Is equality necessarily transitive? [duplicate]

Question

I want to introduce three definitions into the philosophy of logic for the purpose of improving first order logic. Consider the following three definitions.

Definitions

C is an arbitrary constant iff ∀x[x=C]

C is a specific constant iff ∃!x[x=C]

C is a general constant iff ∀x[x=C] ∨ ∃!x[x=C]

Let the domain of discourse be U, where

∀x[x ∈U]

x is a thing iff x ∈U

Definition ∀x[P(x)]=P(x1) & P(x2) & P(x3) &..

Now suppose C is an arbitrary constant. Then, C=x1 & C=x2 & C=x3 &..., where the symbols x1,x2,x3,..., each denote a unique thing, so they are specific constants.

If equality is necessarily transitive then you can conclude x1=x2=x3=..., which must be interpreted as saying only one thing exists. That's false. Therefore the proposed three definitions are acceptable only if equality isn't necessarily transitive.

So is equality necessarily transitive?

To answer this question place equality '=' in the signature. To understand its meaning use Hao Wang's axiom of Identity.

Hao Wang's Axiom of Identity

∀y[Φ(y) iff ∃x[x=y iff Φ(x)]]

You will need Universal Instantiation, Universal Generalization, Existential Instantiation, and Existential Generalization.

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EDIT-

Derived Rule Universal Instantiation UI:

∀x[P(x)] ⊢ P(x_i), where x_i is a specific constant

1. ∀x[P(x)]
2. P(x1) & P(x2) &... & P(x_i) &... [1;Df]
3. P(x_i) &... [2;i-1 applications of simplification 2]
4. P(x_i) [3; simplification 1]
5. x_i is a specific constant. [Df]

Corollary UI₂:

∀x[P(x)] ⊢ P(α), where α is an arbitrary constant.

Corollary UI₃:

∀x[P(x)] ⊢ P(C), where C is a general constant.

Derived Rule Existential Generalization EG:

P(C) ⊢ ∃x[P(x)], where C is a general constant.

1. P(C)
2. If ∀x[~P(x)] then ~P(C), where C is a general constant. [UI₃]
3. If P(C) then ~∀x[~P(x)] [2; trans]
4. If P(C) then ∃x[P(x)] [3;Df]
5. ∃x[P(x)] [1,4;MP]

Theorem 1 (substitution principle): If A is a specific constant and B is a specific constant, and A=B and P(A) then P(B).

1. A is a specific constant and B is a specific constant, and A=B and P(A) [OSC1]
2. ∀y[P(y) iff ∃x[x=y iff P(x)]] [axiom]
3. P(B) iff ∃x[x=B iff P(x)]] [2;UI]
4. A=B & P(A) [1; simplification]
5. ∃x[x=B & P(x)] [4;EG]
6. P(B) [3,5;MP2]
7. If A is a specific constant and B is a specific constant, and A=B and P(A) then P(B) [1-6;CSC1]

Q.E.D.

Theorem 2 (substitution principle): If α is an arbitrary constant and C is a specific constant, and C=α; and P(α) then P(C).

1. α is an arbitrary constant and C is a specific constant, and α=C; and P(α) [OSC1]
2. ∀y[P(y) iff ∃x[x=y iff P(x)]] [axiom]
3. P(C) iff ∃x[x=C iff P(x)]] [2;UI]
4. α=C & P(α) [1; simplification]
5. ∃x[x=C & P(x)] [4;EG]
6. P(C) [3,5;MP2]
7. If α is an arbitrary constant and C is a specific constant, and C=α; and P(α) then P(C) [1-6;CSC1]

Q.E.D.

Derived Rule Universal Generalization UG:

P(α) ⊢ ∀x[P(x)], where α is an arbitrary constant.

1. P(α)
2. α is an arbitrary constant.
3. ∀x[x=α] [2;Df]
4. x1=α & x2=α & x3=α &... [3;Df]
5. P(x1) & P(x2) & P(x3) &... [1,4; substitution]
6. ∀x[P(x)] [5;Df]

Theorem 3: Let α be an arbitrary constant.

P(α) iff ∀x[P(x)]

1. P(α) [OSC1]
2. ∀x[P(x)] [1;UG]
3. If P(α) then ∀x[P(x)] [1-2;CSC1]
4. ∀x[P(x)] [OSC1]
5. P(α) [4;UI₂]
6. If ∀x[P(x)] then P(α) [4-5;CSC1]
7. P(α) iff ∀x[P(x)] [3,6; Df]

Q.E.D.

Theorem 4: ∃x[P(x)] = P(x1) ∨ P(x2) ∨... ∨ P(x_i) ∨...

1. ~∀x[~P(x)] = ~[~P(x1) & ~P(x2) &... ] [Df]
2. ~∀x[~P(x)] = P(x1) ∨ P(x2) ∨... [1;prop]
3. ∃x[P(x)] = P(x1) ∨ P(x2) ∨... [2;Df]

Q.E.D.

Derived Rule Existential Instantiation EI:

∃x[P(x)] ⊢ P(x_i), where x_i is a specific constant.

1. ∃x[P(x)]
2. ∃x[P(x)] = P(x1) ∨ P(x2) ∨... [Th.4]
3. P(x1) ∨ P(x2) ∨... ∨ P(x_i) ∨... [1,2; substitution]
4. [[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] ∨ P(x_i) [3;prop]

Line 4 is true, and has a truth value that is constant in time, so it is necessarily true, by my understanding of temporal modal logic.

5. ☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] ∨ P(x_i) } [4; TML]

If it is possible that all the P(x_i)'s have a truth value that varies in time, then they could all simultaneously be false, in which case line 5 would be false. Therefore at least one of the P(x_i)'s has a truth value that is constant in time. Let it be P(x_i). Therefore

6. If ☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] ∨ P(x_i) } then ☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] } ∨ P(x_i) [5; TML]

7. ☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] } ∨ P(x_i) [5,6; MP]

It is possible that all the P(x_i)'s except P(x_i) have a truth value that varies in time, so they could all be simultaneously false. Therefore

8. ◊¬{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] }
9. ~☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] } [8;Df]
10. P(x_i) [7,9;DS]

Theorem 5 (transitivity of equality):

Let A,B,C be specific constants.

∀A∀B∀C[if A=B & B=C then A=C]

1. a,b,c are arbitrary specific constants. [OSC1]
2. ∀w[P(w) iff ~(w = c) [Df]
3. P(a) iff ~(a=c) [2;UI]
4. P(b) iff ~(b=c) [2;UI]
5. If a = b & P(a) then P(b) [substitution]
6. If a=b & ~(a=c) then ~(b=c) [3,4,5; rule of replacement]
7. If a=b then (if ~(a=c) then ~(b=c) ) [6; EXPORTATION]
8. If a=b then (if b=c then a=c) [7;trans]
9. If a=b and b=c then a=c [8;IMPORTATION]
10. ∀A∀B∀C[if A=B & B=C then A=C], where A,B,C are specific constants [9;UG]

Q E.D.

Theorem 6: If C is an arbitrary constant then C isn't a specific constant.

1. C is an arbitrary constant and C is a specific constant. [OSC1]
2. ∀x[x=C] ∧ ∃!x[x=C] [1;Df]
3. x1=C & x2=C & x3=C &... [2;Df]
4. ∃x[x=C ∧ ∀y[if y=C then y=x] [2;Df]
5. x_i=C ∧ ∀y[if y=C then y=x_i] [4;EI]
6. x_i is a specific constant. [Df]
7. If x1=C then x1= x_i, and if x2=C then x2= x_i, and ... [5;Df]
8. x1= x_i, and x2= x_i, and x3=x_i, and... [3,7;MP]
9. x1=x2=x3=... [8; transitivity of equality]
10. ∃!x[x ∈U] [meaning of 9]
11. ~ ∃!x[x ∈U] [Metatheorem]
12. ∃!x[x ∈U] ∧ ~ ∃!x[x ∈U] [10,11;conj]
13. If C is an arbitrary constant and C is a specific constant then contradiction. [1-12;CSC1]
14. Not(C is an arbitrary constant and C is a specific constant) [13;RAA]
15. If C is an arbitrary constant then C isn't a specific constant. [14; prop]

Q.E.D.

Answer 1

I think all this formalism is hiding the basic intuition that drives the definition of equality. Without transitivity I fail to see how equality has retains its usual meaning. You're basically inventing a new concept and simply calling it "equality" but I don't think it can capture the spirit of equality if it is not transitive. I think @mauro showed that you can derive transitivity from other axioms, but I think it's one of those situations where "pick n of m axioms and you can derive the rest" -- doubtless some axiomatizations are more parsimonious but all ind up implying the same set of properties.

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One interesting case where I think your concept of non-transitive equality is useful is describing speciation, which is a particular type of sorties paradox that comes up in evolutionary processes.

Definition of Species: We say that organisms X and Y are part of the same species S() (i.e. S(X)=S(Y)) if they can reproduce with one another in nature and produce fertile offspring (assuming X and Y are both have properly functioning reproductive systems) (1).

Thus if we think about the lineage of a currently existing species, we will find that they have an unbroken chain of ancestor species. E.g., humans and chimpanzees share a common ancestor species if you go back far enough.

This creates an apparent paradox in evolutionary theory: how do we get speciation if reproduction is necessary for driving evolution?

The paradox comes about because we mistake the if in the definition for an iff. We assume that every member of a species must be able to successfully breed with every other member with the same probability of success. However, we know this cannot be the case because we observe speciation which respects the above definition. Recent research (and here) is also exploring the nature of this.

Although the definition above is perfectly fine, it hides the fact that while organism X and Y can conceive fertile offspring, the probability of that happening can vary widely between individual of a given species. Over time, the (non-random) processes of mate preferences and gamete-level compatibility means that you get statistical separation long before you get speciation. Essentially, you get subgroups of a species that all satisfy our definition of a species but which contrive to induce divergent genetic evolution.

Speciation is usually accelerated by geographic separation, extreme events, etc but nonetheless, we have the same issue that equality is not transitive:

If we have a chain of offspring: X1 -> X2 -> X3.... the following pairwise equalities will also hold: S(X1)=S(X2), S(X2)=S(X3), S(X3)=S(X4), yet at some point T, we will have a situation where S(X1) != S(XT)!

In this sense, I see a non-transitive equality as being useful, it hinges on the fact that A can successfully reproduce with B depends on more than one gene, factor, etc. That means you can accumulate differences that do not impact ability to reproduce, just likelihood. Over time, there are too many differences and we get speciation.

Sorry for mini-bio essay but seemed like an interesting example of your idea.

Answer 2

You ask:

So is equality necessarily transitive?

Yes. Equality can be generalized as an equivalence relation (along with congruence, similarity, identity, etc.) which requires three properties: reflexivity, symmetry, and transitivity. No transitivity, no equivalence. From WP:

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality.