Priest (12.0) Beyond the Limits of Thought, ‘Introduction [to chapter 12],’ summary

 

by Corry Shores

 

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Graham Priest

 

Beyond the Limits of Thought

 

Part 4 Language and Its Limits

 

Ch.12 The Unity of Thought

 

12.0 Introduction

 

 

Brief summary:

There are contradictions that lie at the limits of thought, and in this chapter we look at such contradictions as they manifest in Frege’s and (early) Wittgenstein’s theories of meaning.

 

 

Summary

 

Priest notes that the main issue of this book is the limits of thought. He recalls that “We have seen how contradictions at the limits of thought that we have been concerned with take a very sharp form in the inclosure contradictions of self-reference” (197). We saw in the book’s second part that for Kant it was a matter of the limits of reason. But in the twentieth century, language takes the center stage in philosophical debates. So we look now at how the problem of the limits of thought has manifested in twentieth century philosophy in terms of language (197).

 

Priest says that “It is natural for a theory of language to have implications about what can and what cannot be expressed” (Priest 197). Priest says that in fact modern theories of language are somehow able to “render some very important things – usually themselves – beyond the limit of expression” (197). [I am not sure, but perhaps the “usually themselves” part means that these very theories or systems themselves are somehow self-referentially able to express themselves within themselves.] Priest says that the “contradictions at the limits of thought” will “appear in a new guise” in this context. So now we will examine some modern accounts of meaning, looking for these contradictions at the limits of thought. Priest will not try to evaluate the theories; rather, he will “examine their consequences” (197).

 

Priest will begin with Frege then move to Wittgenstein’s Tractatus, all the while focusing on the unity of thought (197).

 

 

 

 

Graham Priest. Beyond the Limits of Thought. Cambridge: Cambridge University, 1995.

 

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Priest. Beyond the Limits of Thought, entry directory

 

by Corry Shores

 

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Entry Directory for

 

Graham Priest

 

Beyond the Limits of Thought

 

Part 4 Language and Its Limits

 

Ch.12 The Unity of Thought

 

12.0 Introduction

 

12.1 Frege, Concept and Object

 

12.2 The Concept Horse

 

 

 

 

Graham Priest. Beyond the Limits of Thought. Cambridge: Cambridge University, 1995.

Rules and Strategies for Logic Proofs (Alger, Nolt)

 

by Corry Shores

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Rules and Strategies for Logic Proofs

(Agler, Nolt)

 

 

 

David Agler

 

Symbolic Logic

 

Agler’s PD+ (11 Int-Elim Propositional Derivation Rules plus Additional Rules)

 

(Agler 242-244)

 

 

Agler’s Proof Strategies

 

SP#1(E+): First, eliminate any conjunctions with ‘∧E,’ disjunctions with DS or ‘∨E,’ conditionals with ‘→E’ or MT, and biconditionals with ‘↔E.’ Then, if necessary, use any introduction rules to reach the desired conclusion.

SP#2(B): First, work backward from the conclusion using introduction rules (e.g., ‘∧I,’‘∨I,’‘→I,’‘↔I’). Then, use SP#1(E).

SP#3(EQ+): Use DeM on any negated disjunctions or negated conjunctions, and then use SP#1(E). Use IMP on negated conditionals, then use DeM, and then use SP#1(E).

SA#1(P,¬Q): If the conclusion is an atomic proposition (or a negated proposition), assume the negation of the proposition (or the non-negated form of the negated proposition), derive a contradiction, and then use ‘¬I’ or ‘¬E.’

SA#2(→): If the conclusion is a conditional, assume the antecedent, derive the consequent, and use ‘→I.’

SA#3(∧): If the conclusion is a conjunction, you will need two steps. First, assume the negation of one of the conjuncts, derive a contradiction, and then use ‘¬I’ or ‘¬E.’ Second, in a separate subproof, assume the negation of the other conjunct, derive a contradiction, and then use ‘¬I’ or ‘¬E.’ From this point, a use of ‘∧I’ will solve the proof.

SA#4(∨): If the conclusion is a disjunction, assume the negation of the whole disjunction, derive a contradiction, and then use ‘¬I’ or ‘¬E.’

(Agler 199; 216)

 

 

 

Agler’s Quantification Rules

 

Universal Elimination (∀E) From any universally quantified proposition ‘(∀x)P,’ we can derive a substitution instance ‘P(a/x)’ in which all bound variables are consistently replaced with any individual constant (name).

(∀x)P P(a/x)

∀E

 

Existential Introduction (∃Ι) From any possible substitution instance ‘P(a/x),’ an existentially quantified proposition ‘(∃x)P’ can be derived by consistently replacing at least one individual constant (name) with an existentially quantified variable.

P(a/x) (∃x)P

∃I

 

Universal Introduction (∀Ι) A universally quantified proposition ‘(∀x)P’ can be derived from a possible substitution instance ‘P(a/x)’ provided (1) ‘a’ does not occur as a premise or as an assumption in an open subproof, and (2) ‘a’ does not occur in ‘(∀x)P.’

P(a/x) (∀x)P

∀I

 

Existential Elimination (∃E) From an existentially quantified expression ‘(∃x)P,’ an expression ‘Q’ can be derived from the derivation of an assumed substitution instance ‘P(a/x)’ of ‘(∃x)P’ provided (1) the individuating constant ‘a’ does not occur in any premise or in an active proof (or subproof) prior to its arbitrary introduction in the assumption ‘P(a/x),’ and (2) the individuating constant ‘a’ does not occur in proposition ‘Q’ discharged from the subproof.

(∃x)P    | P(a/x)    | .    | .    | .    | Q Q

∃E

(Agler 357-358 See Symbolic Logic section 8.1)

 

 

Agler’s Additional Strategic Rules for Quantification

 

SQ#1(∀E): When using (∀E), the choice of substitution instances ‘P(a/x)’ should be guided by the individual constants (names) already occurring in the proof and any individual constants (names) occurring in the conclusion.
(Agler 328)

SQ#2(∃I): When using (∃I), aim at deriving a substitution instance ‘P(a/x)’ such that a use of (∃I) will result in the desired conclusion. (In other words, if the ultimate goal is to derive ‘(∃x)Px,’ aim to derive a substitution instance of ‘(∃x)Px,’ like ‘Pa,’ ‘Pb,’ ‘Pr,’ so that a use of (∃I) will result in ‘(∃x)Px.’)

