Pure Inductive Logic and Spectrum
Exchangeability
Jeff Paris
School of Mathematics, University of Manchester
– in collaboration with Liz Howarth, Jürgen Landes,
Chris Nix, Alena Vencovská
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
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Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
W.E.Johnson (1858-1931)
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Rudolf Carnap (1891-1970)
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Pure Inductive Logic Framework
Imagine an agent inhabiting a structure M for a first order
language L with just finitely many relation symbols
P(x), P1 (x), P2 (x), R(x, y ) . . . etc.
and countably constant symbols a1 , a2 , a3 , . . . which name
every individual in the universe, and no function symbols
nor equality.
This agent is assumed to have no further knowledge about M
Let SL denote the set of first order sentences of L
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Pure Inductive Logic Framework
Imagine an agent inhabiting a structure M for a first order
language L with just finitely many relation symbols
P(x), P1 (x), P2 (x), R(x, y ) . . . etc.
and countably constant symbols a1 , a2 , a3 , . . . which name
every individual in the universe, and no function symbols
nor equality.
This agent is assumed to have no further knowledge about M
Let SL denote the set of first order sentences of L
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Pure Inductive Logic Framework
Imagine an agent inhabiting a structure M for a first order
language L with just finitely many relation symbols
P(x), P1 (x), P2 (x), R(x, y ) . . . etc.
and countably constant symbols a1 , a2 , a3 , . . . which name
every individual in the universe, and no function symbols
nor equality.
This agent is assumed to have no further knowledge about M
Let SL denote the set of first order sentences of L
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Pure Inductive Logic Framework
Imagine an agent inhabiting a structure M for a first order
language L with just finitely many relation symbols
P(x), P1 (x), P2 (x), R(x, y ) . . . etc.
and countably constant symbols a1 , a2 , a3 , . . . which name
every individual in the universe, and no function symbols
nor equality.
This agent is assumed to have no further knowledge about M
Let SL denote the set of first order sentences of L
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
We ask our agent to ‘rationally’ assign a probability w (θ) to
θ ∈ SL being true in this ambient structure M.
Equivalently we’re asking the agent to pick a ‘rational’
probability function w , where
w : SL → [0, 1] is a probability function on L if it satisfies
(P1)
|= θ ⇒ w (θ) = 1
(P2)
θ |= ¬φ ⇒ w (θ ∨ φ) = w (θ) + w (φ)
W
w (∃x ψ(x)) = limn→∞ w ( ni=1 ψ(ai ))
(P3)
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
We ask our agent to ‘rationally’ assign a probability w (θ) to
θ ∈ SL being true in this ambient structure M.
Equivalently we’re asking the agent to pick a ‘rational’
probability function w , where
w : SL → [0, 1] is a probability function on L if it satisfies
(P1)
|= θ ⇒ w (θ) = 1
(P2)
θ |= ¬φ ⇒ w (θ ∨ φ) = w (θ) + w (φ)
W
w (∃x ψ(x)) = limn→∞ w ( ni=1 ψ(ai ))
(P3)
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
We ask our agent to ‘rationally’ assign a probability w (θ) to
θ ∈ SL being true in this ambient structure M.
Equivalently we’re asking the agent to pick a ‘rational’
probability function w , where
w : SL → [0, 1] is a probability function on L if it satisfies
(P1)
|= θ ⇒ w (θ) = 1
(P2)
θ |= ¬φ ⇒ w (θ ∨ φ) = w (θ) + w (φ)
W
w (∃x ψ(x)) = limn→∞ w ( ni=1 ψ(ai ))
(P3)
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
State Descriptions
A state description Θ(a1 , a2 , . . . , an ) for a1 , a2 , . . . , an is a
conjunction make up of one of ±R(ai1 , ai2 , . . . , aim ) for each
m-ary relation symbol R of L and i1 , i2 , . . . , im ≤ n
E.g. if L has just the binary relation symbol R then the
conjunction of
R(a1 , a1 ) ¬R(a1 , a2 ) R(a1 , a3 )
R(a1 , a4 )
R(a2 , a1 ) ¬R(a2 , a2 ) R(a2 , a3 ) ¬R(a2 , a4 )
R(a3 , a1 ) ¬R(a3 , a2 ) R(a3 , a3 )
R(a3 , a4 )
R(a4 , a1 )
R(a4 , a4 )
R(a4 , a2 ) R(a4 , a3 )
is a state description for a1 , a2 , a3 , a4 .
