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
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 Ax(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, . . . , 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. (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 . . . ai . . . an ⊢ ∃xPa1 . . . x . . . an. (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 . . . bn occurs on the branch. Of course, if ¬Pb1 . . . bn 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.
[Advantages of Free Logics]
[The Negativity Constraint of Negative Free Logics]
[The Tableau Rules for Negative Free Logics. Example Tableau 1.]
[Example Tableau 2. Counter-Models. Example Counter-Model.]
[The Soundness and Completeness of Positive Free Logics and Negative Free Logics]
[Problems with Negative Free Logics]
[Neutral Free Logics]
Summary
[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 Ax(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 Ax(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:
Ax(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)
[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, . . . , 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.]
[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, . . . , dn⟩ ∈ v(P) then d1 ∈ v(ℭ), and …and dn ∈ v(ℭ)
We will call (*) the Negativity Constraint. Logics that impose this constraint are called negative free logics.
(293)
[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 . . . ai . . . an ⊢ ∃xPa1 . . . x . . . an.]
[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 . . . ai . . . an ⊢ ∃xPa1 . . . x . . . an
(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 . . . ai . . . an ⊢ ∃xPa1 . . . x . . . an:
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)
[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 . . . bn occurs on the branch. Of course, if ¬Pb1 . . . bn 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 . . . bn occurs on the branch. Of course, if ¬Pb1 . . . bn 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)
[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)
[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)
[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)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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