing that the name ‘Sherlock Holmes’ does not denote anything, it would seem that ‘Sherlock Holmes exists’ is false, not truth-valueless.

(463-464)

[contents]

 

 

 

 

 

 

21.6.5

[Existence Gaps and Future Contingents]

 

[Another possibility is to say that existence statements can be neither true nor false when they state the existence of something bound up with a future contingency. So, we might say, “‘The first Pope of the 25th century will exist (but does not yet)’ or ‘Hilary will exist’ – where ‘Hilary’ rigidly designates the first Pope of the 25th century – is neither true nor false. But this seems wrong. If there is such a Pope, this is true” (464).]

 

[(ditto)]

Aristotle argued that statements about a future state of affairs that is not, as yet, determined are neither true nor false (see 7.9). If this is correct then, arguably, ‘The first Pope of the 25th century will exist (but does not yet)’ or ‘Hilary will exist’ – where ‘Hilary’ rigidly designates the first Pope of the 25th century – is neither true nor false. But this seems wrong. If there is such a Pope, this is true.

(464)

[contents]

 

 

 

 

 

 

21.6.6

[Existence Gaps and Verificationism]

 

[There is a stronger argument for truth-valueless existence statements, namely, ones that call for verificationism. So if “one can verify neither ‘a exists’ nor its negation, for some suitable a, then this statement is neither true nor false. Thus, for example, ‘The author of the Dao De Ching in fact existed’, or ‘Laozi in fact existed’ might be of this kind” (464).]

 

[(ditto)]

Better arguments can be found if one subscribes to verificationism of some kind. This might be a philosophy of mathematics which identifies mathematical truth with provability; or it might be a philosophy of science which identifies truth with empirical verifiability. If one subscribes to such a view, and one can verify neither ‘a exists’ nor its negation, for some suitable a, then this statement is neither true nor false. Thus, for example, ‘The author of the Dao De Ching in fact existed’, or ‘Laozi in fact existed’ might be of this kind.

(464)

[contents]

 

 

 

 

 

 

21.6.7

[Dying as Involving a Vague, Valueless Existential Predication]

 

[Another way that we can have valueless existence statements would be borderline ranges of vague predicates, as for example during the gradual process of death where during a certain period some but not all vital bodily functions have ceased and thus when there is “a grey area where it is vague as to whether or not someone exists”.]

 

[(ditto)]

As another example: some have argued that statements about the borderline range of some vague predicate are neither true nor false (see 11.3.6, 11.3.7). Thus, ‘Dana is an adult’, said of Dana around puberty, might be thought to be neither true nor false. But can existence be a vague predicate? Certainly: when people die they go out of existence (let us suppose). But dying can be a gradual process. Bodily functions do not normally all cease at once; there can therefore be a grey area where it is vague as to whether or not someone exists.

(464)

[contents]

 

 

 

 

 

 

21.6.8

[Borderline Existence Statements as Both True and False]

 

[We can also think of borderline existence cases as involving the value i with the sense of both true and false. For, “What intuition tells us, after all, is that the statement in question seems to be as true as it is false, as false as it is true; and, as far as that goes, the symmetric positions, both and neither, would seem to be as good as each other. Hence, borderline cases of existence might deliver existence statements that are both true and false” (464).]

 

