## Summary of Paradoxes of Identity by Deutsch

Outline:

• The Paradox of 101 Dalmatians
• The Ship of Theseus Paradox

Summary:

• Some Definitions
• Paradoxes: arguments from apparently undeniable premises to obviously unacceptable conclusions.
• LL: x=y=>[f(x)->f(y)].
• NI: if a=b is true, then it is necessarily true.
• ND: if a!=b is true, then it is necessarily true.
• Photo A: old dog with a gray muzzle
• Photo B: young dog without a gray muzzle.
• A and B are photographs of the same dog Oscar.
• Dog in B has a property that the dog in A lacks => A and B are not photos of the same dog. Contradiction.
• What are the solutions?
• (1) Properties are relations to times: t, and t’.
• (2) Extend the object in time as well as space: photos capture distinct temporal parts.
• What are the objections?
• To (1): Properties are complicated relations, not only relations to times.
• To (2): Objects are not “wholly present” at any given time.
• In time t’, Oscar loses tail.
• In time t<t’, Oscar-minus (the whole dog minus his tail) and Oscar are distinct.
• In time t’, by ND, Oscar and Oscar-minus are distinct, since Oscar has a property at t’ that Oscar-minus lacks. Contradiction.
• What are the 4 responses?
• Deny there are such things as Oscar-minus.
• The parts of an object are essential.
• Objects of different kinds can occupy the same space at the same time, but objects of the same kind cannot.
• Temporal parts of distinct objects can occupy the same space at the same time.
• The Paradox of 101 Dalmatians
• If Oscar-minus is a dog, then Oscar minus a hair is also a dog.
• There are at least 101 dogs (actually many more).
• What are the responses/solutions?
• Maximality principle: no proper part of a dog is a dog.
• Lewis: deny that the “many” are Dalmatians or deny that Dalmatians are many.
• Day 1: statue s1 is made from clay c, so c is identical to s1.
• Day 2: statue s2 is made from clay c, so c is identical to s2.
• Day 3: a part of s2 is replaced by a new piece of clay, c’.
• s1 is identical to c on day 1, s2 is identical to c on day 2 => s1 is identical to s2 =>  s2 exists on day 2. Contradiction.
• s2 is identical to c on day 3 => c is c’, contradicting NI.
• What are the solutions?
• c and  are not identical:
• c exists prior to the existence of .
• s1 possesses the property of being destroyed while c does not.
•  possesses the property of being squeezed into a ball in the future while does not.
• Frame the issue in terms of c and s that (partially) coincide throughout their entire existence.
• c is a temporally extended object whose day 1 stage is identical to  and whose day 2 stage is identical to s2 => since day 1 and day 2 are different, s1 and s2 are not NI.
• Counterpart theory: different concepts are associated with different counterpart relations and hence with different criteria of trans-world identity.
• The Ship of Theseus Paradox
• A wooden ship restored by replacing all its planks and beams.
• Does the ship remain same?
• What are the views?
• The restored ship appears to qualify equally to be the original, but not the same ship.
• Hobbes.
• Wiggins (1967) and Parfit (1984): brain duplication scenarios.
• The restored ship is identical to the original one, since it exhibits a greater degree of spatio-temporal continuity with the original (Wiggins 1967).
• Problem of intuition: identity is preserved by spatio-temporal continuity v.s. identity is preserved in the process of dismantlement and reassembly.
• The restored ship is not identical to the original one.
• Kripke (1980): Table T is made out of wood H; In world w, T is made out of H’; In world w’, T is made out of H, and T’ is made out of H’. Since T and T’ are not identical in w’, table made out of H’ in w is not T.
• Analogy: In actual world: original ship O, and remodeled ship S; In world w, S’ is built out of the same parts of S. Since S’ and O are different in w, S is not identical to O. (Assumption: S and S’ are the same ship.)
• Additional paradox from Kripkean argument: S eventuates from O by replacing one part of O one day at a time. By transitivity of identity, O and S are the same.
• Kripke reply: whether O could change in S is irrelevant.
• Connections between two issues: the ship of Theseus problem & the question of the necessity of origin.
• Modified ship of Theseus problem:
• Two ships: O and O’, and O’ never sets sail; Planks are removed from O’ and used to replace corresponding planks of O, resulting in S. Do O and O’ have equal claim to be S?
• Criticisms:
• Conflict with the common sense principle (1) that the material of an object can be totally replenished or replaced without affecting its identity (Salmon 1979).
• Conflict with the additional common sense principle (2) that replacement by a single part or small portion preserves identity.
• Counter example of (2) :
• Two exactly similar sandals A and B;
• A is brand new and B is worn out;
• Parts of A and B are exchanged => A’ and B’;
• By ND, A and A’ are distinct, and B and B’ are distinct;

## Cosmological Argument and Argument from Design

In this paper, I will first explain the difference between a deductive argument and an inductive argument. I will compare two arguments which both explain the existence of theistic God, the cosmological argument and the argument from design. I will show that the former is deductive; while the latter is inductive. However, I will argue that the cosmological argument also rests on inductive reasoning.

