by Corry Shores
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[Logic and Semantics, entry directory]
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[The following is summary. You will find typos and other distracting mistakes, because I have not finished proofreading. Bracketed commentary is my own. Please consult the original text, as my summaries could be wrong.]
Summary of
Graham Priest
One: Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness
Part 1:
Unity
Ch.2
Identity and Gluons
2.3
Material Equivalence — Paraconsistent Style
Brief summary:
(2.3.1) Since gluons are inconsistent and since paraconsistent logics are the ones that allow for inconsistency, we need to use a paraconsistent logic for accounting for gluons. We furthermore need to understand how material equivalence works in paraconsistent logics. (2.3.2) In classical logic, formulas are either just in the true “zone” or just in the false “zone.” Whenever two formulas are in the same zone, then their material equivalence is true, and it is false otherwise. (2.3.3) In paraconsistent logics, formulas can be in both the true and the false zones. This means that a formula might be materially equivalent to a formula in the opposite zone, if at least one of the two are in both zones. But in that case, their material equivalence will be in both the true and the false zones as well. (2.3.4) Material equivalence in paraconsistent logic is reflexive and symmetric, but not transitive. (2.3.5) An “inference is valid (⊨) just if in every situation where all the premises are true (though they may be false as well), so is the conclusion” (19). Priest then provides a list of important valid inferences, along with an important invalid one (transitivity of material equivalence).
• ⊨ A ≡ A
• A ≡ B ⊨ B ≡ A
• A, B ⊨ A ≡ B
• ¬A, ¬B ⊨ A ≡ B
• A, ¬B ⊨ ¬(A ≡ B)
• B, ¬B ⊨ A ≡ B
• A ≡ B ⊨ ¬A ≡ ¬B
• ¬A ≡ ¬B ⊨ A ≡ B
• A ≡ B, B ≡ C ⊨ (A ≡ C) ∨ (B ∧ ¬B)
• A ≡ B, B ≡ C ⊭ A ≡ C
[Turning to Material Equivalence in Paraconsistent Logics]
[Material Equivalence in Classical Logic]
[Material Equivalence in Paraconsistent Logic]
[Material Equivalence as Non-Transitive Although Reflexive and Symmetric]
[Validity and Important Valid and Invalid Inferences]
Summary
[Turning to Material Equivalence in Paraconsistent Logics]
[Since gluons are inconsistent and since paraconsistent logics are the ones that allow for inconsistency, we need to use a paraconsistent logic for accounting for gluons. We furthermore need to understand how material equivalence works in paraconsistent logics.]
[Recall from from the brief summary of section 2.1 that:
(2.1.1) The gluon is the factor that binds parts into a unity. It has the contradictory properties of both being and not being an object. We now will see how gluons bind parts into unities, which involves breaking the Bradley regress. (2.1.2) The binding action of gluons involves non-transitive identity.
(Brief summary of section 2.1)
And from section 2.2:
(2.2.1) To explain how gluons bind parts into a unified whole, we need to break the Bradley regress, which prevents gluons from simply being object-parts. (2.2.2) We might name the parts of a unified object with letters, as for example, a, b, c, and d. The gluon, symbolized 中, is what binds all the other parts into the unified whole. If the gluon were distinct from the other parts (in the sense of not being identical to them), there would always be room for another gluon to intervene between the first gluon and the given parts, which leads to the Bradley regress. To avoid it, we say that the gluon is identical to each of the parts, thereby closing those “gaps”. (2.2.3) The gluon is non-transitively identical with each and every part. That means that although each part is identical to the gluon, they are not thereby identical to one another. And, parts can themselves be composed of parts by means of another internal gluon.
x
xxxxb
xxxx||
ax=x中x=xc
xxxx||
xxxxd
xxxx
(2.2.4) Gluonic unity involves non-transitive identity, meaning that a = 中 and 中 = c, but not thereby a = c.
(Brief summary of section 2.2)
So gluons are the binding factor that unify objects, but they themselves are contradictory objects, being that they both are and are not objects. Now recall from section P.5 that paraconsistent logics allow for contradictions in that they do not enable us to derive any arbitrary formula we want, so we will need such a logic for our account of gluons. We saw also in section P.5 that in paraconsistent logics, negation has different logical properties. In classical logic, whenever negation operates on a formula, the formula it operates on will be simply true or simply false, and the negation operator will flip that value. But in paraconsistent logics, formulas can take both true and false values. Thus the negation of such a formula is also both true and false. Priest says that now to further understand the paraconsistent logic of gluons, we need to understand the logical properties of material equivalence in paraconsistent logic.]
