Paradoxes and the structure of reasoning David Ripley University of - - PowerPoint PPT Presentation

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Paradoxes and the structure of reasoning David Ripley University of - - PowerPoint PPT Presentation

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Paradoxes and the structure of reasoning David Ripley University of Connecticut http://davewripley.rocks (UConn logo, 1959) Paradoxes Introduction Paradoxes Introduction Think about testing a hypothesis. Paradoxes Introduction A

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Paradoxes and the structure of reasoning

David Ripley

University of Connecticut http://davewripley.rocks (UConn logo, 1959)

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Paradoxes

Introduction

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Paradoxes Introduction

Think about testing a hypothesis.

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Paradoxes Introduction

A simplified picture:

  • 1. Suppose the hypothesis is true.
  • 2. Figure out what else would follow.
  • 3. Check whether those other things are really true.
  • 4. If not, the hypothesis was wrong.
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Paradoxes Introduction

  • 2. Figure out what else would follow.

Here, some basic assumptions are helpful, like:

  • things either are a certain way or they’re not,
  • if things are one way, they’re not also any incompatible way,
  • for something to be true is for things to be as it says they are,
  • we can think about collections of things that are a certain way,

and so on.

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Paradoxes Introduction

If those basic assumptions aren’t trustworthy, the whole project falls apart.

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Paradoxes

Examples

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Paradoxes Examples

Liar paradox

“This sentence is not true.” If it’s true, then it’s not true. So if it’s true, it’s both true and not true. But that’s a contradiction! So it’s not true after all. But that’s what it says! So it is true. It’s both true and not true. We have a contradiction.

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Paradoxes Examples

Liar paradox

“This sentence is not true.” If it’s true, then it’s not true. So if it’s true, it’s both true and not true. But that’s a contradiction! So it’s not true after all. But that’s what it says! So it is true. It’s both true and not true. We have a contradiction.

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Paradoxes Examples

Liar paradox

“This sentence is not true.” If it’s true, then it’s not true. So if it’s true, it’s both true and not true. But that’s a contradiction! So it’s not true after all. But that’s what it says! So it is true. It’s both true and not true. We have a contradiction.

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Paradoxes Examples

Curry paradox

“If this sentence is true, then 2 + 2 = 5.” If it’s true, then if it’s true, then 2 + 2 = 5. So if it’s true, 2 + 2 does = 5. But that’s what it says! So it is true. So 2 + 2 = 5.

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Paradoxes Examples

Curry paradox

“If this sentence is true, then 2 + 2 = 5.” If it’s true, then if it’s true, then 2 + 2 = 5. So if it’s true, 2 + 2 does = 5. But that’s what it says! So it is true. So 2 + 2 = 5.

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Paradoxes Examples

Curry paradox

“If this sentence is true, then 2 + 2 = 5.” If it’s true, then if it’s true, then 2 + 2 = 5. So if it’s true, 2 + 2 does = 5. But that’s what it says! So it is true. So 2 + 2 = 5.

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Paradoxes Examples

Russell paradox (1/2)

Some collections don’t contain themselves. Others do. Think of the collection of all limes, and the collection of everything else.

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Paradoxes Examples

Russell paradox (2/2)

Now, think of the collection of all collections that don’t contain themselves. (It contains, among other things, the collection of all limes.) But does it contain itself? If it does, it doesn’t. If it doesn’t, it does. It’s a lot like the liar sentence; it leads to contradiction in the same way.

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Paradoxes Examples

The basic assumptions we use to investigate anything seem to be broken.

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Paradoxes Examples

If it follows from the mere existence of a Curry sentence that 2 + 2 = 5, what right do we have to say the Earth isn’t flat?

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Paradoxes Examples

We have a choice:

  • Give up. Our reasoning really is broken.

Maybe we can find another way to learn. Maybe not. OR

  • Push on. Find some trustworthy reasoning,

even if it’s not what we’re used to.

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Paradoxes Examples

We have a choice:

  • Give up. Our reasoning really is broken.