SQ#3(∀I): When the goal proposition is a universally quantified proposition ‘(∀x)P,’ derive a substitution instance ‘P(a/x)’ such that a use of (∀I) will result in the desired conclusion.

SQ#4(∃E) Generally, when deciding upon a substitution instance ‘P(a/x)’ to assume for a use of (∃E), choose one that is foreign to the proof.

(Agler 358)

 

 

 

 

John Nolt

 

Logics

 

 

Nolt’s 10 Rules of Inference

 

~E	Negation Elimination (Double Negation)	From ~~Φ, infer Φ.
~I	Negation Introduction (Reductio ad Absurdum, Indirect Proof	Given a hypothetical derivation of any formula of the form (Ψ & ~Ψ) from Φ, end the derivation and infer ~Φ.
&E	Conjunction Elimination (Simplification)	From (Φ & Ψ), infer either Φ or Ψ.
&I	Conjunction Introduction (Conjunction)	From Φ and Ψ, infer (Φ & Ψ).
∨E	Disjunction Elimination (Constructive Dilemma)	From (Φ ∨ Ψ), (Φ→ Θ), and (Ψ→ Θ), infer Θ.
∨I	Disjunction Introduction (Addition)	From Φ, infer either (Φ ∨ Ψ) or (Ψ ∨ Φ).
~E	Conditional Elimination (Modus Ponens)	Given (Φ→ Ψ) and Φ, infer Ψ.
→I	Conditional Introduction (Conditional Proof)	Given a hypothetical derivation of Ψ from Φ, end the derivation and infer (Φ→ Ψ).
↔E	Biconditional Elimination	From (Φ ↔ Ψ), infer either (Φ→Ψ) or (Ψ→Φ).
↔I	Biconditional Introduction	From (Φ→ Ψ ) and (Ψ→Φ), infer (Φ ↔ Ψ).

(Nolt 102)

 

 

 

Nolt’s Important Derived Rules

 

MT	Modus Tollens	From Φ → Ψ and ~Ψ, infer ~Φ.
CP	Contraposition	From Φ → Ψ, infer ~Ψ → ~Φ.
↔MP	Biconditional Modus Ponens	From Φ ↔ Ψ and Φ, infer Ψ. And from Φ ↔ Ψ and Ψ, infer Φ.
↔MT	Biconditional Modus Tollens	From Φ ↔ Ψ and ~Ψ, infer ~Φ. And from Φ ↔ Ψ and ~Φ, infer ~Ψ.
DS	Disjunctive Syllogism	From Φ ∨ Ψ and ~Φ, infer Ψ. And from Φ ∨ Ψ and ~Ψ, infer Φ.
HS	Hypothetical Syllogism	From Φ → Ψ and Ψ → Θ, infer Φ → Θ.
DN	Double Negation	From Φ, infer ~~Φ.
DM	DeMorgan’s Law’s	From ~(Φ ∨ Ψ), infer ~Φ & ~Ψ. From ~Φ & ~Ψ, infer ~(Φ ∨ Ψ). From ~(Φ & Ψ), infer ~Φ ∨ ~Ψ. From ~Φ ∨ ~Ψ, infer ~(Φ & Ψ). From Φ ∨ Ψ, infer ~(~Φ & ~Ψ). From ~(~Φ & ~Ψ), infer Φ ∨ Ψ. From Φ & Ψ, infer ~(~Φ ∨ ~Ψ). From ~(~Φ ∨ ~Ψ), infer Φ & Ψ.
COM	Commutation	From Φ & Ψ, infer Ψ & Φ. From Φ ∨ Ψ, infer Ψ ∨ Φ.
ASSOC	Association	From (Φ & Ψ) & Θ, infer Φ & (Ψ & Θ). From Φ & (Ψ & Θ), infer (Φ & Ψ) & Θ. From (Φ ∨ Ψ) ∨ Θ, infer Φ ∨ (Ψ ∨ Θ). From Φ ∨ (Ψ ∨ Θ), infer (Φ ∨ Ψ) ∨ Θ.
DIST	Distribution	From (Φ & Ψ) ∨ Θ, infer (Φ ∨ Θ) & (Ψ ∨ Θ). From (Φ ∨ Θ) & (Ψ ∨ Θ), infer (Φ & Ψ) ∨ Θ. From (Φ ∨ Ψ) & Θ, infer (Φ & Θ) ∨ (Ψ & Θ). From (Φ & Θ) ∨ (Ψ & Θ), infer (Φ ∨ Ψ) & Θ.
MI	Material Implication	From Φ → Ψ, infer ~Φ ∨ Ψ. From ~Φ ∨ Ψ, infer Φ → Ψ. From Φ → Ψ, infer ~(Φ & ~Ψ). From ~(Φ & ~Ψ), infer Φ → Ψ.
EFQ	Ex Falso Quodlibet	From Φ and ~Φ, infer any formula Ψ.

(Nolt 102)

 

 

 

Nolt’s Proof Strategies

 

If the conclusion or subconclusion you are trying to prove is of the form:	Then try this strategy:
~Φ	Hypothesize Φ and work toward a subconclusion of the form Ψ & ~Ψ in order to obtain ~Φ by ~I.
Φ & Ψ	Prove the subconclusions Φ and Ψ separately and then join them by &I.
Φ ∨ Ψ	If either Φ or Ψ is a premise, simply apply ∨I to obtain Φ ∨ Ψ. Otherwise, if there is a disjunctive premise Θ ∨ Δ, try proving the two conditionals Θ → (Φ ∨ Ψ) and Δ → ( Φ ∨ Ψ) as subconclusions and then using ∨E to obtain Φ ∨ Ψ. If neither of these strategies works, then hypothesize ~(Φ ∨ Ψ) and work toward a subconclusion of the form Θ & ~Θ in order to obtain Φ ∨ Ψ by ~I and~ E.
Φ → Ψ	Hypothesize Φ and work toward the subconclusion Ψ in order to obtain the conditional by →I.
Φ ↔ Ψ	Prove the subconclusions Φ → Ψ and Ψ → Φ; then use ↔I to obtain Φ ↔ Ψ.

