Mixing
Difference between $models$ and $Rightarrow$
$begingroup$
What is the difference between $models$ and $Rightarrow$ in propositional logic?
logic propositional-calculus
edited Nov 19 '13 at 11:37
Adi Dani
15.3k32246
asked Nov 19 '13 at 11:34
user1090614user1090614
2471411
$endgroup$
add a comment |
$begingroup$
What is the difference between $models$ and $Rightarrow$ in propositional logic?
logic propositional-calculus
edited Nov 19 '13 at 11:37
Adi Dani
15.3k32246
asked Nov 19 '13 at 11:34
user1090614user1090614
2471411
$endgroup$
1
$begingroup$
It depends on what you mean by $Rightarrow$. Some authors essentially take it to mean $models$, others take it to mean material implication. Can you say which particular book you are using?
$endgroup$
– Carl Mummert
Nov 19 '13 at 11:38
1
$begingroup$
Dont know if it helps, I am reading "Modeling and reasoning with bayesian networks". I have always believed they were the same, but I am beginning to think that the author does not. What is the potential difference?
$endgroup$
– user1090614
Nov 19 '13 at 11:40
add a comment |
$begingroup$
What is the difference between $models$ and $Rightarrow$ in propositional logic?
logic propositional-calculus
edited Nov 19 '13 at 11:37
Adi Dani
15.3k32246
asked Nov 19 '13 at 11:34
user1090614user1090614
2471411
$endgroup$
What is the difference between $models$ and $Rightarrow$ in propositional logic?
logic propositional-calculus
logic propositional-calculus
edited Nov 19 '13 at 11:37
Adi Dani
15.3k32246
asked Nov 19 '13 at 11:34
user1090614user1090614
2471411
edited Nov 19 '13 at 11:37
Adi Dani
15.3k32246
asked Nov 19 '13 at 11:34
user1090614user1090614
2471411
edited Nov 19 '13 at 11:37
Adi Dani
15.3k32246
edited Nov 19 '13 at 11:37
Adi Dani
15.3k32246
edited Nov 19 '13 at 11:37
Adi Dani
15.3k32246
15.3k32246
asked Nov 19 '13 at 11:34
user1090614user1090614
2471411
asked Nov 19 '13 at 11:34
user1090614user1090614
2471411
asked Nov 19 '13 at 11:34
user1090614user1090614
2471411
2471411
1
$begingroup$
It depends on what you mean by $Rightarrow$. Some authors essentially take it to mean $models$, others take it to mean material implication. Can you say which particular book you are using?
$endgroup$
– Carl Mummert
Nov 19 '13 at 11:38
1
$begingroup$
Dont know if it helps, I am reading "Modeling and reasoning with bayesian networks". I have always believed they were the same, but I am beginning to think that the author does not. What is the potential difference?
$endgroup$
– user1090614
Nov 19 '13 at 11:40
add a comment |
1
$begingroup$
It depends on what you mean by $Rightarrow$. Some authors essentially take it to mean $models$, others take it to mean material implication. Can you say which particular book you are using?
$endgroup$
– Carl Mummert
Nov 19 '13 at 11:38
1
$begingroup$
Dont know if it helps, I am reading "Modeling and reasoning with bayesian networks". I have always believed they were the same, but I am beginning to think that the author does not. What is the potential difference?
$endgroup$
– user1090614
Nov 19 '13 at 11:40
1
1
$begingroup$
It depends on what you mean by $Rightarrow$. Some authors essentially take it to mean $models$, others take it to mean material implication. Can you say which particular book you are using?
$endgroup$
– Carl Mummert
Nov 19 '13 at 11:38
$begingroup$
It depends on what you mean by $Rightarrow$. Some authors essentially take it to mean $models$, others take it to mean material implication. Can you say which particular book you are using?
$endgroup$
– Carl Mummert
Nov 19 '13 at 11:38
1
1
$begingroup$
Dont know if it helps, I am reading "Modeling and reasoning with bayesian networks". I have always believed they were the same, but I am beginning to think that the author does not. What is the potential difference?