[(ditto) (Note: the argument of gaps for dying from section 21.6.7 above seems odd to me, and the gluts version here seems much more reasonable. If it is neither true nor false that one is alive (exists), and it is neither true nor false that one is not alive, than what can we say about the person’s state of being? Can we say that it is true it is some third state? To me it seems more reasonable to say it is both true and false that one exists when in the transitional process of dying, as there is not really another predicate I can think of that applies here that would be just true (and surely some predicate or other regarding its state of being should hold, because we cannot simply say they are dead, but the dying person is still there in some state of being that is also not life, under these gap assumptions). The gap thinking seems to be the following. Can we say the dying person is dead (does not exist)? No, because they are not dead enough to be such. Can we say they are alive? No, because they are not alive enough to be such. In other words, there is a window during which neither the predicate “exists” nor its negation holds. But, as I pointed out, there is still a person there in some state of being, and presumably that state can be given a name and serve as a predication to the dying person. Now, under this gap reasoning, that predicate cannot be “exists”. But I have two problems with that. If it is not “exists,” then you are saying it is false that they exist. That to me seems like you are saying that “they exist” is false and not valueless (otherwise you might be saying they exist only partly, but then you are using a fuzzy value or maybe even a glut, which is not what we are assuming here for gaps). My other problem is that if you insist that “they exist” is neither true nor false, but that moments later after they fully die, “they exist” is false, then, as I noted, something true can be predicated of their existential state which is between existence and non-existence. So my final point here is that to be in a state between existence and non-existence would not be like jumping into a third state that is completely different from existence and non-existence but would rather seem to have certain properties of existence and certain properties of non-existence. For, it is a continuous variation from the one state to the other. So for example, you may have consciousness but not cell life-sustenance on account of stopped blood flow, or maybe you do not have consciousness but you have blood flow. I am not sure what really is involved in death processes. At any rate, it seems to me that existence in the dying transition phase would seem to be a glut, in that it is both true that you exist (in that you have enough functions at this very moment to say that you have not completely passed out of existence and thus that you still are existing, even if barely so) but it is also true to say that you do not exist (as you have a lack of certain functions that will sustain your existence for much longer and thus you are practically dead. To put it phenomenologically, when you are passing away, you will be consciousness of your own fading consciousness, and thus in one instant you will experience both your state of existence and non-existence simultaneously).)]

What of a logic where i is interpreted as both true and false. Could a sentence of the form ℭa be both true and false? Some have suggested that the statements about the borderline range of some vague predicate are both true and false. What intuition tells us, after all, is that the statement in question seems to be as true as it is false, as false as it is true; and, as far as that goes, the symmetric positions, both and neither, would seem to be as good as each other. Hence, borderline cases of existence might deliver existence statements that are both true and false.

(464)

[contents]

 

 

 

 

 

 

 

21.6.9

[Paradoxes of Self-Reference Involving Existence Statements, Like Berry’s Paradox, as Both True and False]

 

[There are existence statements involving paradoxical self-reference that can be considered both true and false. Priest gives the example of Berry’s paradox. “Consider all those (whole) numbers that can be specified in English by a (context-independent) description with less than, say, 100 words. There is a finite number of these, so there are many numbers that cannot be so specified. There must therefore be a least. But there cannot be such a number, since if it did exist it would be specified by the description ‘the least (whole) number that cannot be specified in English by a description with less than 100 words’. The least whole number that cannot be specified in English by a description with less than 100 words both does and does not, therefore, exist” (465).]

 

[(ditto)]

One final example. Some have argued that paradoxical sentences generated by the paradoxes of self-reference are both true and false (see 7.7). Some of these can be existence statements, as in Berry’s paradox, which is as follows. Consider all those (whole) numbers that can be specified in English by a (context-independent) description with less than, say, 100 words. There is a finite number of these, so there are many numbers that cannot be so specified. There must therefore be a least. But there cannot be such a number, since if it did exist it would be specified by the description ‘the least (whole) number that cannot be specified in English by a description with less than 100 words’. The least whole number that cannot be specified in English by a description with less than 100 words both does and does not, therefore, exist. So paradoxes of self-reference may deliver existence statements that are both true and false.

(465)

[contents]

 

 

 

 

 

 

 

From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

 

 

Priest (13.4) An Introduction to Non-Classical Logic, ‘Free Logics: Positive, Negative and Neutral,’ summary

 

by Corry Shores

 

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[Central Entry Directory]

[Logic and Semantics, entry directory]

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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

 

13.