(Nolt 99)

 

 

 

Nolt’s Quantifier Rules

 

∃I	Existential Introduction	Let Φ be any formula containing some name α and Φβ/α be the result of replacing at least one occurrence of α in Φ by some variable β not already in Φ. Then from Φ infer ∃βΦβ/α.
∃E	Existential Elimination	Let ∃βΦ be any existential formula, Ψ any formula, and α any name that occurs neither in Φ nor in Ψ. And let Φα/β be the result of replacing all occurrences of the variable β in Φ by α. Then, given a derivation of Ψ from the hypothesis Φα/β, end the hypothetical derivation and from ∃βΦ infer Ψ, provided that α does not occur in any other hypothesis whose hypothetical derivation has not ended or in any assumption.
∀E	Universal Elimination	Let ∀βΦ be any universally quantified formula and Φα/β be the result of replacing all occurrences of the variable β in Φ by some name α. Then from ∀βΦ infer Φα/β.
∀I	Universal Introduction	Let Φ be a formula containing a name α, and let Φβ/α be the result of replacing all occurrences of α in Φ by some variable β not already in Φ. Then from Φ infer ∀β Φβ/α, provided that α does not occur in any hypothesis whose hypothetical derivation has not yet ended or in any assumption.

(Nolt 225, 229, 233, 236)

 

 

 

Nolt’s Quantifier Exchange Rules (QE)

 

From ∀βΦ, infer ~∃β~Φ.	From ∃βΦ, infer ~∀β~Φ.
From ~∀βΦ, infer ∃β~Φ.	From ~∃βΦ, infer ∀β~Φ.
From ∀β~Φ, infer ~∃βΦ.	From ∃β~Φ, infer ~∀βΦ.
From ~∀β~Φ, infer ∃βΦ.	From ~∃β~Φ, infer ∀βΦ.

(Nolt 238)

 

 

 

Quantifier Proof Strategies

 

If the conclusion or subconclusion you are trying to prove is of the form:	Then try this strategy:
∃βΦ	Work toward a subconclusion of the form Φα/β, where α is a name that does not occur in Φ, in order to obtain ∃βΦ by ∃I. If there is an existential premise, it is likely that the subconclusion Φα/β will have to be derived hypothetically after a representative instance of this premise has been hypothesized for ∃E. In that case, to avoid misusing ∃E, obtain ∃βΦ by ∃I before ending the hypothetical derivation with ∃E. If all else fails, hypothesize ~∃βΦ and work toward a subconclusion of the form Θ & ~Θ in order to obtain ∃βΦ by ~I and ~E.
∀βΦ	Work toward a subconclusion of the form Φα/β, where α is a name that does not occur in Φ or in any assumption or hypothesis whose hypothetical derivation has not yet ended, in order to obtain ∀βΦ by ∀I. If there are universal premises, it is likely that some or all of them will have to be instantiated with a by ∀E before this can be done.

(Nolt 239)

 

 

 

Nolt’s Rules for Identity

 

=I	Identity Introduction	Where α is any name, assert α = α.
= E	Identity Elimination	From a premise of the form α = β and a formula Φ containing either α or β, infer any formula which results from replacing one or more occurrences of either of these names by the other in Φ.

(Nolt 241-242)

 

 

 

Nolt’s Leibnizian Modal Logic Rules

 

DUAL	Duality	From either ◊Φ and ~□~Φ, infer the other; from either □Φ and ~◊~Φ, infer the other.
K	K rule	From □(Φ → Ψ), infer (□Φ → □Ψ).
T	T rule	From □Φ, infer Φ.
S4	S4 rule	From □Φ, infer □□Φ
B	Brouwer rule	From Φ, infer □◊Φ.
N	Necessitation	If Φ has previously been proven as a theorem, then any formula of the form □Φ may be introduced at any line of a proof.
□=	Necessity of identity	From α = β, infer □α = β.

(Nolt 328)

 

 

Free Logic Quantification Inference Rules

Free Existential Introduction (F∃I) Let Φ be any formula containing some name α and Φβ/α be the result of replacing at least one occurrence of α in Φ by some variable β not already in Φ. Then from Φ and ∃x x = α infer ∃βΦβ/α.

Free Existential Elimination (F∃E) Let ∃βΦ be any existential formula, Ψ any formula, and α any name that occurs neither in Φ nor in Ψ. And let Φα/β be the result of replacing all occurrences of the variable β in Φ by α. Then, given a derivation of Ψ from the hypothesis Φα/β & ∃x x = α, end the hypothetical derivation and from ∃βΦ infer Ψ, provided that α does not occur in any other hypothesis whose hypothetical derivation has not ended or in any assumption.

Free Universal Introduction (F∀I) Let Φ be a formula containing a name α, and let Φβ/α be the result of replacing all occurrences of α in Φ by some variable β not already in Φ. Then given a derivation of Φ from the hypothesis ∃x x = α, end the hypothetical derivation and infer ∀βΦβ/α, provided that α does not occur in any other hypothesis whose hypothetical derivation has not yet ended or in any assumption.

Free Universal Elimination (F∀E) Let ∀βΦ be any universally quantified formula and Φα/β be the result of replacing all occurrences of the variable β in Φ for some name α. Then from ∀βΦ and ∃x x = α infer Φα/β.

(Nolt 403, see section 15.1)

 

 

From:

Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman &; Littlefield, 2013.

 

Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.

Nolt (8.5) Logics, ‘Identity,’ summary

 

by Corry Shores

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[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own.]

 

 

 

Summary of

 

John Nolt

 

Logics

 

Part 3: Classical Predicate Logic

 

Chapter 8: Classical Predicate Logic: Inference

 

8.5 Identity

 

 

 

Brief summary:

There are two identity inference rules for making proofs in predicate logic.

 

Identity Introduction (=I)  Where α is any name, assert α = α.

 

Identity Elimination (= E) From a premise of the form α = β and a formula Φ containing either α or β, infer any formula which results from replacing one or more occurrences of either of these names by the other in Φ.