It tells us all ‘atomic’ relations between the a1 , a2 , . . . , an .
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
State Descriptions
A state description Θ(a1 , a2 , . . . , an ) for a1 , a2 , . . . , an is a
conjunction make up of one of ±R(ai1 , ai2 , . . . , aim ) for each
m-ary relation symbol R of L and i1 , i2 , . . . , im ≤ n
E.g. if L has just the binary relation symbol R then the
conjunction of
R(a1 , a1 ) ¬R(a1 , a2 ) R(a1 , a3 )
R(a1 , a4 )
R(a2 , a1 ) ¬R(a2 , a2 ) R(a2 , a3 ) ¬R(a2 , a4 )
R(a3 , a1 ) ¬R(a3 , a2 ) R(a3 , a3 )
R(a3 , a4 )
R(a4 , a1 )
R(a4 , a4 )
R(a4 , a2 ) R(a4 , a3 )
is a state description for a1 , a2 , a3 , a4 .
It tells us all ‘atomic’ relations between the a1 , a2 , . . . , an .
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
State Descriptions
A state description Θ(a1 , a2 , . . . , an ) for a1 , a2 , . . . , an is a
conjunction make up of one of ±R(ai1 , ai2 , . . . , aim ) for each
m-ary relation symbol R of L and i1 , i2 , . . . , im ≤ n
E.g. if L has just the binary relation symbol R then the
conjunction of
R(a1 , a1 ) ¬R(a1 , a2 ) R(a1 , a3 )
R(a1 , a4 )
R(a2 , a1 ) ¬R(a2 , a2 ) R(a2 , a3 ) ¬R(a2 , a4 )
R(a3 , a1 ) ¬R(a3 , a2 ) R(a3 , a3 )
R(a3 , a4 )
R(a4 , a1 )
R(a4 , a4 )
R(a4 , a2 ) R(a4 , a3 )
is a state description for a1 , a2 , a3 , a4 .
It tells us all ‘atomic’ relations between the a1 , a2 , . . . , an .
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Gaifman’s Theorem
A probability function w is completely determined by its values
on state descriptions.
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Gaifman’s Theorem
A probability function w is completely determined by its values
on state descriptions.
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Recap
We ask our agent to ‘rationally’ assign a probability w (θ) to
θ ∈ SL being true in this ambient structure M.
Equivalently we’re asking the agent to pick a ‘rational’
probability function w
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Recap
We ask our agent to ‘rationally’ assign a probability w (θ) to
θ ∈ SL being true in this ambient structure M.
Equivalently we’re asking the agent to pick a ‘rational’
probability function w
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
How can the Agent do it?
By the application of ‘rational principles’ . . .
. . . based on Symmetry, Relevance, Irrelevance, Analogy, . . .
Constant Exchangeability Principle, Ex
For θ(a1 , a2 , . . . , an ) ∈ SL and any distinct j1 , j2 , . . . , jn
w (θ(a1 , a2 , . . . , an )) = w (θ(aj1 , aj2 , . . . , ajn ))
Likewise the Predicate Exchangeability Principle, Px,
where we interchange relations of the same arity and the
Strong Negation Principle, SN where we replace the
relation R everywhere by ¬R.
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
How can the Agent do it?
By the application of ‘rational principles’ . . .
. . . based on Symmetry, Relevance, Irrelevance, Analogy, . . .
Constant Exchangeability Principle, Ex
For θ(a1 , a2 , . . . , an ) ∈ SL and any distinct j1 , j2 , . . . , jn
w (θ(a1 , a2 , . . . , an )) = w (θ(aj1 , aj2 , . . . , ajn ))
Likewise the Predicate Exchangeability Principle, Px,
where we interchange relations of the same arity and the
Strong Negation Principle, SN where we replace the
relation R everywhere by ¬R.
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
How can the Agent do it?
By the application of ‘rational principles’ . . .
. . . based on Symmetry, Relevance, Irrelevance, Analogy, . . .
Constant Exchangeability Principle, Ex
For θ(a1 , a2 , . . . , an ) ∈ SL and any distinct j1 , j2 , . . . , jn
w (θ(a1 , a2 , . . . , an )) = w (θ(aj1 , aj2 , . . . , ajn ))
Likewise the Predicate Exchangeability Principle, Px,
where we interchange relations of the same arity and the
Strong Negation Principle, SN where we replace the
relation R everywhere by ¬R.
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
How can the Agent do it?
By the application of ‘rational principles’ . . .
. . . based on Symmetry, Relevance, Irrelevance, Analogy, . . .
Constant Exchangeability Principle, Ex
For θ(a1 , a2 , . . . , an ) ∈ SL and any distinct j1 , j2 , . . . , jn
w (θ(a1 , a2 , . . . , an )) = w (θ(aj1 , aj2 , . . . , ajn ))
Likewise the Predicate Exchangeability Principle, Px,
where we interchange relations of the same arity and the
Strong Negation Principle, SN where we replace the
relation R everywhere by ¬R.
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
How can the Agent do it?
By the application of ‘rational principles’ . . .
. . . based on Symmetry, Relevance, Irrelevance, Analogy, . . .
Constant Exchangeability Principle, Ex
For θ(a1 , a2 , . . . , an ) ∈ SL and any distinct j1 , j2 , . . . , jn
w (θ(a1 , a2 , . . . , an )) = w (θ(aj1 , aj2 , . . . , ajn ))
Likewise the Predicate Exchangeability Principle, Px,
where we interchange relations of the same arity and the
Strong Negation Principle, SN where we replace the
relation R everywhere by ¬R.
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Adam’s Apples
Suppose that Adam knows that apples of strain A are
good pollinators and apples of strain B are easily pollinated.
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Then he might conclude that were he to plant them next to
each other
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
the result would be fruitful . . .
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Such intuitions however are easily challenged, e.g.
Given
R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )
which of R(a3 , a1 ), ¬R(a3 , a1 ) would you think the more likely?
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Such intuitions however are easily challenged, e.g.
Given
R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )
which of R(a3 , a1 ), ¬R(a3 , a1 ) would you think the more likely?
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Such intuitions however are easily challenged, e.g.
Given
R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )
which of R(a3 , a1 ), ¬R(a3 , a1 ) would you think the more likely?
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
The Completely Independent Probability Function
Assume L has the single binary relation symbol R.
The Completely Independent Probability Function w0 gives
each of the ±R(ai , aj ) probability 1/2 and treats them
all as stochastically independent
E.g.
w0 (R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) =
= (1/2) × (1/2) × (1/2) = 1/8
Trouble is, re our earlier question
w0 (R(a3 , a1 ) | R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) = 1/2
w0 (¬R(a3 , a1 ) | R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) = 1/2
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
The Completely Independent Probability Function
Assume L has the single binary relation symbol R.
The Completely Independent Probability Function w0 gives
each of the ±R(ai , aj ) probability 1/2 and treats them
all as stochastically independent
E.g.
w0 (R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) =
= (1/2) × (1/2) × (1/2) = 1/8
Trouble is, re our earlier question
w0 (R(a3 , a1 ) | R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) = 1/2
w0 (¬R(a3 , a1 ) | R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) = 1/2
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
The Completely Independent Probability Function
Assume L has the single binary relation symbol R.
The Completely Independent Probability Function w0 gives
each of the ±R(ai , aj ) probability 1/2 and treats them
all as stochastically independent
E.g.
w0 (R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) =
= (1/2) × (1/2) × (1/2) = 1/8
Trouble is, re our earlier question
w0 (R(a3 , a1 ) | R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) = 1/2
w0 (¬R(a3 , a1 ) | R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) = 1/2
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
The Completely Independent Probability Function
Assume L has the single binary relation symbol R.
The Completely Independent Probability Function w0 gives
each of the ±R(ai , aj ) probability 1/2 and treats them
all as stochastically independent
E.g.
w0 (R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) =
= (1/2) × (1/2) × (1/2) = 1/8
Trouble is, re our earlier question
w0 (R(a3 , a1 ) | R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) = 1/2
w0 (¬R(a3 , a1 ) | R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) = 1/2
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
The Completely Independent Probability Function
Assume L has the single binary relation symbol R.
The Completely Independent Probability Function w0 gives
each of the ±R(ai , aj ) probability 1/2 and treats them
all as stochastically independent
E.g.
w0 (R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) =
= (1/2) × (1/2) × (1/2) = 1/8
Trouble is, re our earlier question
w0 (R(a3 , a1 ) | R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) = 1/2
w0 (¬R(a3 , a1 ) | R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )) = 1/2
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Spectrum Exchangeability
Given a state description Θ(a1 , a2 , . . . , an ) define the
equivalence relation ∼Θ on {a1 , . . . , an } by:
ai ∼Θ aj ⇐⇒ Θ(a1 , a2 , . . . , an ) ∧ ai = aj is consistent
equivalently iff ai , aj are indistinguishable on the basis of
Θ(a1 , . . . , an ).
The spectrum of Θ(a1 , . . . , an ) is the multiset of sizes of the
equivalence classes according to ∼Θ .
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Spectrum Exchangeability
Given a state description Θ(a1 , a2 , . . . , an ) define the
equivalence relation ∼Θ on {a1 , . . . , an } by:
ai ∼Θ aj ⇐⇒ Θ(a1 , a2 , . . . , an ) ∧ ai = aj is consistent
equivalently iff ai , aj are indistinguishable on the basis of
Θ(a1 , . . . , an ).
The spectrum of Θ(a1 , . . . , an ) is the multiset of sizes of the
equivalence classes according to ∼Θ .
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Spectrum Exchangeability
Given a state description Θ(a1 , a2 , . . . , an ) define the
equivalence relation ∼Θ on {a1 , . . . , an } by:
ai ∼Θ aj ⇐⇒ Θ(a1 , a2 , . . . , an ) ∧ ai = aj is consistent
equivalently iff ai , aj are indistinguishable on the basis of
Θ(a1 , . . . , an ).
The spectrum of Θ(a1 , . . . , an ) is the multiset of sizes of the
equivalence classes according to ∼Θ .
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Spectrum Exchangeability
Given a state description Θ(a1 , a2 , . . . , an ) define the
equivalence relation ∼Θ on {a1 , . . . , an } by:
ai ∼Θ aj ⇐⇒ Θ(a1 , a2 , . . . , an ) ∧ ai = aj is consistent
equivalently iff ai , aj are indistinguishable on the basis of
Θ(a1 , . . . , an ).
The spectrum of Θ(a1 , . . . , an ) is the multiset of sizes of the
equivalence classes according to ∼Θ .
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Example
Suppose Θ(a1 , a2 , a3 , a4 ) is the conjunction of
R(a1 , a1 ) ¬R(a1 , a2 ) R(a1 , a3 )
R(a1 , a4 )
R(a2 , a1 ) ¬R(a2 , a2 ) R(a2 , a3 ) ¬R(a2 , a4 )
R(a3 , a1 ) ¬R(a3 , a2 ) R(a3 , a3 )
R(a3 , a4 )
R(a4 , a1 )
R(a4 , a4 )
R(a4 , a2 ) R(a4 , a3 )
Then the equivalence classes are {a1 , a3 }, {a2 }, {a4 } and the
spectrum is
{2, 1, 1}
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Spectrum Exchangeability
Spectrum Exchangeability, Sx
If the state descriptions Θ(a1 , . . . , an ), Φ(a1 , . . . , an ) have the
same spectrum then
w (Θ(a1 , . . . , an )) = w (Φ(a1 , . . . , an ))
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Spectrum Exchangeability
Spectrum Exchangeability, Sx
If the state descriptions Θ(a1 , . . . , an ), Φ(a1 , . . . , an ) have the
same spectrum then
w (Θ(a1 , . . . , an )) = w (Φ(a1 , . . . , an ))
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
So the conjunctions of the
R(a1 , a1 ) ¬R(a1 , a2 ) R(a1 , a3 )
R(a2 , a1 ) ¬R(a2 , a2 ) R(a2 , a3 )
R(a3 , a1 ) ¬R(a3 , a2 ) R(a3 , a3 )
and of the
¬R(a1 , a1 ) ¬R(a1 , a2 ) R(a1 , a3 )
¬R(a2 , a1 ) ¬R(a2 , a2 ) R(a2 , a3 )
R(a3 , a1 )
R(a3 , a2 ) R(a3 , a3 )
get the same probability as both have spectrum {2, 1}
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
The Promised Land (?)