Free Logics

 

13.4

Free Logics: Positive, Negative and Neutral

 

 

 

 

Brief summary:

(13.4.1) Free logics do not have two problematic inferences in classical first-order logic. They do not have as a logical truth that ∃x(Px ∨ ¬Px), in other words, that it is impossible to have nothing existing. And they do not have the inference A_x(a) ⊨ ∃xA, in other words, that anything that takes a predication must be an existing thing. Free logics avoid these problems by allowing there to be non-existing things in the domain. (13.4.2) 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, . . . , d_n⟩ ∈ v(P) then d₁ ∈ v(ℭ), and …and d_n ∈ 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. (13.4.3) The tableau rules for negative free logics are all those for unrestricted free logic plus the Negativity Constraint Rule (NCR), which allows for the the characteristic inference of negative free logics: Pa₁ . . . a_i . . . a_n ⊢ ∃xPa₁ . . . x . . . a_n. (Below I compile all the rules.)

 

Double Negation Development (¬¬D)

¬¬A ↓ A

 

Conjunction Development (∧D)

A ∧ B ↓ A ↓ B

 

Negated Conjunction Development (¬∧D)

¬(A ∧ B) ↙   ↘ ¬A       ¬B

 

Disjunction Development (∨D)

A ∨ B ↙   ↘ A      B

 

Negated Disjunction Development (¬∨D)

¬(A ∨ B) ↓ ¬A ↓ ¬B

 

Conditional Development (⊃D)

A ⊃ B ↙    ↘ ¬A        B

 

Negated Conditional Development (¬⊃D)

¬(A ⊃ B) ↓ A ↓ ¬B

 

Biconditional Development (≡D)

A ≡ B ↙    ↘ A        ¬A B        ¬B

 

Negated Biconditional Development (≡D)

A ≡ B ↙    ↘ A        ¬A ¬B         B

 

Negated Existential Development (¬∃D)

¬∃xA ↓ ∀x¬A

 

Negated Universal Development (¬∀D)

¬∀xA ↓ ∃x¬A

 

Universal Instantiation Development (UI,D)

∀xA ↙    ↘ ¬ℭa      Ax(a)   where a is any constant on the branch (choosing a new constant only if there are none there already)

 

Particular Instantiation Development (PI,D)

∃xA ↓ ℭc ↓ Ax(c)   where c is a constant new to the branch

 

Negativity Constraint Rule(NCR,D)

Pa1 ... an ↓ ℭa1 ⫶ ℭan

 

(6-8; 266; 291; 293, with names and additional text at the bottom made by me)

 

(13.4.4) Priest next gives an example tableau for an invalid inference, and he states that we construct countermodels in the same way as before: “To read off a counter-model from an open branch, we take a domain which contains a distinct object, ∂b, for every constant, b, on the branch. v(b) is ∂_b. v(P) is the set of n-tuples ⟨∂_b1, . . . , ∂_bn⟩ such that Pb₁ . . . b_n occurs on the branch. Of course, if ¬Pb₁ . . . b_n is on the branch, ⟨∂_b1, . . . , ∂_bn⟩ ∉ v(P), since the branch is open. (If a predicate or constant does not occur on the branch, the value given to it by v is a don’t care condition: it can be anything one likes.)” (p.268). And, E = v(ℭ) (p.292). Priest then provides an example counter-model. (13.4.5) “The tableaux for positive and negative free logics are sound and complete with respect to their semantics” (294). (13.4.6) But negative free logics do not account for why sentences such as “Homer worshipped Zeus” are intuitively true even though negative free logics would make them false on account of the fact that it is a non-existent object that is being predicated. (13.4.7) 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.

 

 

 

 

 

 

Contents

 

13.4.1

[Advantages of Free Logics]

 

13.4.2

[The Negativity Constraint of Negative Free Logics]

 

13.4.3

[The Tableau Rules for Negative Free Logics. Example Tableau 1.]

 

13.4.4

[Example Tableau 2. Counter-Models. Example Counter-Model.]

 

13.4.5

[The Soundness and Completeness of Positive Free Logics and Negative Free Logics]

 

13.4.6

[Problems with Negative Free Logics]

 

13.4.7

[Neutral Free Logics]

 

 

 

 

 

 

Summary

 

13.4.1

[Advantages of Free Logics]

 

[Free logics do not have two problematic inferences in classical first-order logic. They do not have as a logical truth that ∃x(Px ∨ ¬Px), in other words, that it is impossible to have nothing existing. And they do not have the inference A_x(a) ⊨ ∃xA, in other words, that anything that takes a predication must be an existing thing. Free logics avoid these problems by allowing there to be non-existing things in the domain.]

 

[We should first recall some of the ideas from section 12.6. The following comes from the brief summary:

(12.6.2) One problem with classical first-order semantics is the following. A standard interpretation of ∃x is ‘There exists an x such that’. This means that if we have ∃xA, it tells us that something does exist which satisfies A. Furthermore, it is a logical truth in classical first-order semantics that ∃x(A ∨ ¬A). This means that within these semantics, we are forced to hold that something must exist that would satisfy either A or its negation. That furthermore implies that we have to conclude that no matter what, something must exist. But that claim does not seem like a logical truth, because we can think that it is possible that nothing exists. To deal with this problem, we cannot simply allow the domain of quantification to be empty, because that makes us unable to assign constants some denotation. Another solution that we explore later is to make the evaluation function v be a partial function, meaning that it has no value for some constants. (12.6.3) Another problem with first-order classical semantics is that A_x(a) ⊨ ∃xA is valid, meaning that anything that can be predicated must exist. But Pegasus can be predicated (for example as a mythological figure), but it does not in fact exist. (12.6.4) Even if one objects that the problem with the Pegasus example is that we wrongly think that existence is a proper predicate, we can still find true sentences where there are other sorts of predicates for objects that nonetheless denote non-existing things, like “Sherlock Holmes is a character in a work of fiction.” The denotation calls for Holmes to be in the domain, but his fictionality calls for him not to be in the domain.

(from the brief summary of section 12.6)

What we saw in those sections was that we encounter a problem with classical first-order logic. When we think of the particular quantifier as expressing existence (or if we at least think of denotation requiring an object in the domain of existents), then we can have valid inferences that are intuitively not true (often times, we saw, this involves fictional entities.) Now let us look at two particular problematic inferences from section 12.6. Here is quotation for the first one:

It is standard to read ∃x as ‘There exists an x such that’, in which case ∃xA expresses the fact that there exists something that satisfies A. Since the domain of quantification is non-empty, ∃x(A ∨ ¬A) is a logical truth, and expresses the fact that there exists something which satisfies either A or its negation – or simply that something exists. This hardly seems to be a logical truth. It would seem entirely possible that there should be nothing. To avoid this, we could allow the domain of quantification to be empty, but we would then be unable to assign constants any denotation. Perhaps the natural remedy for this is to allow v to be a partial function (so that it may have no value for some constants). We will return to this matter in chapter 21, when we consider logics with truth value gaps.

(p.275, section 12.6.2)

But we saw in section 13.3.3 that this does not hold in free logics:

 

⊬ ∃x(Px ∨ ¬Px)
1. . 2. . 3. . 4. . 5.	¬∃x(Px ∨ ¬Px) ↓ ∀y¬(Px ∨ ¬Px) ↙        ↘     ¬ℭa      ¬(Pa ∨ ¬Pa)               ↓                ¬Pa                ↓                ¬¬Pa                 ×	P . 1¬∃ . 2UI . 3¬∨ . 3¬∨ (5×4) (open) invalid

(p. 292, section 13.3.3; enumeration and step accounting are my own and are probably mistaken)

 

In other words, we can have that ∃x(Px ∨ ¬Px) is false, because we can have a model where no objects exist, even though we have a non-existing thing a for our domain. Here is quotation for another problematic inference:

The fact that the denotation function is always defined also makes the following inference valid:

A_x(a) ⊨ ∃xA

Now, presumably, it is true that Pegasus does not exist. But the conclusion that there exists something that does not exist is certainly false.

(p.275, section 12.6.2)

In other words, in classical first-order logic, anytime something is denoted and predicated, it must exist. So because we have the name “Pegasus” and the predication of it that it is a mythological figure, we have to conclude that there is such an thing in existence. But as we saw from the tableau in section 13.3.3, this inference does not hold in free logics.

 

Pa ⊬ ∃xPx
1. . 2. . 3. . 4.	Pa ↓ ¬∃xPx ↓ ∀y¬Px ↙        ↘ ¬ℭa         ¬Pa             ×	P . P . 2¬∃ . 3UI (4×1) (open) invalid

(enumeration and step accounting are my own and are probably mistaken)

 

So in other words, we can have a predicated denoting thing that nonetheless does not exist, because it is not in the E domain.]