(241-242)

 

 

Summary

 

Nolt will explain how to make proofs in predicate logic when identity is involved. He will give two rules, namely, identity introduction ‘=I’ and identity elimination ‘=E’.

 

Identity Introduction (=I)  Where α is any name, assert α = α.

(241)

 

As we can see, there are not any premises involved, and so it is not really a rule of inference. Rather, “It simply licenses us to assert a logical truth of the form α = α at any line of a proof” (Nolt 241). Nolt says that this rule is valid, because a logical truth is true no matter what the premises, even if there are no premises. [I am not certain, but perhaps the idea is like how you can insert a tautology into any proof, but I am not sure.] But since this rule does not need to be based on any particular line, we do not need to give a line number in the justification part of the proof. Nolt gives the example of a proof for the sequent

∀x∀y(x = y → Rxy) ⊢ Raa

[So let us set it up, by first placing the assumption.

 

1.

∀x∀y(x = y → Rxy)

A

 

In the proofs we made with Agler, we would put a ‘P’ for the premises, but here we place an ‘A’, perhaps to mean ‘assumption’ but I am not sure. With Agler, ‘assumption’ had a different meaning (it was used for starting subproofs). For Nolt, it seems we would use ‘H’ instead of ‘A’ when making a subproof, and perhaps the ‘H’ means ‘hypothesis’. But again I am not sure yet. At any rate, we are going to want to derive Raa. Since we are working with a universal quantifier, that means we will want to use universal elimination. Recall from Agler section 8.1 the rule:

 

Universal Elimination (∀E) From any universally quantified proposition ‘(∀x)P,’ we can derive a substitution instance ‘P(a/x)’ in which all bound variables are consistently replaced with any individual constant (name).

(∀x)P P(a/x)

∀E

(Agler 325)

 

So using this rule and also the new identity introduction rule, we will want to go from ∀x∀y(x = y → Rxy) to obtain Raa. We could go one step at a time to get the constant a into the formulation, using universal elimination. So let us do that.

 

1.	∀x∀y(x = y → Rxy)	A
2.	∀x(a = y → Ray)	1∀E
3.	a = a → Raa	2∀E

 

Now, recall that that the identity introduction rule says that “where α is any name, assert α = α”. So we have the name ‘a’. And were we to also have a = a, then we could use conditional elimination to derive our goal proposition. So let us do that.]

 

1.	∀x∀y(x = y → Rxy)	A
2.	∀x(a = y → Ray)	1∀E
3.	a = a → Raa	2∀E
4.	a = a	=I
5.	Raa	3,4→E

(Nolt 241)

 

Nolt then shows another “typical use of =I” with the proof for

⊢~∃x~x=x

(Nolt 241)

[Recall from section 6.3 that structures of the form ‘~a=b’ should be understood as meaning, ‘a is not identical to b’. So this formulation above is perhaps read, there is no x such that x is not identical to itself, or perhaps also, nothing is not identical to itself (or all things are identical to themselves). Recall from Agler Symbolic Logic section 5.4.2 strategic rule SA#1(P,¬Q):

SA#1(P,¬Q)

If the conclusion is an atomic proposition (or a negated proposition), assume the negation of the proposition (or the non-negated form of the negated proposition), derive a contradiction, and then use ‘¬I’ or ‘¬E.’

(Agler 199)

Since we want to prove ~∃x~x=x, we will assume the non-negated form. Then ultimately we will use negation introduction to derive the goal proposition.

 

1.

| ∃x~x=x

H (for ~I)

 

Now recall from Agler Symbolic Logic section  8.1.4 the rule Existential Elimination.

Existential Elimination (∃E) From an existentially quantified expression ‘(∃x)P,’ an expression ‘Q’ can be derived from the derivation of an assumed substitution instance ‘P(a/x)’ of ‘(∃x)P’ provided (1) the individuating constant ‘a’ does not occur in any premise or in an active proof (or subproof) prior to its arbitrary introduction in the assumption ‘P(a/x),’ and (2) the individuating constant ‘a’ does not occur in proposition ‘Q’ discharged from the subproof.

(∃x)P    | P(a/x)    | .    | .    | .    | Q Q

∃E

(Agler 342)

 

So for this, we assume (or hypothesize) a substitution instance for the existentially quantified formula. We will then want to derive some formula that will go into the main subproof. And we will eventually want there to be a contradiction in the subproof, so that we can derive the goal proposition. Recall that we are going to use negation introduction. Here is Agler’s account of it from Symbolic Logic section  5.3.7.

 

(Agler 181)

 

What Nolt will do is use a rule that we did not get in Agler, namely, Ex Falso Quodlibet (EFQ): From Φ and ~Φ, infer any formula Ψ (Nolt 106). But let us first start the sub-subproof for the purpose of existential elimination.

 

1.	| ∃x~x=x	H (for ~I)
2.	|   | ~a = a	H (for ∃E)

 

So we can see already that we are set up to find a contradiction. We will just expose it by using identity introduction.

 

1.	| ∃x~x=x	H (for ~I)
2.	|   | ~a = a	H (for ∃E)
3.	|   | a = a	=I

 

We said that we need a contradiction in the main subproof. Nolt will then just directly derive a contradiction in the sub-subproof so that we accomplish this in one step in the subproof.

 

1.	| ∃x~x=x	H (for ~I)
2.	|   | ~a = a	H (for ∃E)
3.	|   | a = a	=I
4.	|   | P & ~P	3,3EFQ

 

Now we derive that self-contradictory proposition in the subproof, as planned, using existential elimination.

 

1.	| ∃x~x=x	H (for ~I)
2.	|   | ~a = a	H (for ∃E)
3.	|   | a = a	=I
4.	|   | P & ~P	3,3EFQ
5.	| P & ~P	1,2-4∃E

 

Now with a contradiction in our subproof, we can derive the negation of the original hypothesis, which was our goal proposition.]

