Logic Mistake in Mathematical Logic by Tourlakis
In Tourlakis' Mathematical Logic, he claims that $models A $ if and only if $emptyset models A$. This question is on page 36. The first statement implies the second is correct but the converse is incorrect.
A counterexample would be any contradiction say $A land neg A$.
Am I missing something ?
logic propositional-calculus
asked Dec 1 at 17:42
Nameless
426612
|
show 1 more comment
In Tourlakis' Mathematical Logic, he claims that $models A $ if and only if $emptyset models A$. This question is on page 36. The first statement implies the second is correct but the converse is incorrect.
A counterexample would be any contradiction say $A land neg A$.
Am I missing something ?
logic propositional-calculus
asked Dec 1 at 17:42
Nameless
426612
What's the definition of $vDash A$ given in the book? I'm only familiar with the symbol as a relation.
– jgon
Dec 1 at 17:47
1st Def) If for every state $A$ is true then $models A$($A$ is a tautology). 2nd Def) If the state is true for every element in $Gamma$ then $A$ is also true($Gamma$ tautologically implies $A$).
– Nameless
Dec 1 at 17:52
I don't understand the first definition you wrote. Can you reword it? To me, $models A$ is simply an abbreviation of $varnothing models A$, so they mean the same by definition.
– Git Gud
Dec 1 at 18:44
2
Your counter example doesn't work. Let $varphi$ be a contradiction. By definition $varnothingmodels varphi$ means that for every valuation $v$, it holds that $forall psiin varnothingleft(v(psi)=Tright)implies v(varphi)=T$. The antecedent holds vacuously, but the consequent doesn't.
– Git Gud
Dec 1 at 19:10
1
See the post Problems with using validity symbol ⊨ “vacuously” as well as the post The logical consequence of an empty set of premises.
– Mauro ALLEGRANZA
Dec 1 at 20:19
|
show 1 more comment
In Tourlakis' Mathematical Logic, he claims that $models A $ if and only if $emptyset models A$. This question is on page 36. The first statement implies the second is correct but the converse is incorrect.
A counterexample would be any contradiction say $A land neg A$.
Am I missing something ?
logic propositional-calculus
asked Dec 1 at 17:42
Nameless
426612
In Tourlakis' Mathematical Logic, he claims that $models A $ if and only if $emptyset models A$. This question is on page 36. The first statement implies the second is correct but the converse is incorrect.
A counterexample would be any contradiction say $A land neg A$.
Am I missing something ?
logic propositional-calculus
logic propositional-calculus
asked Dec 1 at 17:42
Nameless
426612
asked Dec 1 at 17:42
Nameless
426612
asked Dec 1 at 17:42
Nameless
426612
asked Dec 1 at 17:42
Nameless
426612
asked Dec 1 at 17:42
Nameless
426612
426612
What's the definition of $vDash A$ given in the book? I'm only familiar with the symbol as a relation.
– jgon
Dec 1 at 17:47
1st Def) If for every state $A$ is true then $models A$($A$ is a tautology). 2nd Def) If the state is true for every element in $Gamma$ then $A$ is also true($Gamma$ tautologically implies $A$).
– Nameless
Dec 1 at 17:52
I don't understand the first definition you wrote. Can you reword it? To me, $models A$ is simply an abbreviation of $varnothing models A$, so they mean the same by definition.
– Git Gud
Dec 1 at 18:44
2
Your counter example doesn't work. Let $varphi$ be a contradiction. By definition $varnothingmodels varphi$ means that for every valuation $v$, it holds that $forall psiin varnothingleft(v(psi)=Tright)implies v(varphi)=T$. The antecedent holds vacuously, but the consequent doesn't.
– Git Gud
Dec 1 at 19:10
1
See the post Problems with using validity symbol ⊨ “vacuously” as well as the post The logical consequence of an empty set of premises.
– Mauro ALLEGRANZA
Dec 1 at 20:19
|
show 1 more comment
What's the definition of $vDash A$ given in the book? I'm only familiar with the symbol as a relation.
– jgon
Dec 1 at 17:47
1st Def) If for every state $A$ is true then $models A$($A$ is a tautology). 2nd Def) If the state is true for every element in $Gamma$ then $A$ is also true($Gamma$ tautologically implies $A$).
– Nameless
Dec 1 at 17:52
I don't understand the first definition you wrote. Can you reword it? To me, $models A$ is simply an abbreviation of $varnothing models A$, so they mean the same by definition.
– Git Gud
Dec 1 at 18:44
2
Your counter example doesn't work. Let $varphi$ be a contradiction. By definition $varnothingmodels varphi$ means that for every valuation $v$, it holds that $forall psiin varnothingleft(v(psi)=Tright)implies v(varphi)=T$. The antecedent holds vacuously, but the consequent doesn't.
– Git Gud
Dec 1 at 19:10
1
See the post Problems with using validity symbol ⊨ “vacuously” as well as the post The logical consequence of an empty set of premises.
– Mauro ALLEGRANZA
Dec 1 at 20:19
What's the definition of $vDash A$ given in the book? I'm only familiar with the symbol as a relation.
– jgon
Dec 1 at 17:47
What's the definition of $vDash A$ given in the book? I'm only familiar with the symbol as a relation.
– jgon
Dec 1 at 17:47
1st Def) If for every state $A$ is true then $models A$($A$ is a tautology). 2nd Def) If the state is true for every element in $Gamma$ then $A$ is also true($Gamma$ tautologically implies $A$).