$endgroup$
– user1090614
Nov 19 '13 at 11:40
$begingroup$
Dont know if it helps, I am reading "Modeling and reasoning with bayesian networks". I have always believed they were the same, but I am beginning to think that the author does not. What is the potential difference?
$endgroup$
– user1090614
Nov 19 '13 at 11:40
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
In the book "Modeling and reasoning with Bayesian networks", page 14, $Rightarrow$ is defined to be the material implication connective. Some other authors write this as $rightarrow$.
Other authors write $Rightarrow$ as a synonym for $models$, which is a connective in the metatheory that indicates logical implication.
Both conventions appear often enough that you have to look at each author's conventions individually.
answered Nov 19 '13 at 11:44
Carl MummertCarl Mummert
67.5k7133251
$endgroup$
$begingroup$
@user1090614: I agree with Carl, although I've often seen $models$ be used for semantic entailment, i.e., $(Amodels B)Rightarrow$ "A and B mean the same thing" while the $Rightarrow$ has been used in the sense of formally implies.
$endgroup$
– user76844
Nov 19 '13 at 14:39
$begingroup$
A connective connects similar kinds of objects. $models$ doesn't do this, and consequently is not a connective.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 14:56
$begingroup$
@Doug Spoonwood: it is common in the metatheory to write $phi models psi$, where $phi$ and $psi$ are both formulas of the same logic. In that case $models$ looks like a connective to me.
$endgroup$
– Carl Mummert
Nov 19 '13 at 14:58
$begingroup$
@CarlMummert In that case I agree once we have parentheses expressed. However, it is also common enough to write things like (p⇒q), (q⇒r) $models$ (p⇒r), where the left hand side gets understood as a set of propositions. A set of propositions is not the same sort of object as a proposition. So, in that case it doesn't look like a connective to me.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 15:09
2
$begingroup$
$models$ gets used in a number of ways. We may write: $phimodelspsi$ with formulae; $Phimodelspsi$ with a set of formulae and a formula; ${cal M}modelsphi$ with a semantic model (whatever that might be in context) and a formula. I think that the last the most general, since the others can be expressed in terms of it. E.g., $Phimodelspsi$ means that each ${cal M}$ such that ${cal M}modelsphi$ for every $phiinPhi$ is such that ${cal M}modelspsi$. $phimodelspsi$ means ${phi}modelspsi$.
$endgroup$
– Joshua Taylor
Nov 19 '13 at 17:12
|
show 3 more comments
$begingroup$
⇒ is a logical connective in the object language. It connects two propositions. It has a specific truth table and/or a characterization by a set of axioms (or axiom schema) under a rule of inference(s).
$models$ happens in the metalanguage and usually refers only to semantic entailment. It doesn't connect individual propositions, but rather relates a proposition or a set of propositions to another proposition or set of propositions. It doesn't have a truth table. With $models$ we can write things likes
$models$ (p⇒q).
p $models$ q.
{p, q, r} $models$ (p⇒(q⇒r)).
There is no meaningful statement like {p, q, r} ⇒ (p⇒(q⇒r)), because ⇒ only relates particular propositions, not sets of propositions.
answered Nov 19 '13 at 15:06
Doug SpoonwoodDoug Spoonwood
8,13212244
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
In the book "Modeling and reasoning with Bayesian networks", page 14, $Rightarrow$ is defined to be the material implication connective. Some other authors write this as $rightarrow$.
Other authors write $Rightarrow$ as a synonym for $models$, which is a connective in the metatheory that indicates logical implication.
Both conventions appear often enough that you have to look at each author's conventions individually.
answered Nov 19 '13 at 11:44
Carl MummertCarl Mummert
67.5k7133251
$endgroup$
$begingroup$
@user1090614: I agree with Carl, although I've often seen $models$ be used for semantic entailment, i.e., $(Amodels B)Rightarrow$ "A and B mean the same thing" while the $Rightarrow$ has been used in the sense of formally implies.
