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Why are logics related to lattices and algebras? What can be said about this relationship? [closed]
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To me it seems very mysterious (and unexpected) that (classical propositional) logic can be "lifted" to powersets and Boolean algebras, in the sense that elements of a powerset behave like propositions (and logical connectives like or and and are "lifted" to set-theoretic union and intersection) and they satisfy the axioms of a Boolean algebra.
And even more mysterious is the observation that intuitionistic propositional logic is the logic of open sets of a topological space and of Heyting algebras.
And then quantum propositional logic seems to be the logic of orthocomplemented lattices of closed subspaces of a Hilbert space (and what algebra?).
And then the modal logic S4 seems to be the logic of interior algebras (and what space?).
And, along this line of thought, David Ellerman has some really interesting stuff about the logic of partitions (of a set), which is dual to the logic of subsets (of a set) and which seems to be related to the fact that subsets are dual to quotient sets (and then he relates partition logic to information theory in the same way that subset logic is related to probability theory).
So logics seem to be related (or, equivalent, even) to various objects in various branches of math. Eg.
beginarrayc
textclassical propositional logic & textsubsets of a set & textBoolean algebra \ hline
textintuitionistic propositional logic & textopen sets of a topological space & textHeyting algebra \ hline
textquantum (propositional?) logic & textclosed subspaces of a Hilbert space & text? \ hline
textS4 & text? & textinterior algebra \ hline
textpartition (propositional?) logic & textpartitions of a set & text? \ hline
textclassical first-order logic & text? & text? \ hline
textclassical $n$th-order logic, for all $n$ & text? & text? \ hline
endarray
- Why are logics related to lattices and algebras? What can be said about this relationship? Are there any books that talk about this relationship (and the table above) in detail?
- How can the $2$nd-to-last row in the table above be filled (ie. for classical first-order logic)?
- Is there a mathematical object that contains all classical logics of all degrees, all at once? Kinda like the exterior algebra of a vector space contains all exterior powers of all degrees, all at once, together with a meaningful way in which the elements of different degrees interact (like graded-commutativity and the exterior product)?
logic model-theory
edited Jul 26 at 7:14
Taroccoesbrocco
3,38941331
asked Jul 26 at 1:21
étale-cohomology
9151617
closed as too broad by Rob Arthan, max_zorn, Brian Borchers, Lord Shark the Unknown, Mostafa Ayaz Jul 28 at 10:45
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |Â
up vote
5
down vote
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To me it seems very mysterious (and unexpected) that (classical propositional) logic can be "lifted" to powersets and Boolean algebras, in the sense that elements of a powerset behave like propositions (and logical connectives like or and and are "lifted" to set-theoretic union and intersection) and they satisfy the axioms of a Boolean algebra.
And even more mysterious is the observation that intuitionistic propositional logic is the logic of open sets of a topological space and of Heyting algebras.
And then quantum propositional logic seems to be the logic of orthocomplemented lattices of closed subspaces of a Hilbert space (and what algebra?).
And then the modal logic S4 seems to be the logic of interior algebras (and what space?).
And, along this line of thought, David Ellerman has some really interesting stuff about the logic of partitions (of a set), which is dual to the logic of subsets (of a set) and which seems to be related to the fact that subsets are dual to quotient sets (and then he relates partition logic to information theory in the same way that subset logic is related to probability theory).
So logics seem to be related (or, equivalent, even) to various objects in various branches of math. Eg.
beginarrayc
textclassical propositional logic & textsubsets of a set & textBoolean algebra \ hline
textintuitionistic propositional logic & textopen sets of a topological space & textHeyting algebra \ hline
textquantum (propositional?) logic & textclosed subspaces of a Hilbert space & text? \ hline
textS4 & text? & textinterior algebra \ hline
textpartition (propositional?) logic & textpartitions of a set & text? \ hline
textclassical first-order logic & text? & text? \ hline
textclassical $n$th-order logic, for all $n$ & text? & text? \ hline
endarray
- Why are logics related to lattices and algebras? What can be said about this relationship? Are there any books that talk about this relationship (and the table above) in detail?
- How can the $2$nd-to-last row in the table above be filled (ie. for classical first-order logic)?
- Is there a mathematical object that contains all classical logics of all degrees, all at once? Kinda like the exterior algebra of a vector space contains all exterior powers of all degrees, all at once, together with a meaningful way in which the elements of different degrees interact (like graded-commutativity and the exterior product)?
logic model-theory
edited Jul 26 at 7:14
Taroccoesbrocco
3,38941331
asked Jul 26 at 1:21
étale-cohomology
9151617
closed as too broad by Rob Arthan, max_zorn, Brian Borchers, Lord Shark the Unknown, Mostafa Ayaz Jul 28 at 10:45
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
4
You might look into algebraic logic, and in particular cylindric algebras for first-order logic.