 

1.	| ∃x~x=x	H (for ~I)
2.	|   | ~a = a	H (for ∃E)
3.	|   | a = a	=I
4.	|   | P & ~P	3,3EFQ
5.	| P & ~P	1,2-4∃E
6.	~∃x~x=x	1-5~I

(Nolt 241)

 

Nolt gives this explanation for the process:

Since the desired conclusion ‘~∃x~x=x’ is negative, we hypothesize ‘∃x~x=x’ for indirect proof at line 1. But this is an existential formula, and so we hypothesize ‘~a = a’, a representative instance of it, for ∃E at line 2. We can now contradict line 2 by applying =I at line 3, but to obtain a contradiction that does not contain the name ‘a’ of the representative individual, we must use EFQ. Steps of ∃E and ~I then complete the proof.

(Nolt 241)

 

Identity Elimination is a matter of substituting identical things for one another:

The elimination rule for identity is familiar from algebra as the rule that allows us to substitute equals for equals – but in logic, since we are not dealing | only with quantities, it is more accurate to say ‘identicals for identicals.’ It is sometimes called the rule of identity substitution.

 

Identity Elimination (= E) From a premise of the form α = β and a formula Φ containing either α or β, infer any formula which results from replacing one or more occurrences of either of these names by the other in Φ.

 

Uses of =E are annotated by citing two line numbers: the number of the line on which α = β occurs and the number of the line on which Φ occurs.

(241-242)

 

Given that proving the validity of =E requires a lot of work, we will postpone such a proof until section 9.1. Until then, we will merely see how it is used. He gives as illustration the proof for:

⊢ (Fa & a = b) → Fb

(242)

[What we need to prove takes the form of a conditional. Recall strategic rule SA#2(→) from Agler Symbolic Logic section 5.4.2.

SA#2(→)

If the conclusion is a conditional, assume the antecedent, derive the consequent, and use ‘→I.’

(Agler 199)

So we will want to begin by hypothesizing the antecedent.

 

1.

| Fa & a = b

H (for →I)

 

And we will want to derive the consequent so that we can use conditional introduction in order to derive the goal proposition. We also said that we would use identity elimination. Look again at the goal proposition: (Fa & a = b) → Fb. As we can see, we have an identity statement that will allow for a substitution that can give us our consequent. So first we will use conjunction elimination to extract the two parts of our hypothesis, which will allow us to make that substitution.

 

1.	| Fa & a = b	H (for →I)
2.	| Fa	1&E
3.	| a = b	1&E

 

Now we will make the substitution using identity elimination to derive the goal proposition’s consequent.

 

1.	| Fa & a = b	H (for →I)
2.	| Fa	1&E
3.	| a = b	1&E
4.	| Fb	2,3=E

 

And finally we can use conjunction introduction to derive the goal proposition.]

 

1.	| Fa & a = b	H (for →I)
2.	| Fa	1&E
3.	| a = b	1&E
4.	| Fb	2,3=E
5.	(Fa & a = b) → Fb	1-4→I

(Nolt 242)

 

Nolt describes the reasoning this way:

The theorem is a conditional, so the overall strategy is conditional proof. With respect to the formal statement of the =E rule, α = β is ‘a = b’ and Φ is ‘Fa’.

(Nolt 242)

 

 

When the equated terms occur more than once in a formula, then we can substitute them in all the variety of ways that maintain the equality. So if we have ‘Laa’ and ‘a = b’, then we can use identity elimination to derive ‘Lab’ or ‘Lba’ or ‘Lbb’. We also can make our substitutions to ‘a = b’, since it also is a formula. Thus we can also derive ‘a = a’, ‘b = b’, and ‘b = a’. However, obtaining ‘b = a’ is not a derivation we can make in one step [because we are applying the rule separately for the first term and for the second]. So:

 

1.	a = b	A
2.	b = b	1,1=E
3.	b = a	1,2=E

(Nolt 242)

 

 

 

From:

Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.

 

 

Or if otherwise noted:

Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman &; Littlefield, 2013.

 

 

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Peirce (CP1.330-1.331) Collected Papers of Charles Sanders Peirce, Vol1/Bk3/Ch2/B/§5, "Polar Distinction and Volition", summary

 

by Corry Shores

 

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[The following is summary. Boldface and bracketed commentary are mine. Proofreading is incomplete, so please forgive my typos.]

 

 

Summary of

 

Charles Sanders Peirce

 

Collected Papers of Charles Sanders Peirce

 

Volume 1: Principles of Philosophy

 

Book 3: Phenomenology

 

Chapter 2: The Categories in Detail

 

B: Secondness

 

§5: Polar Distinction and Volition [1.330-1.331]

 

 

Brief summary:

A polar distinction is one where {a} two equally prevalent things are lacking a third one that is in coordination with them and also where {b} there is a neutrality of some sort separating these two poles. In the natural world, there are a limited number of polar distinctions. For example, there is past and future, the two sexes, and magnetic poles. However, our psychic life is full of polar distinctions, like pleasure and pain and right and wrong. They are always a matter of volition, as they involve us trying to attain one pole while avoiding its polar opposite, as with the case of pleasure and pain for example. Our actions are volitional only if along with our intentions and efforts to effect a change we also perceive that change happening. But if we only just perceive the change without intending and striving to have effected it, our experience is just perception. And if we are intending and striving to make a change, but there is a delay before we see it effected, then until it is effected it is just a longing, and only after when we see it accomplished is the act one of willing. And acts of attention are necessarily volitional.

 

 

 

 

Summary

 

1.330

[A polar distinction is one where there are two equally prevalent parties, without a third related to them, but with a neutrality that separates them. In the natural world there is just a limited number of polar distinctions, like past and future, and the two sexes. However, in psychic life there are many more, like pleasure and pain and right and wrong. These distinctions arise from our volition striving for one thing in avoidance of its polar opposite.]