$endgroup$
– user76844
Nov 19 '13 at 14:39
$begingroup$
A connective connects similar kinds of objects. $models$ doesn't do this, and consequently is not a connective.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 14:56
$begingroup$
@Doug Spoonwood: it is common in the metatheory to write $phi models psi$, where $phi$ and $psi$ are both formulas of the same logic. In that case $models$ looks like a connective to me.
$endgroup$
– Carl Mummert
Nov 19 '13 at 14:58
$begingroup$
@CarlMummert In that case I agree once we have parentheses expressed. However, it is also common enough to write things like (p⇒q), (q⇒r) $models$ (p⇒r), where the left hand side gets understood as a set of propositions. A set of propositions is not the same sort of object as a proposition. So, in that case it doesn't look like a connective to me.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 15:09
2
$begingroup$
$models$ gets used in a number of ways. We may write: $phimodelspsi$ with formulae; $Phimodelspsi$ with a set of formulae and a formula; ${cal M}modelsphi$ with a semantic model (whatever that might be in context) and a formula. I think that the last the most general, since the others can be expressed in terms of it. E.g., $Phimodelspsi$ means that each ${cal M}$ such that ${cal M}modelsphi$ for every $phiinPhi$ is such that ${cal M}modelspsi$. $phimodelspsi$ means ${phi}modelspsi$.
$endgroup$
– Joshua Taylor
Nov 19 '13 at 17:12
|
show 3 more comments
$begingroup$
In the book "Modeling and reasoning with Bayesian networks", page 14, $Rightarrow$ is defined to be the material implication connective. Some other authors write this as $rightarrow$.
Other authors write $Rightarrow$ as a synonym for $models$, which is a connective in the metatheory that indicates logical implication.
Both conventions appear often enough that you have to look at each author's conventions individually.
answered Nov 19 '13 at 11:44
Carl MummertCarl Mummert
67.5k7133251
$endgroup$
$begingroup$
@user1090614: I agree with Carl, although I've often seen $models$ be used for semantic entailment, i.e., $(Amodels B)Rightarrow$ "A and B mean the same thing" while the $Rightarrow$ has been used in the sense of formally implies.
$endgroup$
– user76844
Nov 19 '13 at 14:39
$begingroup$
A connective connects similar kinds of objects. $models$ doesn't do this, and consequently is not a connective.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 14:56
$begingroup$
@Doug Spoonwood: it is common in the metatheory to write $phi models psi$, where $phi$ and $psi$ are both formulas of the same logic. In that case $models$ looks like a connective to me.
$endgroup$
– Carl Mummert
Nov 19 '13 at 14:58
$begingroup$
@CarlMummert In that case I agree once we have parentheses expressed. However, it is also common enough to write things like (p⇒q), (q⇒r) $models$ (p⇒r), where the left hand side gets understood as a set of propositions. A set of propositions is not the same sort of object as a proposition. So, in that case it doesn't look like a connective to me.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 15:09
2
$begingroup$
$models$ gets used in a number of ways. We may write: $phimodelspsi$ with formulae; $Phimodelspsi$ with a set of formulae and a formula; ${cal M}modelsphi$ with a semantic model (whatever that might be in context) and a formula. I think that the last the most general, since the others can be expressed in terms of it. E.g., $Phimodelspsi$ means that each ${cal M}$ such that ${cal M}modelsphi$ for every $phiinPhi$ is such that ${cal M}modelspsi$. $phimodelspsi$ means ${phi}modelspsi$.
$endgroup$
– Joshua Taylor
Nov 19 '13 at 17:12
|
show 3 more comments
$begingroup$
In the book "Modeling and reasoning with Bayesian networks", page 14, $Rightarrow$ is defined to be the material implication connective. Some other authors write this as $rightarrow$.
Other authors write $Rightarrow$ as a synonym for $models$, which is a connective in the metatheory that indicates logical implication.