The difference between a deductive argument and an inductive argument rests on whether the argument has a truth preserving logical structure and whether it creates new knowledge.

A deductively valid argument has a truth preserving logical structure, in other words, if its premises are true, its conclusion must be true. For example:

1. All men are moral (premise);
2. Socrates is a man (premise);
3. Socrates is moral (conclusion).

However, a deductively invalid argument is an argument that its premises do not form the logical necessity to give its conclusion. For example:

1. Poison can kill people (premise);
2. Daniel was killed (premise);
3. Daniel had taken poison (conclusion).

This argument is invalid, as experience tells us that many reasons can cause death, so the conclusion cannot be deduced from the premises.

Furthermore, deductive reasoning does not grant new knowledge, as it draws specific conclusion based on the prior knowledge of its premises, of which the conclusion is merely a subset of existing knowledge.

The cosmological argument is a deductive argument. It states that “(1) every being is a dependent being or an independent being, and (2) not every being can be a dependent being, therefore, (3) there exists a self-existent being”, which can be represented by the following formal logical  structure:

1. P or Q (premise);
2. P is not true (premise);
3. Q holds (conclusion).

We can verify its deductive validity via an intuitive example: an integer is either even or odd, so if an integer is not odd, then it must be even. The conclusion of this argument must be true if the two premises are true.

To the contrary, an inductive argument does not guarantee the truth of its conclusion. It draws general inferences from specific instances, and the truth of its premises provides a reason for accepting the conclusion with some level of probability. For example:

1. The sun has risen every day in the past (premise);
2. The sun will rise in the future (conclusion).

Although there is an infinitesimal possibility that the sun could explode tomorrow, this is unlikely to happen by our current knowledge, so the conclusion has high probability to be true.

Since inductive argument give generalizations of current knowledge, the generalized conclusion can then be applied to other new specific conditions with some confidence level. In this way, the inductive argument makes prediction and thus produces new knowledge.

The argument from design rests on inductive reasoning. It proceeds by Inference to the Best Explanation from the complexity, diversity and the appearance of purposes of design in the realm of living things to the existence of a divine designer, namely the theistic God. It is called “best explanation” because alternative explanations such as magic and randomness seem far less plausible by intuition. For example, the human body is a complex biological machine such that each system functions appropriately for a purpose and harmonious cooperate to carry out vital activities, which is unlikely to be caused by some random or magic power. Therefore, among all the alternatives the argument from design is its only probable explanation.

I will argue that cosmological argument (CA) indeed rests on inductive reasoning, because its premises rest on the Principle of Sufficient Reason (PSR), and PSR is an inductive argument.

First, CA(1) rests on PSRa. CA(1) states that “every being is a dependent being or an independent being”, with “dependence” defined as “can be caused by others or composed of others”, and “independence” defined as “can be caused by itself or composed of itself”. In other words, CA(1) proposes that every being can be explained by its causes or its components, and the cause or component can either be some other being or itself. This is based on PSRa. PSRa states that “there must be an explanation of the existence of any being”, which denies the existence of any unexplained being, implying every being must be either explained by others or explained by itself.

Second, CA(2) rests on PSRb. PSRb states that “there must be an explanation of any positive fact whatever”, which requires an explanation of the existence of the collection of all the dependent beings. The collection is either explained by itself, for which implies it is independent, or explained by others, for which the “other” being must be independent. Therefore, PSRb necessitates the existence of at least one independent being, so CA(2) holds.

Finally, PSR is inductive. It comes from the following reasoning:

1. We observe many beings/facts have had an explanation in the past (premise).
2. Therefore, every being/fact has and will have an explanation (conclusion).

The conclusion, however, does not exclude the possibility, albeit small, of no explanation. For example, the movement of sub-atomic particles whose cause has not fully explored by scientists might not have explanations. In sum, PSR which cosmological argument rests on is inductive.