Dialetheism: An Introduction

How to be a dialetheist. Why to be a ... In fact, every classical logical truth is an LP logical truth, and vice ... Classical logic is only valid over classical (consistent,.
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How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Dialetheism: An Introduction David Ripley University of Melbourne

Vagueness Seminar

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Why to be a dialetheist Vagueness and dialetheism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Jaskowski Truth-functions The independent argument Truth Denial Why to be a dialetheist Vagueness and dialetheism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Jaskowski

• A, ¬A 6` B, but A ∧ ¬A ` B • A, B 6` A ∧ B • Not very useful for a dialetheist—too based on dialogue

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Jaskowski Truth-functions The independent argument Truth Denial Why to be a dialetheist Vagueness and dialetheism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

FDE

T

B

N

F • ∨ is least upper bound, ∧ is greatest lower bound • ¬ switches T and F, and leaves B and N alone • T and B are designated Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

FDE is classicalish

• A conjunction is true iff both conjuncts are, and false iff one

conjunct is • All De Morgan’s laws and double negation laws hold • BUT: • A ∧ ¬A 6` B • B 6` A ∨ ¬A • There are no logical truths • Nothing is unsatisfiable

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

LP

T B F • ∧ is min, ∨ is max • ¬ switches T and F, and leaves B alone • T and B are designated

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

LP is even more classicalish

• Has all the above classical features of FDE plus: • B ` A ∨ ¬A • In fact, every classical logical truth is an LP logical truth,

and vice versa • This includes ¬(A ∧ ¬A) • Still, nothing is unsatisfiable

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Jaskowski Truth-functions The independent argument Truth Denial Why to be a dialetheist Vagueness and dialetheism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

The independent argument

A A ∨ B ?

B

¬A ?

• The ?red step is Disjunctive Syllogism • DS must go • This can stay:

A∨B ¬A B ∨ (A ∧ ¬A)

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Modus ponens

• Note that if we take A ⊃ B := ¬A ∨ B, then modus ponens

for ⊃ is DS • So modus ponens for ⊃ is invalid • Other conditionals?

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Models FDE LP

CL

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Restricted validity

• Classical logic is only valid over classical (consistent,

complete) models • But it is valid there!

• LP is only valid over LP (complete) models • But it is valid there!

• This is not a matter of adding premises

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Jaskowski Truth-functions The independent argument Truth Denial Why to be a dialetheist Vagueness and dialetheism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Options

• You can keep the T-schema (A iff T hAi) • Do you want: ¬A iff ¬T hAi? • If yes, full intersubstitutivity • If no, then more distinctions are available

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Jaskowski Truth-functions The independent argument Truth Denial Why to be a dialetheist Vagueness and dialetheism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Disagreement

• How to express disagreement? • Asserting negation is not sufficient to disagree (FDE: or

necessary) • Different speech act: denial

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Why to be a dialetheist Semantic Paradox Set-theoretic Paradox Change, Borders Vagueness Vagueness and dialetheism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

The Liar

• Consider λ := T h¬λi

[λ] ¬λ λ ∧ ¬λ

Dialetheism

[¬λ] λ λ ∧ ¬λ λ ∧ ¬λ B

λ ∨ ¬λ

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Excluded Middle vs. Explosion

• If we get rid of either LEM or ECQ, the paradox is not as

troubling • Any argument for one is thus an argument against the

other

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Grelling’s Paradox

• H(‘H’) iff ¬ H(‘H’)

[H(‘H’)] ¬ H(‘H’) H(‘H’) ∧¬ H(‘H’)

Dialetheism

¬ [H(‘H’)] H(‘H’) H(‘H’) ∧¬ H(‘H’) H(‘H’) ∧¬ H(‘H’) B

H(‘H’) ∨¬ H(‘H’)

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Why to be a dialetheist Semantic Paradox Set-theoretic Paradox Change, Borders Vagueness Vagueness and dialetheism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Russell’s Paradox

• The set of all pencils is not a pencil, so it’s not

self-membered • The set of all things mentioned on this slide is mentioned

on this slide, so it is self-membered • Consider the set R of all non-self-membered sets • R = {x : x 6∈ x} • y ∈ R iff y 6∈ y • R ∈ R iff R 6∈ R • And we’re off!

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Burali-Forti Paradox

• ∅ is the first ordinal • Any initial set of ordinals is the least ordinal greater than

everything in the set • Consider the set Ω of all ordinals • This is the least ordinal greater than all ordinals

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Dialetheist set theory

• Dialetheist set theory is built on a naive comprehension

axiom: • For any condition A, there is a set of all and only the things

that are A • This is naive, but workable in an appropriate logic

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Why to be a dialetheist Semantic Paradox Set-theoretic Paradox Change, Borders Vagueness Vagueness and dialetheism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Change

• Consider something that changes from being F to not

being F • Suppose the change is instantaneous, and happens at

time t • At t, is the thing F , not F , neither F nor not F , or both F

and not F ? • Symmetry pushes us towards one of the latter two

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Borders

• Similarly, consider the border between France and

Germany • Does the border belong to France, to Germany, to both, or

to neither? • Again, symmetry pushes us towards the latter two

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Why to be a dialetheist Semantic Paradox Set-theoretic Paradox Change, Borders Vagueness Vagueness and dialetheism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Some people are both bald and not bald

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Why to be a dialetheist Vagueness and dialetheism Borderline cases Dialetheism & Contextualism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Dialetheism about vagueness

• For any vague predicate F , there are clear F s and clear

non-F s • What to say in between? • They overlap • Some things are both F and not F

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Outline

How to be a dialetheist Why to be a dialetheist Vagueness and dialetheism Borderline cases Dialetheism & Contextualism

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Contextualism

• Contextualists agree that some things are both F and not F • But they take the two F s to have distinct extensions • The dialetheist takes them to have the same (inconsistent)

extension

Dialetheism

Dave Ripley

How to be a dialetheist

Why to be a dialetheist

Vagueness and dialetheism

Internal context

• Contextualists often point to ‘internal context’: subtle

mental states that can shift from one moment to the next • For example, I might see a patch as closer to green one

moment and closer to blue the next • But does internal context affect the extension of ‘green’

and ‘blue’? And how does it?

Dialetheism

Dave Ripley