Pure Inductive Logic Jeff Paris School of Mathematics, University - - PowerPoint PPT Presentation

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pure inductive logic

Pure Inductive Logic Jeff Paris School of Mathematics, University - - PowerPoint PPT Presentation

Aug 09, 2023 213 likes •758 views

Pure Inductive Logic Jeff Paris School of Mathematics, University of Manchester in collaboration with J urgen Landes, Chris Nix, Alena Vencovsk a Jeff Paris Pure Inductive Logic Page 1 of 1 Jeff Paris Pure Inductive Logic et-

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Pure Inductive Logic

Jeff Paris

School of Mathematics, University of Manchester

– in collaboration with J¨ urgen Landes, Chris Nix, Alena Vencovsk´ a

Jeff Paris Pure Inductive Logic

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Page 1 of 1 et- 3... Jeff Paris Pure Inductive Logic

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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

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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

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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

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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

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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) (P3) w(∃x ψ(x)) = limn→∞ w (n

i=1 ψ(ai))

Jeff Paris Pure Inductive Logic

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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) (P3) w(∃x ψ(x)) = limn→∞ w (n

i=1 ψ(ai))

Jeff Paris Pure Inductive Logic

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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) (P3) w(∃x ψ(x)) = limn→∞ w (n

i=1 ψ(ai))

Jeff Paris Pure Inductive Logic

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How should shehe do it?

By the application of ‘rational principles’ . . . . . . based on Symmetry, Relevance, Irrelevance, Analogy, . . . Example Constant Exchangeability Principle, Ex For θ(x1, x2, . . . , xn) a formula of L not mentioning any constants w(θ(ai1, ai2, . . . , ain)) = w(θ(aj1, aj2, . . . , ajn)) Similarly replacing a relation symbol R everywhere in φ ∈ SL by ¬R should not change the probability (as in the coin toss example) – the Strong Negation Principle

Jeff Paris Pure Inductive Logic

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How should shehe do it?

By the application of ‘rational principles’ . . . . . . based on Symmetry, Relevance, Irrelevance, Analogy, . . . Example Constant Exchangeability Principle, Ex For θ(x1, x2, . . . , xn) a formula of L not mentioning any constants w(θ(ai1, ai2, . . . , ain)) = w(θ(aj1, aj2, . . . , ajn)) Similarly replacing a relation symbol R everywhere in φ ∈ SL by ¬R should not change the probability (as in the coin toss example) – the Strong Negation Principle

Jeff Paris Pure Inductive Logic

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How should shehe do it?

By the application of ‘rational principles’ . . . . . . based on Symmetry, Relevance, Irrelevance, Analogy, . . . Example Constant Exchangeability Principle, Ex For θ(x1, x2, . . . , xn) a formula of L not mentioning any constants w(θ(ai1, ai2, . . . , ain)) = w(θ(aj1, aj2, . . . , ajn)) Similarly replacing a relation symbol R everywhere in φ ∈ SL by ¬R should not change the probability (as in the coin toss example) – the Strong Negation Principle

Jeff Paris Pure Inductive Logic

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How should shehe do it?

By the application of ‘rational principles’ . . . . . . based on Symmetry, Relevance, Irrelevance, Analogy, . . . Example Constant Exchangeability Principle, Ex For θ(x1, x2, . . . , xn) a formula of L not mentioning any constants w(θ(ai1, ai2, . . . , ain)) = w(θ(aj1, aj2, . . . , ajn)) Similarly replacing a relation symbol R everywhere in φ ∈ SL by ¬R should not change the probability (as in the coin toss example) – the Strong Negation Principle

Jeff Paris Pure Inductive Logic

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How should shehe do it?

By the application of ‘rational principles’ . . . . . . based on Symmetry, Relevance, Irrelevance, Analogy, . . . Example Constant Exchangeability Principle, Ex For θ(x1, x2, . . . , xn) a formula of L not mentioning any constants w(θ(ai1, ai2, . . . , ain)) = w(θ(aj1, aj2, . . . , ajn)) Similarly replacing a relation symbol R everywhere in φ ∈ SL by ¬R should not change the probability (as in the coin toss example) – the Strong Negation Principle

Jeff Paris Pure Inductive Logic

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Jeff Paris Pure Inductive Logic

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Jeff Paris Pure Inductive Logic

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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

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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

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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

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Into the Polyadic

For simplicity assume that L has just a single binary relation symbol R. A state description for a1, a2, . . . , an is a quantifier free sentence of the form n

i,j=1 ±R(ai, aj)

State descriptions are where it all happens in this subject because:- Gaifman’s Theorem w is completely determined by its values on state descriptions.