Sx implies Ex, Px, SN.
Recall the question
‘Given
R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )
which of R(a3 , a1 ), ¬R(a3 , a1 ) would you think the more likely?’
Sx implies that ¬R(a3 , a1 ) is at least as likely as R(a3 , a1 )
(so ‘analogy’ wins out)
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
The Promised Land (?)
Sx implies Ex, Px, SN.
Recall the question
‘Given
R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )
which of R(a3 , a1 ), ¬R(a3 , a1 ) would you think the more likely?’
Sx implies that ¬R(a3 , a1 ) is at least as likely as R(a3 , a1 )
(so ‘analogy’ wins out)
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
The Promised Land (?)
Sx implies Ex, Px, SN.
Recall the question
‘Given
R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )
which of R(a3 , a1 ), ¬R(a3 , a1 ) would you think the more likely?’
Sx implies that ¬R(a3 , a1 ) is at least as likely as R(a3 , a1 )
(so ‘analogy’ wins out)
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
The Promised Land (?)
Sx implies Ex, Px, SN.
Recall the question
‘Given
R(a1 , a2 ) ∧ R(a2 , a1 ) ∧ ¬R(a1 , a3 )
which of R(a3 , a1 ), ¬R(a3 , a1 ) would you think the more likely?’
Sx implies that ¬R(a3 , a1 ) is at least as likely as R(a3 , a1 )
(so ‘analogy’ wins out)
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Conformity
Consider the two ‘unary relations’ R(a1 , x) and R(x, x) of L.
Which of the two ‘state descriptions’
R(a1 , a1 ) ∧ R(a1 , a2 ) ∧ ¬R(a1 , a3 ) ∧ R(a1 , a4 )
R(a1 , a1 ) ∧ R(a2 , a2 ) ∧ ¬R(a3 , a3 ) ∧ R(a4 , a4 )
should we think the more likely?
The intuition is that there is no rational reason why
R(a1 , x) and R(x, x) should, in isolation, differ
Hence the above ‘state descriptions’ should get the same
probability . . .
. . . and assuming Sx they do
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Conformity
Consider the two ‘unary relations’ R(a1 , x) and R(x, x) of L.
Which of the two ‘state descriptions’
R(a1 , a1 ) ∧ R(a1 , a2 ) ∧ ¬R(a1 , a3 ) ∧ R(a1 , a4 )
R(a1 , a1 ) ∧ R(a2 , a2 ) ∧ ¬R(a3 , a3 ) ∧ R(a4 , a4 )
should we think the more likely?
The intuition is that there is no rational reason why
R(a1 , x) and R(x, x) should, in isolation, differ
Hence the above ‘state descriptions’ should get the same
probability . . .
. . . and assuming Sx they do
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Conformity
Consider the two ‘unary relations’ R(a1 , x) and R(x, x) of L.
Which of the two ‘state descriptions’
R(a1 , a1 ) ∧ R(a1 , a2 ) ∧ ¬R(a1 , a3 ) ∧ R(a1 , a4 )
R(a1 , a1 ) ∧ R(a2 , a2 ) ∧ ¬R(a3 , a3 ) ∧ R(a4 , a4 )
should we think the more likely?
The intuition is that there is no rational reason why
R(a1 , x) and R(x, x) should, in isolation, differ
Hence the above ‘state descriptions’ should get the same
probability . . .
. . . and assuming Sx they do
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Conformity
Consider the two ‘unary relations’ R(a1 , x) and R(x, x) of L.
Which of the two ‘state descriptions’
R(a1 , a1 ) ∧ R(a1 , a2 ) ∧ ¬R(a1 , a3 ) ∧ R(a1 , a4 )
R(a1 , a1 ) ∧ R(a2 , a2 ) ∧ ¬R(a3 , a3 ) ∧ R(a4 , a4 )
should we think the more likely?