 

[Peirce defines polar distinction as “any distinction between two equally decided characters to which no third seems to be coordinate (although a neutrality separates them).” The natural world does not have many such polar distinctions. Peirce thinks that the list of polar distinctions in nature is limited nearly to the following: the past and the future, the two ways of passing over a line, right-and left-handed spirals and helices, the magnetic poles, the electric poles (the prior set are related), the right and left sides of our bodies, and the two sexes. But in human psychology we have many more polar distinctions, with most of them being matters of volition, for example, pleasure and pain (where the factor of volition is found in the fact that we strive for the one and avoid the other), right and wrong (where again we aim for the one and try not to aim for the other), necessity and impossibility (I am not sure the volitional element here; Peirce says it is apparent when we need to consider their rational modifications), and reasonable and perverse (I also do not follow the volitional element here, but Peirce says they “imply that assent is as free as choice ever is, and so proclaim their volitional strain”. Perhaps the idea is that we volitionally choose and effect our acceptance of the situation when things are necessary or when they are impossible.). Peirce notes that we can find many antonyms in the thesaurus that show how there are various polar psychic sorts of distinctions. Peirce also says that for our purposes here, we do not need to reflect more on the volitional element involved in these psychic cases. But were we do so, we would find that these dichotomies are the result of volition. (I am not sure why, but perhaps the idea is that volition involves choosing and directing ones actions  in accordance with some polar element rather than its opposite.)

 

Calling any distinction between two equally decided characters to which no third seems to be coördinate (although a neutrality separates them) a polar distinction, in the external world polar distinctions are few. That of past and future, with the resulting two ways of passing over a line (and consequent right-and left-handed spirals and helices, whence probably the magnetic and possibly the electric poles – supposing the latter to be truly “polar” in our sense), with the right and left sides of our bodies, and the two sexes, seems pretty much to exhaust the list of them. Yet for the much smaller universe of psychology, polar distinctions abound, most of them referring to volition. Thus, pleasure is any kind of sensation that one immediately seeks, pain any that one immediately shuns. Right and wrong are expressly volitional. Necessity and impossibility so obviously refer to volition that the words often need qualification to show that rational modifications of them are meant. The words reasonable and perverse imply that assent is as free as choice ever is, and so proclaim their volitional strain. Roget’s Thesaurus illustrates the great aptitude of the psychical to polar distinction. Any very close examination of how far this is due to volition would cause us to wander quite away from the subject of this essay. It would show that dichotomy, meaning the fact that the elements that a distinction separates are just two in number, is strikingly often – perhaps that it is presumably always – due to volition ....

(165)

 

 

 

1.331

 

[Our actions are just perceptual if they are only aware of changes that are happening. But they are volitional if we are aware of our role in those changes having taken place. Until the change happens, we are longing for the change rather than willing it. Attention itself is something volitional.]

 

 

[Peirce will now elaborate more on the volition. (I do not follow this paragraph very well, so please consult the quotation to follow.) He says that in the mode of consciousness which is purely perceptual, we are aware just of something having been done. But in the mode of consciousness which is volitional, and that thus involves our willing, we are aware of something being done when it is actually effected (by us). And even if someone is given a physical task that they struggle to effect but cannot, as it is too difficult, it is still a volitional act, as they have commanded their muscles to work and resist with all their force. Peirce then makes a distinction between desire and willing. He describes “table-turning”, which I do not know anything about. (It is explained here at this wikipedia page.) The idea seems to be the Peirce sat around a table with other people. They all were positioned fairly far from the table, but still just close enough that their finger tips touched the table. They did not actively try to move the table with their muscles, but rather they just tried to use the pure force of their will to move the table. Then, after a short period during which they willed with all their might, the table would begin turning. Peirce says that before the table moved, their involvement was merely a longing, but not a willing. However, it becomes a willing when that longing and intention to effect change are met with the perception of the intended change in the world. Peirce then discusses a notion that some psychologists of his time use, namely, the concept of involuntary attention. Peirce says that this is not a very well conceived idea. It could mean either of two things. It could mean unpremeditated attention or it could mean attention influenced by conflicting desires. We are aware of conflicting desires, and we also know how what we desire to do might not accord with what we will to do. In fact, consciousness of this situation lies “at the root of our consciousness of free will.” Peirce thinks that “involuntary attention” is a contradiction in terms. Perhaps his point here is simply that any act of attention requires volition. As I did not grasp these ideas very well, please check the quotation below.]

Although the mode of consciousness we call volition, or willing, contrasts decidedly with the mere perception that something has been done, yet it is not perfected, and perhaps does not take place at all, until something is actually effected. Trying to shove something too heavy for the man to stir nevertheless accomplishes, in considerable measure, the only thing that he directly willed to do – namely, to contract certain muscles. In the days of table-turning we used to be commanded to sit quite away from a table, and “with all our might” to will that the table should move; and since the whole weight of our outstretched arms soon made our finger-tips unconsciously numb (for things are not apt to be consciously unconscious; and there were other concurring physiological effects that we did not suspect), while we were possessed of no other “might” over the table than through our muscles, we used to be speedily rewarded, by a direct consciousness of willing that the table move, accompanied by the vision of its wondrous obedience. Until it moved, we were only longing, not willing. So when certain psychologists write, chiefly in French – a language abounding in exquisite distinctions, but one in which any analytical method of interpretation is so sure to lead to misunderstandings, that the language is not well adapted to psychology or philosophy – about “involuntary attention,” they can only mean one of two things, either unpremeditated attention or attention influenced by conflicting desires. Though “desire” implies a tendency to volition, and though it is a natural hypothesis that a man cannot will to do that which he has no sort of desire to do, yet we all know conflicting desires but too well, and how treacherous they are apt to be; and a desire may perfectly well be discontented with volition, i.e., with what the man will do. The consciousness of that truth seems to me to be the root of our consciousness of free will. “Involuntary attention” involves in correct English a contradiction in adjecto.

(165-166)

 

 

 

Peirce, C.S. Collected Papers of Charles Sanders Peirce, Vol 1: Principles of Philosophy.  In Collected Papers of Charles Sanders Peirce [Two Volumes in One], Vols. 1 and 2. Edited by Charles Hartshorne and Paul Weiss. Cambridge, Massachusetts: 1965 [1931].

 

.

Peirce (CP1.326-1.329) Collected Papers of Charles Sanders Peirce, Vol1/Bk3/Ch2/B/§4, "The Dyad", summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

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[The following is summary. Boldface and bracketed commentary are mine. Proofreading is incomplete, so please forgive my typos.]

 

 

Summary of

 

Charles Sanders Peirce

 

Collected Papers of Charles Sanders Peirce

 

Volume 1: Principles of Philosophy

 

Book 3: Phenomenology

 

Chapter 2: The Categories in Detail

 

B: Secondness

 

§4: The Dyad [1.326-1.329]

 

 

Brief summary:

In a dyad, there are two parts that have entered into a union such that they maintain their individuality while also taking on the property of being in a united partnership. Corresponding to each member is a perspective of the whole that takes as its point of origin one member looking out toward the other, and this perspective is also called a “side” of the dyad. An illustration is when in Genesis God said “Let there be light” and there was light. Here there is the dyad of members, God and the created light. There is no third member, not even the act. For, this act is the bond inherent to the dyad and not somehow external or additional to it. And the two sides can be understood in the following way. One side is God commanding, and thereby causing, light to come into existence, and the other side is light’s appearing, which makes God become its creator through its own arising into existence. The first perspective captures the active, primary, fundamental, side of the dyad, while the other perspective captures the  passive, derivative, secondary side of this dyad. We also notice in this example that there was no delay between God’s fiat and the light coming to be. Were there such a mediation, that would have been a third and thus not a dyad. Such a mediating third could also be a reason or a law that makes one thing follow from another. So there is no such reason or law in dyads. We can also conclude that existence is solely a matter of dyads. We know that existence is not in monads, as they are pure potentialities or “may-bes”. And triads are matters of generalities rather than particular existences. (We also know from section 1.298 that there are no higher forms than triads).  Therefore, existence belongs exclusively to dyads. And to exist also means that there are forces causing something to sustain itself despite opposing forces acting against its existence. We say that monads have Being (in that they, as possibilities, are real and they are what they are), but only dyads can exist.

 

 

Summary

 

1.326

[A dyad is a union of two subjects that are brought into a state of oneness. Somehow the parts maintain their individuality while also taking on the property of partnership. Corresponding to the two parts are two internal perspectives of the dyad. ]

 

[Frege says that a dyad consists of two subjects that have been brought into oneness. They somehow maintain both their individuality while also constituting a unified structure. The dyad somehow retains both the traits of its monads while also having those of the dyad. Furthermore, each monad making up the dyad gains something from the dyad, that is to say, the dyad “imparts a character to each of them,” and this in some sense is like imparting to the monads the character of twoness. On way the dyad can be understood is by taking one member as primarily as or as the origin of our point of view on the whole. This is one “side” of the dyad. the other way to understand the dyad is by taking the perspective of the other member to realize this other “side” of the dyad. So on the one hand there is a pair of subjects in a dyad, which are the two members themselves. But on the other hand there is this pairing as the two “sides” or internal perspectives of the dyad. Peirce says these sides have their mode of union. I am not sure what he means. Perhaps he means either that they have as their mode that of union, or perhaps he is saying they have their own sort of union, distinct from that of the members understood as constituents rather than as sides. Furthermore, each side has a special character resulting from its being a subject of the dyad. (Recall from section 1.303 that a monad cannot be conceived complexly. Its constitution is singular, as it is not made of parts or aspects.) A dyad, then, unlike a monad, has a variety of features that express dyadic relations.]

A dyad consists of two subjects brought into oneness. These subjects have their modes of being in themselves, and they also have their modes of being, as first and second, etc., in connection with each other. They are two, if not really, at least in aspect. There is also some sort of union of them. The dyad is not the subjects; it has the subjects as one element of it. It has, besides, a suchness of monoidal character; and it has suchness, or suchnesses, peculiar to it as a dyad. The dyad brings the subjects together, and in doing so imparts a char- | acter to each of them. Those characters are, in some sense, two. The dyad has also two sides according to which subject is considered as first. These two sides of the dyad form a second pair of subjects attached to the dyad; and they have their mode of union. Each of them also has a special character as a subject of the dyad.

 

This description shows that the dyad, in contrast to the monad, has a variety of features; and all these features present dyadic relations.

(pp.163-164)

 

 

1.327

[Peirce gives as an example of a dyad, God, who is creating light. Here we can only think of these two members with the act of creation being the unifying factor. One aspect is God doing the creating with light being created. The other aspect is light being created and causing God to become the creator.]

 

[Peirce will give an example of a dyad, although it is a bit odd. He notes how God said, let there be light, and then there was light. In order to understand this as a dyad, we must only think of there being God who is creating light by declaring it to be. We must only think of there being the two subjects, God and the light. We cannot include third elements like this being a verse from the book of Genesis (for here the third member is the book of Genesis), nor should we think of it as a proposition that we can either believe or not (for here the third member is we ourselves). We also should not think of there being two parts to the event, namely, the fiat (which would be like a cause) and the coming to be of the light (which would be like the effect). Rather, the fiat and the coming to be of the light are in “one indivisible fact”. Furthermore, although we have the two subjects, God and the light, and although we also have the act of creation relating the two, we should not think of that act of creation a third member. But how Peirce has us conceive it is a little vague. He says that we should think of it “merely as the suchness of connection of God and light”. I am not sure exactly what that means, but perhaps the idea is that the members are united into a dyad and that unity has a certain quality in this case, namely the quality of creation. I am guessing. Peirce then says that the dyad is the fact, and this fact determines the existence of the light and the creatorship of God. Here perhaps he is thinking of the two sides of the dyad, but I am not sure. In that case, the one singular fact of the unity can be seen as having two sides, that of creating and being created. Specifically, one side is God compelling light to come into existence, and the other side is light making God become its creator by it coming into existence through his power. But still we see that one aspect captures the active, primary, fundamental sides while the other aspect captures the, passive, derivative, secondary side of this particular dyadic relation. Peirce might be saying that all dyadic relations have two sides that can be characterized this way, (as having active/passive sides), but I am not entirely sure.]

As an example of a dyad take this: God said, Let there be light, and there was light. We must not think of this as a verse of Genesis, for Genesis would be a third thing. Neither must we think of it as proposed for our acceptance, or as held for true; for we are third parties. We must simply think of God creating light by fiat. Not that the fiat and the coming into being of the light were two facts; but that it is in one indivisible fact. God and light are the subjects. The act of creation is to be regarded, not as any third object, but merely as the suchness of connection of God and light. The dyad is the fact. It determines the existence of the light, and the creatorship of God. The two aspects of the dyad are, first, that of God compelling the existence of the light, and that of the light as, by its coming into existence, making God a creator. This last is in the present example merely a mere point of view, without any reality corresponding to it. That is one of the special features of the particular example chosen. Of the two aspects of the dyad, then, one is in this instance, fundamental, real, and primary, while the other is merely derivative, formal, and secondary.