Both conventions appear often enough that you have to look at each author's conventions individually.
answered Nov 19 '13 at 11:44
Carl MummertCarl Mummert
67.5k7133251
$endgroup$
In the book "Modeling and reasoning with Bayesian networks", page 14, $Rightarrow$ is defined to be the material implication connective. Some other authors write this as $rightarrow$.
Other authors write $Rightarrow$ as a synonym for $models$, which is a connective in the metatheory that indicates logical implication.
Both conventions appear often enough that you have to look at each author's conventions individually.
answered Nov 19 '13 at 11:44
Carl MummertCarl Mummert
67.5k7133251
answered Nov 19 '13 at 11:44
Carl MummertCarl Mummert
67.5k7133251
answered Nov 19 '13 at 11:44
Carl MummertCarl Mummert
67.5k7133251
answered Nov 19 '13 at 11:44
Carl MummertCarl Mummert
67.5k7133251
67.5k7133251
$begingroup$
@user1090614: I agree with Carl, although I've often seen $models$ be used for semantic entailment, i.e., $(Amodels B)Rightarrow$ "A and B mean the same thing" while the $Rightarrow$ has been used in the sense of formally implies.
$endgroup$
– user76844
Nov 19 '13 at 14:39
$begingroup$
A connective connects similar kinds of objects. $models$ doesn't do this, and consequently is not a connective.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 14:56
$begingroup$
@Doug Spoonwood: it is common in the metatheory to write $phi models psi$, where $phi$ and $psi$ are both formulas of the same logic. In that case $models$ looks like a connective to me.
$endgroup$
– Carl Mummert
Nov 19 '13 at 14:58
$begingroup$
@CarlMummert In that case I agree once we have parentheses expressed. However, it is also common enough to write things like (p⇒q), (q⇒r) $models$ (p⇒r), where the left hand side gets understood as a set of propositions. A set of propositions is not the same sort of object as a proposition. So, in that case it doesn't look like a connective to me.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 15:09
2
$begingroup$
$models$ gets used in a number of ways. We may write: $phimodelspsi$ with formulae; $Phimodelspsi$ with a set of formulae and a formula; ${cal M}modelsphi$ with a semantic model (whatever that might be in context) and a formula. I think that the last the most general, since the others can be expressed in terms of it. E.g., $Phimodelspsi$ means that each ${cal M}$ such that ${cal M}modelsphi$ for every $phiinPhi$ is such that ${cal M}modelspsi$. $phimodelspsi$ means ${phi}modelspsi$.
$endgroup$
– Joshua Taylor
Nov 19 '13 at 17:12
|
show 3 more comments
$begingroup$
@user1090614: I agree with Carl, although I've often seen $models$ be used for semantic entailment, i.e., $(Amodels B)Rightarrow$ "A and B mean the same thing" while the $Rightarrow$ has been used in the sense of formally implies.
$endgroup$
– user76844
Nov 19 '13 at 14:39
$begingroup$
A connective connects similar kinds of objects. $models$ doesn't do this, and consequently is not a connective.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 14:56
$begingroup$
@Doug Spoonwood: it is common in the metatheory to write $phi models psi$, where $phi$ and $psi$ are both formulas of the same logic. In that case $models$ looks like a connective to me.
$endgroup$
– Carl Mummert
Nov 19 '13 at 14:58
$begingroup$
@CarlMummert In that case I agree once we have parentheses expressed. However, it is also common enough to write things like (p⇒q), (q⇒r) $models$ (p⇒r), where the left hand side gets understood as a set of propositions. A set of propositions is not the same sort of object as a proposition. So, in that case it doesn't look like a connective to me.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 15:09
2
$begingroup$
$models$ gets used in a number of ways. We may write: $phimodelspsi$ with formulae; $Phimodelspsi$ with a set of formulae and a formula; ${cal M}modelsphi$ with a semantic model (whatever that might be in context) and a formula. I think that the last the most general, since the others can be expressed in terms of it. E.g., $Phimodelspsi$ means that each ${cal M}$ such that ${cal M}modelsphi$ for every $phiinPhi$ is such that ${cal M}modelspsi$. $phimodelspsi$ means ${phi}modelspsi$.