The intuition is that there is no rational reason why
R(a1 , x) and R(x, x) should, in isolation, differ
Hence the above ‘state descriptions’ should get the same
probability . . .
. . . and assuming Sx they do
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Conformity
Consider the two ‘unary relations’ R(a1 , x) and R(x, x) of L.
Which of the two ‘state descriptions’
R(a1 , a1 ) ∧ R(a1 , a2 ) ∧ ¬R(a1 , a3 ) ∧ R(a1 , a4 )
R(a1 , a1 ) ∧ R(a2 , a2 ) ∧ ¬R(a3 , a3 ) ∧ R(a4 , a4 )
should we think the more likely?
The intuition is that there is no rational reason why
R(a1 , x) and R(x, x) should, in isolation, differ
Hence the above ‘state descriptions’ should get the same
probability . . .
. . . and assuming Sx they do
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Conformity
Consider the two ‘unary relations’ R(a1 , x) and R(x, x) of L.
Which of the two ‘state descriptions’
R(a1 , a1 ) ∧ R(a1 , a2 ) ∧ ¬R(a1 , a3 ) ∧ R(a1 , a4 )
R(a1 , a1 ) ∧ R(a2 , a2 ) ∧ ¬R(a3 , a3 ) ∧ R(a4 , a4 )
should we think the more likely?
The intuition is that there is no rational reason why
R(a1 , x) and R(x, x) should, in isolation, differ
Hence the above ‘state descriptions’ should get the same
probability . . .
. . . and assuming Sx they do
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Inseparability
Suppose that w satisfies Sx and is not equal to w0 .
Then, given a state description Θ(a1 , a2 , . . . , an ) in which
a1 , a2 are indistinguishable (i.e. a1 ∼Θ a2 ) there is a non-zero
probability according to w that they will remain forever
indistinguishable.
BUT the probability according to w that a1 , a2 will be forever
indistinguishable but be distinguishable from each of
a3 , a4 , a5 , . . . is zero
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Inseparability
Suppose that w satisfies Sx and is not equal to w0 .
Then, given a state description Θ(a1 , a2 , . . . , an ) in which
a1 , a2 are indistinguishable (i.e. a1 ∼Θ a2 ) there is a non-zero
probability according to w that they will remain forever
indistinguishable.
BUT the probability according to w that a1 , a2 will be forever
indistinguishable but be distinguishable from each of
a3 , a4 , a5 , . . . is zero
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Inseparability
Suppose that w satisfies Sx and is not equal to w0 .
Then, given a state description Θ(a1 , a2 , . . . , an ) in which
a1 , a2 are indistinguishable (i.e. a1 ∼Θ a2 ) there is a non-zero
probability according to w that they will remain forever
indistinguishable.
BUT the probability according to w that a1 , a2 will be forever
indistinguishable but be distinguishable from each of
a3 , a4 , a5 , . . . is zero
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Inseparability
Suppose that w satisfies Sx and is not equal to w0 .
Then, given a state description Θ(a1 , a2 , . . . , an ) in which
a1 , a2 are indistinguishable (i.e. a1 ∼Θ a2 ) there is a non-zero
probability according to w that they will remain forever
indistinguishable.
BUT the probability according to w that a1 , a2 will be forever
indistinguishable but be distinguishable from each of
a3 , a4 , a5 , . . . is zero
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Link with the Unary
Suppose that the probability function w on a unary language
satisfies Sx and can be extended to all (finite) unary languages
whilst preserving Sx. Then w can be extended to all languages
whilst preserving Sx.
Not all polyadic probability functions satisfying Sx arise in this
way, but,
every probability function satisfying Sx is of the form
(1 + λ)w1 − λw2 ,
with 0 ≤ λ ≤ 1,
where w1 , w2 do arise in this way.
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Link with the Unary
Suppose that the probability function w on a unary language
satisfies Sx and can be extended to all (finite) unary languages
whilst preserving Sx. Then w can be extended to all languages
whilst preserving Sx.