For a start, gluons, we know, are contradictory objects, and so the account needs to be given in a paraconsistent logic, where contradictions do not explode. In Section P.5, we saw how negation works in a paraconsistent context. What we need to know now is how material equivalence (having the same truth value) works in this context.
(18)
[Material Equivalence in Classical Logic]
[In classical logic, formulas are either just in the true “zone” or just in the false “zone.” Whenever two formulas are in the same zone, then their material equivalence is true, and it is false otherwise.]
[In classical logic, we have two truth-value zones, one for truth and one for falsehoods. They are mutually exclusive (meaning that no formula can be found in both) and they are mutually exhaustive (meaning that a formula must be in one or the other, but not neither). This means we will find formulas A, B, C, D, etc. in one or the other zone. Now, if any two are found in the same zone, whether that be true or false, then their material equivalence will be found in the true zone. So if A and C are in the true zone, then so too is A ≡ C, and if B and D are in the false zone, then B ≡ D is then in the true zone, even though B and D are false. And since A and D are in different zones (A is in true and D is in false), then A ≡ D is in the false zone. ]
Classically, every situation partitions sentences of the language into two zones, the truths (ℑ) and the falsehoods (ℱ), the two zones being mutually exclusive and exhaustive:
ℑ ℱ
____________ ____________
/ A \ / B \
x/ C \ / D \
| A≡C | | A≡D |
x\ B≡D / \ B≡C /
\____________/ \____________/
Sentences, A, B, C, . . .therefore find themselves in exactly one or other of the zones. If two sentences are both in the same zone, their material equivalence is in the ℑ zone; whilst if one is in one zone, and the other is in the other zone, their material equivalence is in the ℱ zone. (See the diagram above.)
(18)
[Material Equivalence in Paraconsistent Logic]
[In paraconsistent logics, formulas can be in both the true and the false zones. This means that a formula might be materially equivalent to a formula in the opposite zone, if at least one of the two are in both zones. But in that case, their material equivalence will be in both the true and the false zones as well.]
[In paraconsistent logic, one same formula can be found in both the true and the false zones. That can be depicted by overlapping the zones, and formulas in the overlap are thought to be in both zones equally. Now, the material equivalence of two formulas will be in the true zone still if both are in the same zone, and it still will be in the false zone if both are in the false zone. But it gets a bit complicated. Suppose A is in the true zone, B is in the false zone, and C is in both zones. The material equivalence of A ≡ B is straightforward. It is in the false zone, because A and B are in different zones exclusively. But since C is in both zones, its material equivalences are more complicated. Since C is at least in the true zone, and since A is entirely in the true zone, then A ≡ C will at least be in the true zone. But since C is also at least in the false zone too, then A ≡ C will at least be in the false zone as well. That means A ≡ C is in both the true and the false zones, and thus it is placed in the depicted overlap (see the diagram below.) The same for B and C.]
In paraconsistent logic, everything is the same except that the ℑ and the ℱ zones may overlap.3 Thus we have the following picture:
ℑ ℱ
_________________ ________________
/ \ / \
/ \ / \
/ A / \ B \
x/ / C \ A≡B \
| | A≡C | |
x\ \C≡B/ /
\ \ / /
\ / \ /
\_________________/ \_________________/
As before, the material equivalence of two sentences is in the ℑ zone if both are in the same zone (ℑ or ℱ ), and in the ℱ zone if they are in different zones, but now a sentence can be in both zones.
(18)
3. In some logics, they may underlap as well, so that there are things that are in neither ℑ nor ℱ; but in the logic we will be using, this is not the case.
(18)
[Material Equivalence as Non-Transitive Although Reflexive and Symmetric]
[Material equivalence in paraconsistent logic is reflexive and symmetric, but not transitive.]
[In paraconsistent logic, material equivalence is reflexive and symmetric (so if A is in a particular zone, then A ≡ A is in the true zone (because A will always be in the same zone as itself), and if A ≡ B is in a particular zone, then so too is B ≡ A in that zone. (Suppose A ≡ B is in the false zone. That means A is in one zone and B is in another. So B ≡ A is also in the false zone. Suppose A ≡ B is in the true zone. That means both A and B are in the same zone. Thus So B ≡ A is in the true zone.)), but it is not transitive. Let us look again at Priest’s diagram to see why. Were material equivalence to be transitive, that would mean that if A ≡ C and C ≡ B are in the same zone, then so too should A ≡ B be in the true zone. In our counter-model, we suppose that A is just in true.
