## TPTP Axioms File: LCL016^0.ax

```%------------------------------------------------------------------------------
% File     : LCL016^0 : TPTP v7.5.0. Released v6.1.0.
% 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 (   1 unit;  24 type;  22 defn)
%            Number of atoms       :  231 (  23 equality;  64 variable)
%            Maximal formula depth :   10 (   5 average)
%            Number of connectives :   52 (   5   ~;   3   |;   4   &;  37   @)
%                                         (   1 <=>;   2  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&;   0  !!;   0  ??)
%            Number of type conns  :  137 ( 137   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   29 (  24   :)
%            Number of variables   :   55 (   3 sgn;   6   !;   4   ?;  45   ^)
%                                         (  55   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% 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.
%------------------------------------------------------------------------------
%----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 ) ) )).

%------------------------------------------------------------------------------
```