TPTP Axioms File: LCL016^0.ax
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% File : LCL016^0 : TPTP v9.0.0. Released .0.
% Domain : Logic Calculi (Second Order Modal Logic)
% Axioms : Embedding of second order modal logic in simple type theory
% Version : [Ben13] axioms.
% English : An embedding of second order monomodal logic into simple type
% theory. The concrete logic is base logic K.
% Refs : [Ben13] Benzmueller (2013), Email to Geoff Sutcliffe
% : [BP13] Benzmueller & Paulson (2013), Quantified Multimodal Lo
% Source : [Ben13]
% Names :
% Status : Satisfiable
% Syntax : Number of formulae : 46 ( 22 unt; 24 typ; 22 def)
% Number of atoms : 51 ( 23 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 52 ( 5 ~; 3 |; 4 &; 37 @)
% ( 1 <=>; 2 =>; 0 <=; 0 <~>)
% Maximal formula depth : 1 ( 1 avg; 37 nst)
% Number of types : 3 ( 1 usr)
% Number of type conns : 137 ( 137 >; 0 *; 0 +; 0 <<)
% Number of symbols : 26 ( 23 usr; 2 con; 0-3 aty)
% Number of variables : 55 ( 45 ^ 6 !; 4 ?; 55 :)
% SPC : TH0_SAT_EQU
% Comments : In order to obtain other logics such B or S5 one can further
% restrict the accessibility relation. E.g. for B one can simply
% add the axiom of symmetry for rel. For S5 one would additionally
% postulate reflexivity and transitivity of rel.
% : Quantifiers are provided for individuals, sets or individuals
% (properties), and propositions. We here assume and implement
% constant domain semantics. Respective quantifiers for varying
% domains and cumulative domains can easily be added. An explicit
% "existInWorlds" predicate can be introduced for this, and the
% quantifiers would then be relativized using this predicate. The
% generic operators mbox_generic and mdia_generic can be applied to
% a particular accessibility relation rel to turn these generic
% modal operators turn into a particular mbox and mdia operator for
% rel. Hence, this axiomatization supports multimodal logics, and
% for stating bridge rules there are different options: conditions
% on the accessibility relations can be stated or usual bridge
% rules can be stated unsing propositional quantification.
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%----Declaration of additional base type mu
thf(mu_type,type,
mu: $tType ).
%----Equality on individuals
thf(meq_ind_type,type,
meq_ind: mu > mu > $i > $o ).
thf(meq_ind,definition,
( meq_ind
= ( ^ [X: mu,Y: mu,W: $i] : ( X = Y ) ) ) ).
%----Modal operators mtrue, mfalse, mnot, mor, mand, mimplies, mequiv, ...
thf(mtrue_type,type,
mtrue: $i > $o ).
thf(mtrue,definition,
( mtrue
= ( ^ [W: $i] : $true ) ) ).
thf(mfalse_type,type,
mfalse: $i > $o ).
thf(mfalse,definition,
( mfalse
= ( ^ [W: $i] : $false ) ) ).
thf(mnot_type,type,
mnot: ( $i > $o ) > $i > $o ).
thf(mnot,definition,
( mnot
= ( ^ [Phi: $i > $o,W: $i] :
~ ( Phi @ W ) ) ) ).
thf(mor_type,type,
mor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mor,definition,
( mor
= ( ^ [Phi: $i > $o,Psi: $i > $o,W: $i] :
( ( Phi @ W )
| ( Psi @ W ) ) ) ) ).
thf(mand_type,type,
mand: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mand,definition,
( mand
= ( ^ [Phi: $i > $o,Psi: $i > $o,W: $i] :
( ( Phi @ W )
& ( Psi @ W ) ) ) ) ).
thf(mimplies_type,type,
mimplies: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mimplies,definition,
( mimplies
= ( ^ [Phi: $i > $o,Psi: $i > $o,W: $i] :
( ( Phi @ W )
=> ( Psi @ W ) ) ) ) ).
thf(mimplied_type,type,
mimplied: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mimplied,definition,
( mimplied
= ( ^ [Phi: $i > $o,Psi: $i > $o,W: $i] :
( ( Psi @ W )
=> ( Phi @ W ) ) ) ) ).
thf(mequiv_type,type,
mequiv: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mequiv,definition,
( mequiv
= ( ^ [Phi: $i > $o,Psi: $i > $o,W: $i] :
( ( Phi @ W )
<=> ( Psi @ W ) ) ) ) ).
thf(mxor_type,type,
mxor: ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mxor,definition,
( mxor
= ( ^ [Phi: $i > $o,Psi: $i > $o,W: $i] :
( ( ( Phi @ W )
& ~ ( Psi @ W ) )
| ( ~ ( Phi @ W )
& ( Psi @ W ) ) ) ) ) ).
%----Universal quantification: individuals
thf(mforall_ind_type,type,
mforall_ind: ( mu > $i > $o ) > $i > $o ).
thf(mforall_ind,definition,
( mforall_ind
= ( ^ [Phi: mu > $i > $o,W: $i] :
! [X: mu] : ( Phi @ X @ W ) ) ) ).
%----Universal quantification: sets of individuals (properties)
thf(mforall_indset_type,type,
mforall_indset: ( ( mu > $i > $o ) > $i > $o ) > $i > $o ).
thf(mforall_indset,definition,
( mforall_indset
= ( ^ [Phi: ( mu > $i > $o ) > $i > $o,W: $i] :
! [X: mu > $i > $o] : ( Phi @ X @ W ) ) ) ).
%----Universal quantification: propositions
thf(mforall_prop_type,type,
mforall_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(mforall_prop,definition,
( mforall_prop
= ( ^ [Phi: ( $i > $o ) > $i > $o,W: $i] :
! [P: $i > $o] : ( Phi @ P @ W ) ) ) ).
%----Existential quantification: individuals
thf(mexists_ind_type,type,
mexists_ind: ( mu > $i > $o ) > $i > $o ).
thf(mexists_ind,definition,
( mexists_ind
= ( ^ [Phi: mu > $i > $o,W: $i] :
? [X: mu] : ( Phi @ X @ W ) ) ) ).
%----Existential quantification: sets of individuals (properties)
thf(mexists_indset_type,type,
mexists_indset: ( ( mu > $i > $o ) > $i > $o ) > $i > $o ).
thf(mexists_indset,definition,
( mexists_indset
= ( ^ [Phi: ( mu > $i > $o ) > $i > $o,W: $i] :
? [X: mu > $i > $o] : ( Phi @ X @ W ) ) ) ).
%----Existential quantification: propositions
thf(mexists_prop_type,type,
mexists_prop: ( ( $i > $o ) > $i > $o ) > $i > $o ).
thf(mexists_prop,definition,
( mexists_prop
= ( ^ [Phi: ( $i > $o ) > $i > $o,W: $i] :
? [P: $i > $o] : ( Phi @ P @ W ) ) ) ).
%----Generic mbox operator
thf(mbox_generic_type,type,
mbox_generic: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mbox_generic,definition,
( mbox_generic
= ( ^ [R: $i > $i > $o,Phi: $i > $o,W: $i] :
! [V: $i] :
( ~ ( R @ W @ V )
| ( Phi @ V ) ) ) ) ).
%----Generic mdia operator
thf(mdia_generic_type,type,
mdia_generic: ( $i > $i > $o ) > ( $i > $o ) > $i > $o ).
thf(mdia_generic,definition,
( mdia_generic
= ( ^ [R: $i > $i > $o,Phi: $i > $o,W: $i] :
? [V: $i] :
( ( R @ W @ V )
& ( Phi @ V ) ) ) ) ).
%----The accessibility relation rel
thf(rel_type,type,
rel: $i > $i > $o ).
%----The mbox operator instantiated for rel (further mbox operators
%----for other accessibility relations can be introduced analogously)
thf(mbox_type,type,
mbox: ( $i > $o ) > $i > $o ).
thf(mbox,definition,
( mbox
= ( mbox_generic @ rel ) ) ).
%----The mdia operator instantiated for rel (further mdia operators
%----for other accessibility relations can be introduced analogously)
thf(mdia_type,type,
mdia: ( $i > $o ) > $i > $o ).
thf(mdia,definition,
( mdia
= ( mdia_generic @ rel ) ) ).
%----The notion of validity
thf(mvalid_type,type,
mvalid: ( $i > $o ) > $o ).
thf(mvalid,definition,
( mvalid
= ( ^ [Phi: $i > $o] :
! [W: $i] : ( Phi @ W ) ) ) ).
%----Definition of invalidity
thf(minvalid_type,type,
minvalid: ( $i > $o ) > $o ).
thf(minvalid,definition,
( minvalid
= ( ^ [Phi: $i > $o] :
! [W: $i] :
~ ( Phi @ W ) ) ) ).
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