Mixing
Puzzling axiom for modal cylindric algebras
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I've been working through James Freeman's "Algebraic Semantics for Modal Predicate Logic" (1976), where he introduces modal cylindric algebras -- essentially (special) cylindric algebras augmented with a modal operator. One of the axioms he gives, however, has me a bit puzzled. The axiom is
$$ c_k (*c_k x) = *c_k x,$$
where the asterisk is the algebraic equivalent of a modal diamond. What has me confused is the import this has for quantified modal logic (QML).
I believe the translation of this axiom in QML is
$$ exists x Diamond exists x phi leftrightarrow Diamond exists x phi.$$
In the right-to-left direction this seems to be just an instance of the following axiom/theorem of first-order logic:
$$ phi rightarrow exists x phi.$$
So it would seem that the left-to-right direction must be what holds modal interest and ensures some desirable connection between the modal operators and the quantifiers.
The left-to-right direction is
$$ exists x Diamond exists x phi rightarrow Diamond exists x phi,$$
which is also equivalent to
$$ Box forall x phi rightarrow forall x Box forall x phi.$$
But then this looks like it's just the axiom of first-order logic sometimes called "vacuous quantification":
$$ phi rightarrow forall x phi.$$
Is there something of modal significance to the above axiom for modal cylindric algebras, or is it just an axiom establishing conditions on quantifiers/cylindrifications in which the modal operator appears mysteriously? Or did I blunder somewhere?
quantifiers modal-logic
asked Jan 26 at 2:10
DennisDennis
1,118621
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add a comment |
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I've been working through James Freeman's "Algebraic Semantics for Modal Predicate Logic" (1976), where he introduces modal cylindric algebras -- essentially (special) cylindric algebras augmented with a modal operator. One of the axioms he gives, however, has me a bit puzzled. The axiom is
$$ c_k (*c_k x) = *c_k x,$$
where the asterisk is the algebraic equivalent of a modal diamond. What has me confused is the import this has for quantified modal logic (QML).
I believe the translation of this axiom in QML is
$$ exists x Diamond exists x phi leftrightarrow Diamond exists x phi.$$
In the right-to-left direction this seems to be just an instance of the following axiom/theorem of first-order logic:
$$ phi rightarrow exists x phi.$$
So it would seem that the left-to-right direction must be what holds modal interest and ensures some desirable connection between the modal operators and the quantifiers.
The left-to-right direction is
$$ exists x Diamond exists x phi rightarrow Diamond exists x phi,$$
which is also equivalent to
$$ Box forall x phi rightarrow forall x Box forall x phi.$$
But then this looks like it's just the axiom of first-order logic sometimes called "vacuous quantification":
$$ phi rightarrow forall x phi.$$
Is there something of modal significance to the above axiom for modal cylindric algebras, or is it just an axiom establishing conditions on quantifiers/cylindrifications in which the modal operator appears mysteriously? Or did I blunder somewhere?
quantifiers modal-logic
asked Jan 26 at 2:10
DennisDennis
1,118621
$endgroup$
$begingroup$
Isn't this just (as you say) a way of encoding the principle of vacuous quantification in a modal cylindric-algebraic context?
$endgroup$
– anygivenpoint
Feb 28 at 9:36
$begingroup$
(That sounds more aggressive than intended - I mean that I can't see what distinctively modal significance this has either.)
$endgroup$
– anygivenpoint
Feb 28 at 9:37
add a comment |
$begingroup$
I've been working through James Freeman's "Algebraic Semantics for Modal Predicate Logic" (1976), where he introduces modal cylindric algebras -- essentially (special) cylindric algebras augmented with a modal operator. One of the axioms he gives, however, has me a bit puzzled. The axiom is
$$ c_k (*c_k x) = *c_k x,$$
where the asterisk is the algebraic equivalent of a modal diamond. What has me confused is the import this has for quantified modal logic (QML).
I believe the translation of this axiom in QML is
$$ exists x Diamond exists x phi leftrightarrow Diamond exists x phi.$$
In the right-to-left direction this seems to be just an instance of the following axiom/theorem of first-order logic:
$$ phi rightarrow exists x phi.$$
So it would seem that the left-to-right direction must be what holds modal interest and ensures some desirable connection between the modal operators and the quantifiers.