As we saw in 12.6.1–12.6.4, if the particular quantifier is interpreted as expressing existence, classical first-order logic shows to be valid inferences that are intuitively not so. We saw in 13.3.3 that free logic does not have the same problematic consequences: particular generalisation fails, since a constant can denote a non-existent object; and the logic is not committed to the logical truth that something exists, for there are interpretations where E is the empty set.

(293)

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13.4.2

[The Negativity Constraint of Negative Free Logics]

 

[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, . . . , d_n⟩ ∈ v(P) then d₁ ∈ v(ℭ), and …and d_n ∈ 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.]

 

[We next need to understand the concept of “positive properties,” but I do not have a strong grasp on it. Positive properties seem to be ones that are tangibly there. We can see, kick, or run past an existing object, because it has positive properties (which seem to be real physical properties that we can physically interact with.) Some might have the intuition that existing things can have positive properties, but non-existing ones cannot. In the example of ‘Sherlock Holmes lived in Baker St.’ (section 12.6.4), someone might say that while Baker Street is physically and tangibly real, we cannot say that Sherlock Holmes lived there, because he has no physical presence with which to interact physically with the Street such that it can be said that he was ever living there. (I am guessing here.) I also do not entirely grasp the first solution for this, but maybe it is the following. We will say that there can be non-existing objects in our domain and that names can denote them, like “Sherlock Homes.” But we will disallow non-existent things from having “positive properties.” So I guess that means we can have Sherlock Homes, but he cannot be assigned any predicates. That seems to be what is going on with this “negativity constraint” and with the “negative free logics” it generates. The constraint seems to stipulate that anything in the denotation of a predicate must be an existing thing. (But this all seems quite odd to me. If you cannot predicate a non-existing object, how can they be distinguished in the first place? Surely Homes is different from Watson, but without being able to take any predications, they would seem to be indiscernible and thus according to one prominent definition of identity, they would be one and the same entity (see Nolt section 14.1). Let me quote:)]

The semantics we have been considering allow for non-existent objects to have positive properties (that is, they may satisfy Px, Qxy, or other atomic formulas). Thus, for example, it is not hard to construct an interpretation that makes ¬ℭa ∧ Pa true. Free logics of this kind are called positive free logics. Some have felt it intuitively implausible that a non-existent object can have positive properties. One can see or kick or run past an existent object, but one cannot see or kick or run past a non-existent object. The condition that non-existent objects have no positive properties can be enforced by adding the following constraint on all interpretations. For any n, and n-place predicate, P:

(*) If ⟨d₁, . . . , d_n⟩ ∈ v(P) then d₁ ∈ v(ℭ), and …and d_n ∈ v(ℭ)

We will call (*) the Negativity Constraint. Logics that impose this constraint are called negative free logics.

(293)

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13.4.3

[The Tableau Rules for Negative Free Logics. Example Tableau 1.]

 

[The tableau rules for negative free logics are all those for unrestricted free logic plus the Negativity Constraint Rule (NCR), which allows for the the characteristic inference of negative free logics: Pa₁ . . . a_i . . . a_n ⊢ ∃xPa₁ . . . x . . . a_n.]

 

[The tableau rules for negative free logics it seems contain all those of free logics that we have seen, plus the Negativity Constraint Rule.

(Below I compile all the rules.)