Maybe we can find another way to learn. Maybe not. OR

  • Push on. Find some trustworthy reasoning,

even if it’s not what we’re used to.

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Paradoxes

A bit of formalism

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Paradoxes A bit of formalism

A stock of symbols:

¬ Negation, not & Conjunction, and T True λ the liar sentence ⊢ Entailment, follows from The liar sentence λ is ¬Tλ.

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Paradoxes A bit of formalism

We can derive a contradiction from Tλ: Tλ Tλ λ ¬Tλ Tλ & ¬Tλ So by reductio, we can conclude ¬Tλ.

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Paradoxes A bit of formalism

We can go on to derive a contradiction from ¬Tλ (which we have now proved!): ¬Tλ λ Tλ ¬Tλ Tλ & ¬Tλ So by explosion, everything follows.

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Paradoxes A bit of formalism

Here are the steps we’ve used: ¬Tλ λ λ ¬Tλ A TA TA A A B A & B [A] . . . B & ¬B ¬A A & ¬A B

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Paradoxes A bit of formalism

Here are the steps we’ve used: ¬Tλ λ λ ¬Tλ A TA TA A A B A & B [A] . . . B & ¬B ¬A A & ¬A B

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Paradoxes A bit of formalism

Here are the steps we’ve used: ¬Tλ λ λ ¬Tλ A TA TA A A B A & B [A] . . . B & ¬B ¬A A & ¬A B

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Paradoxes A bit of formalism

Here are the steps we’ve used: ¬Tλ λ λ ¬Tλ A TA TA A A B A & B [A] . . . B & ¬B ¬A A & ¬A B

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Paradoxes A bit of formalism

Here are the steps we’ve used: ¬Tλ λ λ ¬Tλ A TA TA A A B A & B [A] . . . B & ¬B ¬A A & ¬A B

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Paradoxes A bit of formalism

Here are the steps we’ve used: ¬Tλ λ λ ¬Tλ A TA TA A A B A & B [A] . . . B & ¬B ¬A A & ¬A B

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Vocabulary?

Negation?

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Vocabulary? Negation?

One way to undermine this argument is to focus on negation. Two main flavours:

  • Maybe reductio is the problem?
  • Maybe explosion is the problem?
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Vocabulary? Negation?

That is, maybe it’s wrong to think that something leading to a contradiction must be false. Or maybe it’s wrong to think that things can’t be two incompatible ways. Or maybe it’s wrong to think that not being a certain way is incompatible with being that way.

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Vocabulary? Negation?

Problem 1: What, then, makes negation negation?

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Vocabulary? Negation?

Problem 2: Paradoxes that have nothing to do with negation.

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Vocabulary?

Truth?

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Vocabulary? Truth?

Another way to undermine the argument is to focus on truth. Two main flavours:

  • Maybe Tλ doesn’t really entail λ?
  • Maybe λ doesn’t really entail Tλ?
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Vocabulary? Truth?

That is, maybe being true is something different from telling it like it is.

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Vocabulary? Truth?

Problem 1: We can just rebuild the paradoxes with ‘tells it like it is’ instead of ‘is true’. The thought must be that ‘telling it like it is’ is incoherent.

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Vocabulary? Truth?

But this undermines inquiry even more directly than the paradoxes originally did!

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Vocabulary? Truth?

Problem 2: Paradoxes that have nothing to do with truth.

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Vocabulary? Truth?

Pseudo-Scotus: God exists Therefore, this argument is invalid. Suppose God exists, and suppose the argument is valid. Then the argument must be invalid. So it is both valid and invalid; contradiction. Thus, if God exists the argument must be invalid. But this is to prove its conclusion from its premise, so it really is valid! Since the argument is valid, if God exists, then it is invalid. But this would be a contradiction. So God does not exist.

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Vocabulary?

The problem

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Vocabulary? The problem

Solutions that focus on particular vocabulary are limited to paradoxes where that vocabulary plays some role.

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Vocabulary? The problem

As long as we consider only the liar, solutions focusing on negation or truth can seem plausible.