$endgroup$
– Joshua Taylor
Nov 19 '13 at 17:12
$begingroup$
@user1090614: I agree with Carl, although I've often seen $models$ be used for semantic entailment, i.e., $(Amodels B)Rightarrow$ "A and B mean the same thing" while the $Rightarrow$ has been used in the sense of formally implies.
$endgroup$
– user76844
Nov 19 '13 at 14:39
$begingroup$
@user1090614: I agree with Carl, although I've often seen $models$ be used for semantic entailment, i.e., $(Amodels B)Rightarrow$ "A and B mean the same thing" while the $Rightarrow$ has been used in the sense of formally implies.
$endgroup$
– user76844
Nov 19 '13 at 14:39
$begingroup$
A connective connects similar kinds of objects. $models$ doesn't do this, and consequently is not a connective.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 14:56
$begingroup$
A connective connects similar kinds of objects. $models$ doesn't do this, and consequently is not a connective.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 14:56
$begingroup$
@Doug Spoonwood: it is common in the metatheory to write $phi models psi$, where $phi$ and $psi$ are both formulas of the same logic. In that case $models$ looks like a connective to me.
$endgroup$
– Carl Mummert
Nov 19 '13 at 14:58
$begingroup$
@Doug Spoonwood: it is common in the metatheory to write $phi models psi$, where $phi$ and $psi$ are both formulas of the same logic. In that case $models$ looks like a connective to me.
$endgroup$
– Carl Mummert
Nov 19 '13 at 14:58
$begingroup$
@CarlMummert In that case I agree once we have parentheses expressed. However, it is also common enough to write things like (p⇒q), (q⇒r) $models$ (p⇒r), where the left hand side gets understood as a set of propositions. A set of propositions is not the same sort of object as a proposition. So, in that case it doesn't look like a connective to me.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 15:09
$begingroup$
@CarlMummert In that case I agree once we have parentheses expressed. However, it is also common enough to write things like (p⇒q), (q⇒r) $models$ (p⇒r), where the left hand side gets understood as a set of propositions. A set of propositions is not the same sort of object as a proposition. So, in that case it doesn't look like a connective to me.
$endgroup$
– Doug Spoonwood
Nov 19 '13 at 15:09
2
2
$begingroup$
$models$ gets used in a number of ways. We may write: $phimodelspsi$ with formulae; $Phimodelspsi$ with a set of formulae and a formula; ${cal M}modelsphi$ with a semantic model (whatever that might be in context) and a formula. I think that the last the most general, since the others can be expressed in terms of it. E.g., $Phimodelspsi$ means that each ${cal M}$ such that ${cal M}modelsphi$ for every $phiinPhi$ is such that ${cal M}modelspsi$. $phimodelspsi$ means ${phi}modelspsi$.
$endgroup$
– Joshua Taylor
Nov 19 '13 at 17:12
$begingroup$
$models$ gets used in a number of ways. We may write: $phimodelspsi$ with formulae; $Phimodelspsi$ with a set of formulae and a formula; ${cal M}modelsphi$ with a semantic model (whatever that might be in context) and a formula. I think that the last the most general, since the others can be expressed in terms of it. E.g., $Phimodelspsi$ means that each ${cal M}$ such that ${cal M}modelsphi$ for every $phiinPhi$ is such that ${cal M}modelspsi$. $phimodelspsi$ means ${phi}modelspsi$.
$endgroup$
– Joshua Taylor
Nov 19 '13 at 17:12
|
show 3 more comments
$begingroup$
⇒ is a logical connective in the object language. It connects two propositions. It has a specific truth table and/or a characterization by a set of axioms (or axiom schema) under a rule of inference(s).