Not all polyadic probability functions satisfying Sx arise in this
way, but,
every probability function satisfying Sx is of the form
(1 + λ)w1 − λw2 ,
with 0 ≤ λ ≤ 1,
where w1 , w2 do arise in this way.
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Link with the Unary
Suppose that the probability function w on a unary language
satisfies Sx and can be extended to all (finite) unary languages
whilst preserving Sx. Then w can be extended to all languages
whilst preserving Sx.
Not all polyadic probability functions satisfying Sx arise in this
way, but,
every probability function satisfying Sx is of the form
(1 + λ)w1 − λw2 ,
with 0 ≤ λ ≤ 1,
where w1 , w2 do arise in this way.
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Commitments
Suppose our agent adopts the Principle Sx
Does that commit the agent to any categorical beliefs
which are not tautologies?
Is
Theory(Sx) = {θ ∈ SL | w (θ) = 1 for all w satisfying Sx}
more than just the set of tautologies?
Yes
Theory(Sx) = {θ ∈ SL | M |= θ for all finite structures M for L}
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Commitments
Suppose our agent adopts the Principle Sx
Does that commit the agent to any categorical beliefs
which are not tautologies?
Is
Theory(Sx) = {θ ∈ SL | w (θ) = 1 for all w satisfying Sx}
more than just the set of tautologies?
Yes
Theory(Sx) = {θ ∈ SL | M |= θ for all finite structures M for L}
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Commitments
Suppose our agent adopts the Principle Sx
Does that commit the agent to any categorical beliefs
which are not tautologies?
Is
Theory(Sx) = {θ ∈ SL | w (θ) = 1 for all w satisfying Sx}
more than just the set of tautologies?
Yes
Theory(Sx) = {θ ∈ SL | M |= θ for all finite structures M for L}
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Commitments
Suppose our agent adopts the Principle Sx
Does that commit the agent to any categorical beliefs
which are not tautologies?
Is
Theory(Sx) = {θ ∈ SL | w (θ) = 1 for all w satisfying Sx}
more than just the set of tautologies?
Yes
Theory(Sx) = {θ ∈ SL | M |= θ for all finite structures M for L}
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Commitments
Suppose our agent adopts the Principle Sx
Does that commit the agent to any categorical beliefs
which are not tautologies?
Is
Theory(Sx) = {θ ∈ SL | w (θ) = 1 for all w satisfying Sx}
more than just the set of tautologies?
Yes
Theory(Sx) = {θ ∈ SL | M |= θ for all finite structures M for L}
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Equivalently the agent should give zero belief to any
sentence which does not have a finite model.
E.g. the conjunction of
∀x ¬R(x, x)
∀x, y , z ((R(x, y ) ∧ R(y , z)) → R(x, z))
∀x ∃y R(x, y )
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Equivalently the agent should give zero belief to any
sentence which does not have a finite model.
E.g. the conjunction of
∀x ¬R(x, x)
∀x, y , z ((R(x, y ) ∧ R(y , z)) → R(x, z))
∀x ∃y R(x, y )
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Product Recommendation
If you are ever in the agent’s position assume Sx
This no way ‘logically’ determines all probabilities
but it sheds light on symmetry, relevance, irrelevance,
analogy and it is arguably ‘rational’
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Product Recommendation
If you are ever in the agent’s position assume Sx
This no way ‘logically’ determines all probabilities
but it sheds light on symmetry, relevance, irrelevance,
analogy and it is arguably ‘rational’
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Product Recommendation
If you are ever in the agent’s position assume Sx
This no way ‘logically’ determines all probabilities
but it sheds light on symmetry, relevance, irrelevance,
analogy and it is arguably ‘rational’
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Product Recommendation
If you are ever in the agent’s position assume Sx
This no way ‘logically’ determines all probabilities
but it sheds light on symmetry, relevance, irrelevance,
analogy and it is arguably ‘rational’
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability
Product Recommendation
If you are ever in the agent’s position assume Sx
This no way ‘logically’ determines all probabilities
but it sheds light on symmetry, relevance, irrelevance,
analogy and it is arguably ‘rational’
Jeff Paris
Pure Inductive Logic and Spectrum Exchangeability