(Peirce 164)

 

 

1.328

[In the example of God creating light, there is an instantaneous act with no mediation and thus no third between God and the coming to be of the light. Dyad’s cannot be governed by reason or law, as this would also be a third (and thirdness should in fact be understood as such a mediation). Existence is not in monads, as they are potentialities. And triads involve generalities rather that existences. So since dyads are immediate, existence is purely dyadic.]

 

[Peirce explains that he chose this example because there is no intermediating time or process between God’s fiat and the coming to be of the light. The creation is instantaneous. Were there some time or intervening process, then there would be a third member and thus it would not be a dyad. He adds that thirdness can be understood as mediation. This furthermore means that a dyadic event cannot be guided by any reason or law, as this would mediate between the members and serve to connect them. Rather, the event needs to be an act of “arbitrary will or of blind force”. So there is no generality to be understood in a dyad. It is an individual fact and not somehow expressive of a general law. (Recall also from section 1.303 that the monad cannot be understood as an object, because that would require conceiving it in terms of a second thing in relation to which it is an object. And recall also from section 1.304 that the pure quality of feeling, which is monadic, exists as a “may-be” in the sense that it could be said to inhere in some object or be a part of some experience, but if it is not in either of these, it exists no less. In this section I wrote in summary of this idea (the following between ellipsis is copied from that entry). ...

the pure quality of feeling of hearing the train whistle is something that somehow is to be understood as existing apart of our experience of it. So they do not merely exist as being something concretely experienced in the present. Rather, they exist as “may-bes”, because in some way they are distinct from the concrete experiences that may or may not be experiences of these qualities. He seems to demonstrate this distinction with an odd example. He writes, “the word red means something when I say that the precession of the equinoxes is no more red than it is blue, and that it means just what it means when I say that aniline red is red.” But normally we think that qualities are qualities of one thing or another. But qualities of feeling are not to be thought of as inhering in anything. He contrasts qualities of feeling with things like laws. We cannot think of the law of gravity without also thinking that it would have to involve some physical objects with mass. In other words, how can the law of gravity exist if there were not things which could behave in accordance with it? However, we can think of pure qualities of feeling existing without them inhering in some object.

... As such, we might think of the monad in some sense as a potentiality. I am not sure that I follow still, but the idea for Peirce’s next point seems to be that we are not talking about monads in general but rather with monadic qualities, which are potentialities of existence rather than having existence. But a feature of being dyadic is that it exists, perhaps because it is necessarily immediate and given all on its own. Let me quote, as I am not absolutely certain.]

I chose this instance because it is represented as instantaneous. Had there been any process intervening between the causal act and the effect, this would have been a medial, or third, element. Thirdness, in the sense of the category, is the same as mediation. For that reason, pure dyadism is an act of arbitrary will or of blind force; for if there is any reason, or law, governing it, that mediates between the two subjects and brings about their connection. The dyad is an individual fact, as it existentially is; and it has no generality in it. The being of a monadic quality is a mere potentiality, without existence. Existence is purely dyadic.

(164)

 

 

1.329

[Being is monadic, as monads are real (even if they are merely potentialities) and thus have being ,but they do not exist, as they cannot be said to be in an oppositional relation with anything else. Existence, as something only dyads can be said to have, is a matter of oppositional forces, as in order for something to existence in some place at some time, it must be able to hold itself together despite other forces acting against it then and there.]

 

[The next paragraph is very dense and complex, so please refer to the quotation to follow. Let us still try to work through it. We should first try to distinguish Being from existence. Monads (and thus I assume pure qualities) have Being. We do not deny that they are what they are, or that they are real, but we say their are “may-bes” or potentialities. They are what they are and they are real without needing to actually exist in some determinate way. So we can speak of a pure quality, and affirm that it is real, but we do not necessarily say that the quality exists. In order to do so, we would need to place it into an oppositional relation with other actual things, which would mean that it becomes a dyad and thus can no longer be said to be a monad. Now, for things to exist, it is not enough to just be a dyad. The dyadic relations that constitute existence are competitive in some way. For something to exist, there must be some forces which allow it to survive in the face of other forces opposing it and working against its existence. Peirce will use the quote, “The very hyssop that grows on the wall exists in that chink because the whole universe could not prevent it”. We also said that dyads and thus existence do not involve mediating thirds like laws. So Peirce will also say that no law can make something exist. (Something does not exist because some law made it exist but rather it exists because it has been forced to exist or it forces itself to exist despite opposing forces). The next point is not so clear to me. Peirce then says that existence is a presence in some experiential universe. If we are talking about bare physics, and he uses the example of atoms, then it would seem odd to consider such a physical world an experiential one, as if rocks had experiences. I am not sure how we are to interpret this, but we might consider the following things. Peirce we saw previously (in section 1.311 and section 1.313) has a panpsychic perspective of the physical world. So perhaps here he means that things like rocks do have experiences insofar as they are affected by other things. Or perhaps we put aside the idea of panpsychism, and we merely just consider any “dynamical reaction” of one thing and another as constituting an experience of some sort, even if there are no inner states of awareness involved.]

It is to be noted that existence is an affair of blind force. “The very hyssop that grows on the wall exists in that chink because the whole universe could not prevent it.” No law determines any atom to exist. Existence is presence in some experiential universe – whether the universe of material things now existing, or that of laws, or that of phenomena, or that of feelings – and this presence implies that each existing thing is in dynamical reaction with every other in that universe. Existence, therefore, is dyadic; though Being is monadic.

(p.165)

 

 

Peirce, C.S. Collected Papers of Charles Sanders Peirce, Vol 1: Principles of Philosophy.  In Collected Papers of Charles Sanders Peirce [Two Volumes in One], Vols. 1 and 2. Edited by Charles Hartshorne and Paul Weiss. Cambridge, Massachusetts: 1965 [1931].

 

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