$models$ happens in the metalanguage and usually refers only to semantic entailment. It doesn't connect individual propositions, but rather relates a proposition or a set of propositions to another proposition or set of propositions. It doesn't have a truth table. With $models$ we can write things likes
$models$ (p⇒q).
p $models$ q.
{p, q, r} $models$ (p⇒(q⇒r)).
There is no meaningful statement like {p, q, r} ⇒ (p⇒(q⇒r)), because ⇒ only relates particular propositions, not sets of propositions.
answered Nov 19 '13 at 15:06
Doug SpoonwoodDoug Spoonwood
8,13212244
$endgroup$
add a comment |
$begingroup$
⇒ is a logical connective in the object language. It connects two propositions. It has a specific truth table and/or a characterization by a set of axioms (or axiom schema) under a rule of inference(s).
$models$ happens in the metalanguage and usually refers only to semantic entailment. It doesn't connect individual propositions, but rather relates a proposition or a set of propositions to another proposition or set of propositions. It doesn't have a truth table. With $models$ we can write things likes
$models$ (p⇒q).
p $models$ q.
{p, q, r} $models$ (p⇒(q⇒r)).
There is no meaningful statement like {p, q, r} ⇒ (p⇒(q⇒r)), because ⇒ only relates particular propositions, not sets of propositions.
answered Nov 19 '13 at 15:06
Doug SpoonwoodDoug Spoonwood
8,13212244
$endgroup$
add a comment |
$begingroup$
⇒ is a logical connective in the object language. It connects two propositions. It has a specific truth table and/or a characterization by a set of axioms (or axiom schema) under a rule of inference(s).
$models$ happens in the metalanguage and usually refers only to semantic entailment. It doesn't connect individual propositions, but rather relates a proposition or a set of propositions to another proposition or set of propositions. It doesn't have a truth table. With $models$ we can write things likes
$models$ (p⇒q).
p $models$ q.
{p, q, r} $models$ (p⇒(q⇒r)).
There is no meaningful statement like {p, q, r} ⇒ (p⇒(q⇒r)), because ⇒ only relates particular propositions, not sets of propositions.
answered Nov 19 '13 at 15:06
Doug SpoonwoodDoug Spoonwood
8,13212244
$endgroup$
⇒ is a logical connective in the object language. It connects two propositions. It has a specific truth table and/or a characterization by a set of axioms (or axiom schema) under a rule of inference(s).
$models$ happens in the metalanguage and usually refers only to semantic entailment. It doesn't connect individual propositions, but rather relates a proposition or a set of propositions to another proposition or set of propositions. It doesn't have a truth table. With $models$ we can write things likes
$models$ (p⇒q).
p $models$ q.
{p, q, r} $models$ (p⇒(q⇒r)).
There is no meaningful statement like {p, q, r} ⇒ (p⇒(q⇒r)), because ⇒ only relates particular propositions, not sets of propositions.
answered Nov 19 '13 at 15:06
Doug SpoonwoodDoug Spoonwood
8,13212244
edited Nov 19 '13 at 15:30
answered Nov 19 '13 at 15:06
Doug SpoonwoodDoug Spoonwood
8,13212244
answered Nov 19 '13 at 15:06
Doug SpoonwoodDoug Spoonwood
8,13212244
answered Nov 19 '13 at 15:06
Doug SpoonwoodDoug Spoonwood
8,13212244
8,13212244
add a comment |
add a comment |
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1
$begingroup$
It depends on what you mean by $Rightarrow$. Some authors essentially take it to mean $models$, others take it to mean material implication. Can you say which particular book you are using?
$endgroup$
– Carl Mummert
Nov 19 '13 at 11:38
1
$begingroup$
Dont know if it helps, I am reading "Modeling and reasoning with bayesian networks". I have always believed they were the same, but I am beginning to think that the author does not. What is the potential difference?
$endgroup$
– user1090614
Nov 19 '13 at 11:40