 

Double Negation Development (¬¬D)

¬¬A ↓ A

 

Conjunction Development (∧D)

A ∧ B ↓ A ↓ B

 

Negated Conjunction Development (¬∧D)

¬(A ∧ B) ↙   ↘ ¬A       ¬B

 

Disjunction Development (∨D)

A ∨ B ↙   ↘ A      B

 

Negated Disjunction Development (¬∨D)

¬(A ∨ B) ↓ ¬A ↓ ¬B

 

Conditional Development (⊃D)

A ⊃ B ↙    ↘ ¬A        B

 

Negated Conditional Development (¬⊃D)

¬(A ⊃ B) ↓ A ↓ ¬B

 

Biconditional Development (≡D)

A ≡ B ↙    ↘ A        ¬A B        ¬B

 

Negated Biconditional Development (≡D)

A ≡ B ↙    ↘ A        ¬A ¬B         B

 

Negated Existential Development (¬∃D)

¬∃xA ↓ ∀x¬A

 

Negated Universal Development (¬∀D)

¬∀xA ↓ ∃x¬A

 

Universal Instantiation Development (UI,D)

∀xA ↙    ↘ ¬ℭa      Ax(a)   where a is any constant on the branch (choosing a new constant only if there are none there already)

 

Particular Instantiation Development (PI,D)

∃xA ↓ ℭc ↓ Ax(c)   where c is a constant new to the branch

 

Negativity Constraint Rule(NCR,D)

Pa1 ... an ↓ ℭa1 ⫶ ℭan

 

(6-8; 266; 291; 293, with names and additional text at the bottom made by me)

Priest says that this rule allows us to derive an inference that gives the characteristic feature of negative free logics:

Pa₁ . . . a_i . . . a_n ⊢ ∃xPa₁ . . . x . . . a_n

(It seems to mean that if something is predicated, it must exist. But unlike classical first-order logics, it may not also require that all objects in the domain ((or all denoted objects)) be existent.) He then gives the tableau.]

To obtain tableaux for negative free logics, we add the rule:

 

Negativity Constraint Rule(NCR,D)

Pa1 ... an ↓ ℭa1 ⫶ ℭan

(with title box by me)

|

which we will call the Negativity Constraint Rule (NCR). This gives the characteristic inference of negative free logics, Pa₁ . . . a_i . . . a_n ⊢ ∃xPa₁ . . . x . . . a_n:

 

Pa1 ... ai ... an ⊢ ∃xPa1 ... x ... an
1. . 2. . 3. . 4. . 5.	Pa1 ... ai ... an ↓ ¬∃xPa1 ... x ... an ↓ ℭai ↓ ∀x¬Pa1 ... x ... an ↙             ↘   ¬ℭai        ¬Pa1 ... ai ... an ×               ×	P . P . 1NCR . 2¬∀ . 4UI (5a×3) (5b×1) valid

(enumeration and step accounting are my own and are probably mistaken)

 

The NCR is applied at line three.

(294)

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13.4.4

[Example Tableau 2. Counter-Models. Example Counter-Model.]

 

[Priest next gives an example tableau for an invalid inference, and he states that we construct countermodels in the same way as before: “To read off a counter-model from an open branch, we take a domain which contains a distinct object, ∂b, for every constant, b, on the branch. v(b) is ∂_b. v(P) is the set of n-tuples ⟨∂_b1, . . . , ∂_bn⟩ such that Pb₁ . . . b_n occurs on the branch. Of course, if ¬Pb₁ . . . b_n is on the branch, ⟨∂_b1, . . . , ∂_bn⟩ ∉ v(P), since the branch is open. (If a predicate or constant does not occur on the branch, the value given to it by v is a don’t care condition: it can be anything one likes.)” (p.268). And, E = v(ℭ) (p.292). Priest then provides an example counter-model.]

 

[Priest next gives a tableau example for an invalid inference (see below in the quotation.) We read off counter-models as before. Recall that from section 13.3.4:

To read off a counter-model from an open branch, we take a domain which contains a distinct object, ∂b, for every constant, b, on the branch. v(b) is ∂_b. v(P) is the set of n-tuples ⟨∂_b1, . . . , ∂_bn⟩ such that Pb₁ . . . b_n occurs on the branch. Of course, if ¬Pb₁ . . . b_n is on the branch, ⟨∂_b1, . . . , ∂_bn⟩ ∉ v(P), since the branch is open. (If a predicate or constant does not occur on the branch, the value given to it by v is a don’t care condition: it can be anything one likes.)