– Noah Schweber
Jul 26 at 4:19
2
Well logic naturally deals with the consequence relation, and when restrcited to pairs of formulas, you naturally get an order on formulas, and there's often a $land$ and an $lor$; which might explain the connection with lattices (an orders more generally)
– Max
Jul 26 at 12:15
Your question is far too broad. An answer would require an essay. What work have you done for yourself to answer your question? E.g., on understanding how topological spaces give rise to Heyting algebras. Just saying something seems "mysterious" is no evidence of any serious attempt to understand it.
– Rob Arthan
Jul 27 at 23:12
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
To me it seems very mysterious (and unexpected) that (classical propositional) logic can be "lifted" to powersets and Boolean algebras, in the sense that elements of a powerset behave like propositions (and logical connectives like or and and are "lifted" to set-theoretic union and intersection) and they satisfy the axioms of a Boolean algebra.
And even more mysterious is the observation that intuitionistic propositional logic is the logic of open sets of a topological space and of Heyting algebras.
And then quantum propositional logic seems to be the logic of orthocomplemented lattices of closed subspaces of a Hilbert space (and what algebra?).
And then the modal logic S4 seems to be the logic of interior algebras (and what space?).
And, along this line of thought, David Ellerman has some really interesting stuff about the logic of partitions (of a set), which is dual to the logic of subsets (of a set) and which seems to be related to the fact that subsets are dual to quotient sets (and then he relates partition logic to information theory in the same way that subset logic is related to probability theory).
So logics seem to be related (or, equivalent, even) to various objects in various branches of math. Eg.
beginarrayc
textclassical propositional logic & textsubsets of a set & textBoolean algebra \ hline
textintuitionistic propositional logic & textopen sets of a topological space & textHeyting algebra \ hline
textquantum (propositional?) logic & textclosed subspaces of a Hilbert space & text? \ hline
textS4 & text? & textinterior algebra \ hline
textpartition (propositional?) logic & textpartitions of a set & text? \ hline
textclassical first-order logic & text? & text? \ hline
textclassical $n$th-order logic, for all $n$ & text? & text? \ hline
endarray
- Why are logics related to lattices and algebras? What can be said about this relationship? Are there any books that talk about this relationship (and the table above) in detail?
- How can the $2$nd-to-last row in the table above be filled (ie. for classical first-order logic)?
- Is there a mathematical object that contains all classical logics of all degrees, all at once? Kinda like the exterior algebra of a vector space contains all exterior powers of all degrees, all at once, together with a meaningful way in which the elements of different degrees interact (like graded-commutativity and the exterior product)?
logic model-theory
edited Jul 26 at 7:14
Taroccoesbrocco
3,38941331
asked Jul 26 at 1:21
étale-cohomology
9151617
To me it seems very mysterious (and unexpected) that (classical propositional) logic can be "lifted" to powersets and Boolean algebras, in the sense that elements of a powerset behave like propositions (and logical connectives like or and and are "lifted" to set-theoretic union and intersection) and they satisfy the axioms of a Boolean algebra.
And even more mysterious is the observation that intuitionistic propositional logic is the logic of open sets of a topological space and of Heyting algebras.
And then quantum propositional logic seems to be the logic of orthocomplemented lattices of closed subspaces of a Hilbert space (and what algebra?).
And then the modal logic S4 seems to be the logic of interior algebras (and what space?).
And, along this line of thought, David Ellerman has some really interesting stuff about the logic of partitions (of a set), which is dual to the logic of subsets (of a set) and which seems to be related to the fact that subsets are dual to quotient sets (and then he relates partition logic to information theory in the same way that subset logic is related to probability theory).
So logics seem to be related (or, equivalent, even) to various objects in various branches of math. Eg.
beginarrayc
textclassical propositional logic & textsubsets of a set & textBoolean algebra \ hline
textintuitionistic propositional logic & textopen sets of a topological space & textHeyting algebra \ hline
textquantum (propositional?) logic & textclosed subspaces of a Hilbert space & text? \ hline
textS4 & text? & textinterior algebra \ hline
textpartition (propositional?) logic & textpartitions of a set & text? \ hline
textclassical first-order logic & text? & text? \ hline
textclassical $n$th-order logic, for all $n$ & text? & text? \ hline
endarray
- Why are logics related to lattices and algebras? What can be said about this relationship? Are there any books that talk about this relationship (and the table above) in detail?
