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