(p.268)

And, E = v(ℭ)

(p.292, section 13.3.4)

In our example tableau (see below), we have as our constants a, b, and c. So: D = {∂_a, ∂_b, ∂_c}. Since we have ℭa, ℭb, but ¬ℭc, that means: E = {∂_a, ∂_b} = v(ℭ). We have predicate Q, and Qab, so: v(Q) = {⟨∂_a, ∂_b⟩}. But since we only have for predicate S, ¬Sac, that means: v(S) = φ. So let us see how that can make the formula not true.

⊬ (Qab ∧ ¬Sac) ⊃ ℭc

It is a conditional. The consequent is false, because only a and b exist, but not c. Qab is true, because v(Q) = {⟨∂_a, ∂_b⟩}. And ¬Sac is true, because v(S) = φ. Thus the antecedent is true but the consequent false, making the whole conditional false.]

Here is another to show that ⊬ (Qab ∧ ¬Sac) ⊃ ℭc:

 

⊬ (Qab ∧ ¬Sac) ⊃ ℭc
1. . 2. . 3. . 4. . 5. . 6. . 7.	¬((Qab ∧ ¬Sac) ⊃ ℭc) ↓ Qab ∧ ¬Sac ↓ ¬ℭc ↓ Qab ↓ ¬Sac ↓ ℭa ↓ ℭb	P . 1¬⊃ . 1¬⊃ . 2∧ . 2∧ . 4NCR . 2∧ (open) invalid

(enumeration and step accounting are my own and are probably mistaken)

 

The last two lines are given by the NCR. We read off a counter-model as before. Thus, D = {∂_a, ∂_b, ∂_c}, E = {∂_a, ∂_b} = v(ℭ), v(Q) = {⟨∂_a, ∂_b⟩}, and v(S) = φ. It is routine to check that this interpretation satisfies the Negativity Constraint, and that it is a counter-model.

(294)

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13.4.5

[The Soundness and Completeness of Positive Free Logics and Negative Free Logics]

 

[“The tableaux for positive and negative free logics are sound and complete with respect to their semantics” (294).]

 

[(ditto)]

The tableaux for positive and negative free logics are sound and complete with respect to their semantics (as proved in 13.7).

(294)

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13.4.6

[Problems with Negative Free Logics]

 

[But negative free logics do not account for why sentences such as “Homer worshipped Zeus” are intuitively true even though negative free logics would make them false on account of the fact that it is a non-existent object that is being predicated.]

 

[Recall from section 12.6.4 one of our example sentences that are intuitively true but are problematic in classical first-order logic: ‘I am thinking about Sherlock Holmes’. Priest adds some more examples here: ‘Homer worshipped Zeus’ and ‘Little Johnny fears Gollum (whom he believes to exist)’. Priest’s point seems to be that this examples pose problems for negative free logics. I may not grasp the reason why, but it seems to be the following. In a negative free logics, only existing things can be predicated. Now, Zeus is a non-existent thing. But we can predicate it that Zeus was worshipped by Homer. And although Gollum does not exist, we can predicate it that Gollum is feared by little Johnny. Priest next writes, “From this perspective, the verbs ‘kicks’ and ‘runs past’ of 13.4.2 look like special cases.” I may have this wrong, but maybe the idea is the following. Perhaps he is saying that some predicates should apply, like intentional sorts of ones, but not these physically interactive ones, like “kicks” and “runs past”, which would then be a special case of predicates. Let me quote, as I may have that wrong:]

Negative free logics are not without their philosophical problems. In 12.6.4 we noted some apparent counter-examples to the Negativity Constraint. One was ‘I am thinking about Sherlock Holmes’. Others of the same kind are: ‘Homer worshipped Zeus’, ‘Little Johnny fears Gollum (whom he believes to exist)’. From this perspective, the verbs ‘kicks’ and ‘runs past’ of 13.4.2 look like special cases.

(294)

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13.4.7

[Neutral Free Logics]

 

[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.]

 

[(ditto)]

It has been suggested by some that sentences (in particular, atomic sentences) that contain names that do not refer to existent objects should not be uniformly false, but uniformly neither true nor false. Logics which enforce this idea are often referred to as neutral free logics. To do justice to the idea one needs a logic with truth value gaps; we will return to the matter in chapter 21.

(295)

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From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.