ℑ ℱ
_________________ ________________
/ \ / \
/ \ / \
/ A / \ \
x/ / \ \
| | | |
x\ \ / /
\ \ / /
\ / \ /
\_________________/ \_________________/
And B is just in false.
ℑ ℱ
_________________ ________________
/ \ / \
/ \ / \
/ A / \x B \
x/ / \ \
| | | |
x\ \ / /
\ \ / /
\ / \ /
\_________________/ \_________________/
But C is in both true and false:
ℑ ℱ
_________________ ________________
/ \ / \
/ \ / \
/ A / \ B \
x/ / C \ \
| | | |
x\ \ / /
\ \ / /
\ / \ /
\_________________/ \_________________/
Now recall from section 2.3.2 above that:
If two sentences are both in the same zone, their material equivalence is in the ℑ zone; whilst if one is in one zone, and the other is in the other zone, their material equivalence is in the ℱ zone. (See the diagram above.)
(p.18, section 2.3.2 )
Since A is in the true zone and C is at least in the true zone, then their material equivalence is at least in the true zone. But since C is also at least in the false zone, that means they at least are also in different zones, and so their material equivalence is at least also in the false zone. In other words, it will be in the overlap zone, as it is both true and false.
ℑ ℱ
_________________ ________________
/ \ / \
/ \ / \
/ A / \ B \
x/ / C \ \
| | A≡C | |
x\ \ / /
\ \ / /
\ / \ /
\_________________/ \_________________/
The same thing applies for C and B :
ℑ ℱ
_________________ ________________
/ \ / \
/ \ / \
/ A / \ B \
x/ / C \ \
| | A≡C | |
x\ \C≡B/ /
\ \ / /
\ / \ /
\_________________/ \_________________/
Now, are A and B in the same zone? No. That means their material equivalence is false.
ℑ ℱ
_________________ ________________
/ \ / \
/ \ / \
/ A / \ B \
x/ / C \ A≡B \
| | A≡C | |
x\ \C≡B/ /
\ \ / /
\ / \ /
\_________________/ \_________________/
But this is unlike the classical situation. We have A ≡ C as at least true and C ≡ B also as at least true, but A ≡ B will not even be at least true. So the material equivalence does not transfer through the shared term.]
A ≡ A will always be in the ℑ zone, since A is always in the same zone as itself. If A ≡ B is in the ℑ zone, then so is B ≡ A, since these are just ways of saying that A and B are in the same zone. So equivalence is reflexive and symmetric; but it is not transitive. A and C may be in the same zone, and C and B may be in the same zone, though A and B are not, because C is in the overlap. Hence, we may have A ≡ C and C ≡ B being in the ℑ zone, without A ≡ B being so (see the diagram above). Note also that detachment for ≡ may fail: we can have C and C ≡ B in the ℑ zone without B being in it (same diagram).
[Validity and Important Valid and Invalid Inferences]
[An “inference is valid (⊨) just if in every situation where all the premises are true (though they may be false as well), so is the conclusion” (19). Priest then provides a list of important valid inferences, along with an important invalid one (transitivity of material equivalence).]
[Priest next reminds us that an inference is semantically validi, symbolized as ⊨, when the all premises are at least true and so is the conclusion. (I am guessing that the conclusion need only be at least true and thus it can be both true and false under this criteria, given that i is thought to be the designated value in glut logics. See Priest’s Introduction to Non-Classical Logic, section 7.4. Priest lastly provides a list of important valid inferences, along with an important invalid one (transitivity of material equivalence).]
For the record, here are some paraconsistent facts concerning negation, equivalence, and validity. As we noted in Section P.5, an inference is valid (⊨) just if in every situation where all the premises are true (though they may be false as well), so is the conclusion. Bearing this in mind, and remembering that a formula is in the ℑ zone iff its negation is in the ℱ zone, it is easy to check the details.
• ⊨ A ≡ A
• A ≡ B ⊨ B ≡ A
• A, B ⊨ A ≡ B
• ¬A, ¬B ⊨ A ≡ B
• A, ¬B ⊨ ¬(A ≡ B)
• B, ¬B ⊨ A ≡ B
• A ≡ B ⊨ ¬A ≡ ¬B
• ¬A ≡ ¬B ⊨ A ≡ B
• A ≡ B, B ≡ C ⊨ (A ≡ C) ∨ (B ∧ ¬B)
• A ≡ B, B ≡ C ⊭ A ≡ C
Of course, in a paraconsistent context, the truth of ¬A is compatible with that of A. So even if it is the case that ¬(A ≡ B) is true, it can still be the case that A ≡ B is true as well.
From:
Priest, Graham. 2014. One: Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness. Oxford: Oxford University.
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