Jeff Paris Pure Inductive Logic

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Into the Polyadic

For simplicity assume that L has just a single binary relation symbol R. A state description for a1, a2, . . . , an is a quantifier free sentence of the form n

i,j=1 ±R(ai, aj)

State descriptions are where it all happens in this subject because:- Gaifman’s Theorem w is completely determined by its values on state descriptions.

Jeff Paris Pure Inductive Logic

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Into the Polyadic

For simplicity assume that L has just a single binary relation symbol R. A state description for a1, a2, . . . , an is a quantifier free sentence of the form n

i,j=1 ±R(ai, aj)

State descriptions are where it all happens in this subject because:- Gaifman’s Theorem w is completely determined by its values on state descriptions.

Jeff Paris Pure Inductive Logic

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Into the Polyadic

For simplicity assume that L has just a single binary relation symbol R. A state description for a1, a2, . . . , an is a quantifier free sentence of the form n

i,j=1 ±R(ai, aj)

State descriptions are where it all happens in this subject because:- Gaifman’s Theorem w is completely determined by its values on state descriptions.

Jeff Paris Pure Inductive Logic

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Into the Polyadic

For simplicity assume that L has just a single binary relation symbol R. A state description for a1, a2, . . . , an is a quantifier free sentence of the form n

i,j=1 ±R(ai, aj)

State descriptions are where it all happens in this subject because:- Gaifman’s Theorem w is completely determined by its values on state descriptions.

Jeff Paris Pure Inductive Logic

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The Completely Independent Probability Function

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, to 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))

Jeff Paris Pure Inductive Logic

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The Completely Independent Probability Function

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, to 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))

Jeff Paris Pure Inductive Logic

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The Completely Independent Probability Function

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, to 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))

Jeff Paris Pure Inductive Logic

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The Completely Independent Probability Function

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, to 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))

Jeff Paris Pure Inductive Logic

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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

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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

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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

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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

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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, a2) R(a4, a3) R(a4, a4) Then the equivalence classes are {a1, a3}, {a2}, {a4} and the spectrum is {2, 1, 1}

Jeff Paris Pure Inductive Logic

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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

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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

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So the conjunctions of 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 ¬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

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The Promised Land (?)

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 the ¬R(a3, a1) is at least as likely as R(a3, a1) (so ‘analogy’ wins out)

Jeff Paris Pure Inductive Logic

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The Promised Land (?)

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 the ¬R(a3, a1) is at least as likely as R(a3, a1) (so ‘analogy’ wins out)

Jeff Paris Pure Inductive Logic

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The Promised Land (?)

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 the ¬R(a3, a1) is at least as likely as R(a3, a1) (so ‘analogy’ wins out)

Jeff Paris Pure Inductive Logic

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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. Assuming Sx they do!

Jeff Paris Pure Inductive Logic

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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. Assuming Sx they do!

Jeff Paris Pure Inductive Logic

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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. Assuming Sx they do!

Jeff Paris Pure Inductive Logic

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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. Assuming Sx they do!

Jeff Paris Pure Inductive Logic

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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. Assuming Sx they do!

Jeff Paris Pure Inductive Logic

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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

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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

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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

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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

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Carnap & Stegm¨ uller’s Analogieschluss

Suppose that w satisfies Sx and Θ( a) is the state description of L′ ⊂ L satisfied by

a. Then according

to w the most probable state description(s) of L satisfied by a have the same spectrum as Θ( a).

Jeff Paris Pure Inductive Logic

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Sx looks the business . . . but . . .

What is the rational justification for Sx ? Restricted to unary languages Sx can be justified by ‘symmetry’ But can Sx be justified by ‘symmetry’ in the polyadic?

Jeff Paris Pure Inductive Logic

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Sx looks the business . . . but . . .

What is the rational justification for Sx ? Restricted to unary languages Sx can be justified by ‘symmetry’ But can Sx be justified by ‘symmetry’ in the polyadic?

Jeff Paris Pure Inductive Logic

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Sx looks the business . . . but . . .

What is the rational justification for Sx ? Restricted to unary languages Sx can be justified by ‘symmetry’ But can Sx be justified by ‘symmetry’ in the polyadic?

Jeff Paris Pure Inductive Logic

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Sx looks the business . . . but . . .

What is the rational justification for Sx ? Restricted to unary languages Sx can be justified by ‘symmetry’ But can Sx be justified by ‘symmetry’ in the polyadic?

Jeff Paris Pure Inductive Logic