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Vocabulary? The problem

But the Curry and Russell paradoxes have no vocabulary in common at all! Curry: ‘If this sentence is true, then 2 + 2 = 5’. Russell: The collection of all collections that do not contain themselves.

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Vocabulary? The problem

No approach that focuses on particular vocabulary can get at the general phenomenon.

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Structure

Two options

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Structure Two options

If it’s not particular vocabulary, then what is it?

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Structure Two options

There are two main families of response. Both focus not on the steps in our proof, but on its structure.

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Structure

Noncontractive logics

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Structure Noncontractive logics

Return to the derivation of a contradiction from Tλ: Tλ Tλ λ ¬Tλ Tλ & ¬Tλ Note that this uses Tλ twice.

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Structure Noncontractive logics

Tλ on its own does not lead to contradiction. So our reductio should conclude not ¬Tλ outright, but just that if Tλ holds, then ¬Tλ holds. (And we already knew this!)

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Structure Noncontractive logics

Keeping track of number in this way means rejecting contraction: A, A ⊢ C A ⊢ C Two As might suffice for C where one does not.

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Structure Noncontractive logics

Blocking contraction is enough to prevent the paradoxes from causing trouble. The liar and Russell no longer lead to contradiction, Curry no longer leads to 2 + 2 = 5, and so on.

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Structure Noncontractive logics

Valid reasoning now not only uses its premises, but uses them up.

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Structure

Nontransitive logics

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Structure Nontransitive logics

A different approach focuses on the very last step. At this step, we have proved Tλ & ¬Tλ, and then explosion gives us any B at all. That is, we chain together our argument to Tλ & ¬Tλ with the argument from Tλ & ¬Tλ to B.

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Structure Nontransitive logics

The move is a case of transitivity: A ⊢ B B ⊢ C A ⊢ C It’s surprising (and it takes a lot of work to show!) but transitivity is completely dispensable most of the time. It’s a shortcut only; we can do the same things without it. One instance where this is not the case, though, is exactly the crucial link in the paradoxical argument.

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Structure Nontransitive logics

A valid argument from A to C rules out asserting A while denying C. If we’ve ruled out asserting A while denying B, A ⊢ B and ruled out asserting B while denying C, B ⊢ C have we ruled out asserting A while denying C? A ⊢

? C

No! Asserting A while denying C can still be fine, so long as we remain silent about B.

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Structure Nontransitive logics

A valid argument from A to C rules out asserting A while denying C. If we’ve ruled out asserting A while denying B, A ⊢ B and ruled out asserting B while denying C, B ⊢ C have we ruled out asserting A while denying C? A ⊢

? C

No! Asserting A while denying C can still be fine, so long as we remain silent about B.

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Structure Nontransitive logics

Blocking transitivity is a different way to prevent the paradoxes from causing trouble. We still get a contradiction: ⊢ Tλ & ¬Tλ. And we can have A & ¬A ⊢ B. But without transitivity, this is ok!

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Structure

Cumulative reasoning

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Structure Cumulative reasoning

Consider the following procedure for reasoning from some stock of premises: Cumulative reasoning: 1. Start from some stock of premises. 2. Draw conclusions that follow from your stock. 3. Add those conclusions to your original stock, resulting in an expanded stock. 4. Go back to step 2 and repeat. X ⊢ A X, A ⊢ C X ⊢ C

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Structure Cumulative reasoning

X ⊢ A X, A ⊢ C X ⊢ C Noncontractive and nontransitive approaches agree here: this is not ok! For the noncontractivist, this uses X twice to get to C; its conclusion needs to be X, X ⊢ C. For the nontransitivist, this chains things together on A; it cannot be repaired, but much of the time is dispensable.

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Structure Cumulative reasoning

  • Paradoxes seem to show that something

is seriously wrong in our usual practices of inquiry.

  • Solutions that focus on particular vocabulary

like negation or truth miss how widespread paradoxes are.

  • Solutions that focus on the structure of reasoning do better.
  • They have the upshot that cumulative reasoning

is not always trustworthy.