- How can the $2$nd-to-last row in the table above be filled (ie. for classical first-order logic)?
- Is there a mathematical object that contains all classical logics of all degrees, all at once? Kinda like the exterior algebra of a vector space contains all exterior powers of all degrees, all at once, together with a meaningful way in which the elements of different degrees interact (like graded-commutativity and the exterior product)?
logic model-theory
edited Jul 26 at 7:14
Taroccoesbrocco
3,38941331
asked Jul 26 at 1:21
étale-cohomology
9151617
edited Jul 26 at 7:14
Taroccoesbrocco
3,38941331
edited Jul 26 at 7:14
Taroccoesbrocco
3,38941331
edited Jul 26 at 7:14
Taroccoesbrocco
3,38941331
3,38941331
asked Jul 26 at 1:21
étale-cohomology
9151617
asked Jul 26 at 1:21
étale-cohomology
9151617
asked Jul 26 at 1:21
étale-cohomology
9151617
9151617
closed as too broad by Rob Arthan, max_zorn, Brian Borchers, Lord Shark the Unknown, Mostafa Ayaz Jul 28 at 10:45
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as too broad by Rob Arthan, max_zorn, Brian Borchers, Lord Shark the Unknown, Mostafa Ayaz Jul 28 at 10:45
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
4
You might look into algebraic logic, and in particular cylindric algebras for first-order logic.
– Noah Schweber
Jul 26 at 4:19
2
Well logic naturally deals with the consequence relation, and when restrcited to pairs of formulas, you naturally get an order on formulas, and there's often a $land$ and an $lor$; which might explain the connection with lattices (an orders more generally)
– Max
Jul 26 at 12:15
Your question is far too broad. An answer would require an essay. What work have you done for yourself to answer your question? E.g., on understanding how topological spaces give rise to Heyting algebras. Just saying something seems "mysterious" is no evidence of any serious attempt to understand it.
– Rob Arthan
Jul 27 at 23:12
add a comment |Â
4
You might look into algebraic logic, and in particular cylindric algebras for first-order logic.
– Noah Schweber
Jul 26 at 4:19
2
Well logic naturally deals with the consequence relation, and when restrcited to pairs of formulas, you naturally get an order on formulas, and there's often a $land$ and an $lor$; which might explain the connection with lattices (an orders more generally)
– Max
Jul 26 at 12:15
Your question is far too broad. An answer would require an essay. What work have you done for yourself to answer your question? E.g., on understanding how topological spaces give rise to Heyting algebras. Just saying something seems "mysterious" is no evidence of any serious attempt to understand it.
– Rob Arthan
Jul 27 at 23:12
4
4
You might look into algebraic logic, and in particular cylindric algebras for first-order logic.
– Noah Schweber
Jul 26 at 4:19
You might look into algebraic logic, and in particular cylindric algebras for first-order logic.
– Noah Schweber
Jul 26 at 4:19
2
2
Well logic naturally deals with the consequence relation, and when restrcited to pairs of formulas, you naturally get an order on formulas, and there's often a $land$ and an $lor$; which might explain the connection with lattices (an orders more generally)
– Max
Jul 26 at 12:15
Well logic naturally deals with the consequence relation, and when restrcited to pairs of formulas, you naturally get an order on formulas, and there's often a $land$ and an $lor$; which might explain the connection with lattices (an orders more generally)
– Max
Jul 26 at 12:15
Your question is far too broad. An answer would require an essay. What work have you done for yourself to answer your question? E.g., on understanding how topological spaces give rise to Heyting algebras. Just saying something seems "mysterious" is no evidence of any serious attempt to understand it.
– Rob Arthan
Jul 27 at 23:12
Your question is far too broad. An answer would require an essay. What work have you done for yourself to answer your question? E.g., on understanding how topological spaces give rise to Heyting algebras. Just saying something seems "mysterious" is no evidence of any serious attempt to understand it.
– Rob Arthan
Jul 27 at 23:12
add a comment |Â
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4
You might look into algebraic logic, and in particular cylindric algebras for first-order logic.
– Noah Schweber
Jul 26 at 4:19
2
Well logic naturally deals with the consequence relation, and when restrcited to pairs of formulas, you naturally get an order on formulas, and there's often a $land$ and an $lor$; which might explain the connection with lattices (an orders more generally)
– Max
Jul 26 at 12:15
Your question is far too broad. An answer would require an essay. What work have you done for yourself to answer your question? E.g., on understanding how topological spaces give rise to Heyting algebras. Just saying something seems "mysterious" is no evidence of any serious attempt to understand it.
– Rob Arthan
Jul 27 at 23:12