Decidability of the PAL Substitution Core LORI Workshop, ESSLLI 2010 Wes Holliday, Tomohiro Hoshi, and Thomas Icard Logical Dynamics Lab, CSLI Department of Philosophy, Stanford University August 20, 2010 Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 1
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Decidability of the PALSubstitution Core
LORI Workshop, ESSLLI 2010
Wes Holliday, Tomohiro Hoshi, and Thomas IcardLogical Dynamics Lab, CSLI
Department of Philosophy, Stanford University
August 20, 2010
Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 1
Introduction
The substitution core of a logic is the set of formulas all of whosesubstitution instances are valid [van Benthem, 2006].
I Typically the substitution core of a logic coincides with its set ofvalidities, in which case the logic is substitution-closed.
I However, many dynamic logics axiomatized using reduction axiomsare not substitution-closed.
I A classic example is Public Announcement Logic (PAL).
Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 2
Introduction
The substitution core of a logic is the set of formulas all of whosesubstitution instances are valid [van Benthem, 2006].
I Typically the substitution core of a logic coincides with its set ofvalidities, in which case the logic is substitution-closed.
I However, many dynamic logics axiomatized using reduction axiomsare not substitution-closed.
I A classic example is Public Announcement Logic (PAL).
Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 2
Introduction
The substitution core of a logic is the set of formulas all of whosesubstitution instances are valid [van Benthem, 2006].
I Typically the substitution core of a logic coincides with its set ofvalidities, in which case the logic is substitution-closed.
I However, many dynamic logics axiomatized using reduction axiomsare not substitution-closed.
I A classic example is Public Announcement Logic (PAL).
Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 2
Introduction
The substitution core of a logic is the set of formulas all of whosesubstitution instances are valid [van Benthem, 2006].
I Typically the substitution core of a logic coincides with its set ofvalidities, in which case the logic is substitution-closed.
I However, many dynamic logics axiomatized using reduction axiomsare not substitution-closed.
I A classic example is Public Announcement Logic (PAL).
Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 2
Not in the Core...
ExampleThe formula [p]p is valid in PAL.
However, its substitution instance[p ∧ ¬2p](p ∧ ¬2p) is not valid—this is the well-known problem of“unsuccessful” formulas.
Since [ϕ]ϕ is not valid for arbitrary ϕ, [p]p is not “schematically valid.”
It is not in the substitution core.
Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 3
Not in the Core...
ExampleThe formula [p]p is valid in PAL. However, its substitution instance[p ∧ ¬2p](p ∧ ¬2p) is not valid—this is the well-known problem of“unsuccessful” formulas.
Since [ϕ]ϕ is not valid for arbitrary ϕ, [p]p is not “schematically valid.”
It is not in the substitution core.
Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 3
Not in the Core...
ExampleThe formula [p]p is valid in PAL. However, its substitution instance[p ∧ ¬2p](p ∧ ¬2p) is not valid—this is the well-known problem of“unsuccessful” formulas.
Since [ϕ]ϕ is not valid for arbitrary ϕ, [p]p is not “schematically valid.”
It is not in the substitution core.
Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 3
Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 5
In the Core...
We have identified other principles of the substitution core.
Call a formula purely epistemic if every propositional variable andoccurrence of 〈ϕ〉 appears within the scope of a 3. Note that if ϕ ispurely epistemic, so is any substitution instance of ϕ.
Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 6
In the Core...
We have identified other principles of the substitution core.
Call a formula purely epistemic if every propositional variable andoccurrence of 〈ϕ〉 appears within the scope of a 3. Note that if ϕ ispurely epistemic, so is any substitution instance of ϕ.
Wes Holliday, Tomohiro Hoshi, and Thomas Icard: Decidability of the PAL Substitution Core, LORI Workshop, ESSLLI 2010 6
In the Core...
PropositionFormulas 1-3 are schematically valid. In 2 and 3, χ is purely epistemic.