The left-to-right direction is
$$ exists x Diamond exists x phi rightarrow Diamond exists x phi,$$
which is also equivalent to
$$ Box forall x phi rightarrow forall x Box forall x phi.$$
But then this looks like it's just the axiom of first-order logic sometimes called "vacuous quantification":
$$ phi rightarrow forall x phi.$$
Is there something of modal significance to the above axiom for modal cylindric algebras, or is it just an axiom establishing conditions on quantifiers/cylindrifications in which the modal operator appears mysteriously? Or did I blunder somewhere?
quantifiers modal-logic
asked Jan 26 at 2:10
DennisDennis
1,118621
$endgroup$
I've been working through James Freeman's "Algebraic Semantics for Modal Predicate Logic" (1976), where he introduces modal cylindric algebras -- essentially (special) cylindric algebras augmented with a modal operator. One of the axioms he gives, however, has me a bit puzzled. The axiom is
$$ c_k (*c_k x) = *c_k x,$$
where the asterisk is the algebraic equivalent of a modal diamond. What has me confused is the import this has for quantified modal logic (QML).
I believe the translation of this axiom in QML is
$$ exists x Diamond exists x phi leftrightarrow Diamond exists x phi.$$
In the right-to-left direction this seems to be just an instance of the following axiom/theorem of first-order logic:
$$ phi rightarrow exists x phi.$$
So it would seem that the left-to-right direction must be what holds modal interest and ensures some desirable connection between the modal operators and the quantifiers.
The left-to-right direction is
$$ exists x Diamond exists x phi rightarrow Diamond exists x phi,$$
which is also equivalent to
$$ Box forall x phi rightarrow forall x Box forall x phi.$$
But then this looks like it's just the axiom of first-order logic sometimes called "vacuous quantification":
$$ phi rightarrow forall x phi.$$
Is there something of modal significance to the above axiom for modal cylindric algebras, or is it just an axiom establishing conditions on quantifiers/cylindrifications in which the modal operator appears mysteriously? Or did I blunder somewhere?
quantifiers modal-logic
quantifiers modal-logic
asked Jan 26 at 2:10
DennisDennis
1,118621
asked Jan 26 at 2:10
DennisDennis
1,118621
asked Jan 26 at 2:10
DennisDennis
1,118621
asked Jan 26 at 2:10
DennisDennis
1,118621
asked Jan 26 at 2:10
DennisDennis
1,118621
1,118621
$begingroup$
Isn't this just (as you say) a way of encoding the principle of vacuous quantification in a modal cylindric-algebraic context?
$endgroup$
– anygivenpoint
Feb 28 at 9:36
$begingroup$
(That sounds more aggressive than intended - I mean that I can't see what distinctively modal significance this has either.)
$endgroup$
– anygivenpoint
Feb 28 at 9:37
add a comment |
$begingroup$
Isn't this just (as you say) a way of encoding the principle of vacuous quantification in a modal cylindric-algebraic context?
$endgroup$
– anygivenpoint
Feb 28 at 9:36
$begingroup$
(That sounds more aggressive than intended - I mean that I can't see what distinctively modal significance this has either.)
$endgroup$
– anygivenpoint
Feb 28 at 9:37
$begingroup$
Isn't this just (as you say) a way of encoding the principle of vacuous quantification in a modal cylindric-algebraic context?
$endgroup$
– anygivenpoint
Feb 28 at 9:36
$begingroup$
Isn't this just (as you say) a way of encoding the principle of vacuous quantification in a modal cylindric-algebraic context?
$endgroup$
– anygivenpoint
Feb 28 at 9:36
$begingroup$
(That sounds more aggressive than intended - I mean that I can't see what distinctively modal significance this has either.)
$endgroup$
– anygivenpoint
Feb 28 at 9:37
$begingroup$
(That sounds more aggressive than intended - I mean that I can't see what distinctively modal significance this has either.)
$endgroup$
– anygivenpoint
Feb 28 at 9:37
add a comment |
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$begingroup$
Isn't this just (as you say) a way of encoding the principle of vacuous quantification in a modal cylindric-algebraic context?
$endgroup$
– anygivenpoint
Feb 28 at 9:36
$begingroup$
(That sounds more aggressive than intended - I mean that I can't see what distinctively modal significance this has either.)
$endgroup$
– anygivenpoint
Feb 28 at 9:37