– Nameless
Dec 1 at 17:52
1st Def) If for every state $A$ is true then $models A$($A$ is a tautology). 2nd Def) If the state is true for every element in $Gamma$ then $A$ is also true($Gamma$ tautologically implies $A$).
– Nameless
Dec 1 at 17:52
I don't understand the first definition you wrote. Can you reword it? To me, $models A$ is simply an abbreviation of $varnothing models A$, so they mean the same by definition.
– Git Gud
Dec 1 at 18:44
I don't understand the first definition you wrote. Can you reword it? To me, $models A$ is simply an abbreviation of $varnothing models A$, so they mean the same by definition.
– Git Gud
Dec 1 at 18:44
2
2
Your counter example doesn't work. Let $varphi$ be a contradiction. By definition $varnothingmodels varphi$ means that for every valuation $v$, it holds that $forall psiin varnothingleft(v(psi)=Tright)implies v(varphi)=T$. The antecedent holds vacuously, but the consequent doesn't.
– Git Gud
Dec 1 at 19:10
Your counter example doesn't work. Let $varphi$ be a contradiction. By definition $varnothingmodels varphi$ means that for every valuation $v$, it holds that $forall psiin varnothingleft(v(psi)=Tright)implies v(varphi)=T$. The antecedent holds vacuously, but the consequent doesn't.
– Git Gud
Dec 1 at 19:10
1
1
See the post Problems with using validity symbol ⊨ “vacuously” as well as the post The logical consequence of an empty set of premises.
– Mauro ALLEGRANZA
Dec 1 at 20:19
See the post Problems with using validity symbol ⊨ “vacuously” as well as the post The logical consequence of an empty set of premises.
– Mauro ALLEGRANZA
Dec 1 at 20:19
|
show 1 more comment
1 Answer
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I think I see your error. I'm not quite sure what you mean by state, but let me try to informally translate the definitions you gave to symbols.
$models A$ means that
$$forall_{states} A,$$
whereas $Gammamodels A$ for a collection of statements $Gamma$ means
$$forall_{states}left(left(forall_{psiinGamma}psiright)implies Aright).$$
Note the careful parentheses in this second definition, since I'm fairly sure the error is one of misinterpreting the grouping of the quantifiers and symbols here. If I now put $Gamma=varnothing$, then I have $forall_{psiinGamma}psi$ becomes vacuously true, or in other words
$$varnothingmodels A$$
means that
$$forall_{states} mathrm{True}implies A,$$
or
$$forall_{states} A,$$
since $$mathrm{True}implies Atext{ if and only if }A.$$
answered Dec 1 at 18:09
jgon
12.5k21940
I believe "state" in the comment was meant to be "statement."
– Fabio Somenzi
Dec 1 at 18:33
@FabioSomenzi That was my first thought as well, but "If for every statement $A$ is true" doesn't make very much sense either.
– jgon
Dec 1 at 18:38
@FabioSomenzi - a "state" is a truth assignment; see page 26 : "Def 1.3.2. A state $v$ is a function that assigns the value $text f$ or $text t$ to each Boolean variable, while it assigns necessarily the value $text f$ to the constant $bot$ and necessarily the value $text t$ to the constant $top$."
– Mauro ALLEGRANZA
Dec 1 at 20:17
@MauroALLEGRANZA Duly noted. Thanks!
– Fabio Somenzi
Dec 2 at 0:04
add a comment |
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I think I see your error. I'm not quite sure what you mean by state, but let me try to informally translate the definitions you gave to symbols.
$models A$ means that
$$forall_{states} A,$$
whereas $Gammamodels A$ for a collection of statements $Gamma$ means
$$forall_{states}left(left(forall_{psiinGamma}psiright)implies Aright).$$
Note the careful parentheses in this second definition, since I'm fairly sure the error is one of misinterpreting the grouping of the quantifiers and symbols here. If I now put $Gamma=varnothing$, then I have $forall_{psiinGamma}psi$ becomes vacuously true, or in other words
$$varnothingmodels A$$
means that
$$forall_{states} mathrm{True}implies A,$$
or
$$forall_{states} A,$$
since $$mathrm{True}implies Atext{ if and only if }A.$$
answered Dec 1 at 18:09
jgon
12.5k21940
I believe "state" in the comment was meant to be "statement."
– Fabio Somenzi
Dec 1 at 18:33
@FabioSomenzi That was my first thought as well, but "If for every statement $A$ is true" doesn't make very much sense either.
– jgon
Dec 1 at 18:38
@FabioSomenzi - a "state" is a truth assignment; see page 26 : "Def 1.3.2. A state $v$ is a function that assigns the value $text f$ or $text t$ to each Boolean variable, while it assigns necessarily the value $text f$ to the constant $bot$ and necessarily the value $text t$ to the constant $top$."
– Mauro ALLEGRANZA
Dec 1 at 20:17
@MauroALLEGRANZA Duly noted. Thanks!
– Fabio Somenzi
Dec 2 at 0:04
add a comment |
I think I see your error. I'm not quite sure what you mean by state, but let me try to informally translate the definitions you gave to symbols.
$models A$ means that
$$forall_{states} A,$$
whereas $Gammamodels A$ for a collection of statements $Gamma$ means
$$forall_{states}left(left(forall_{psiinGamma}psiright)implies Aright).$$
Note the careful parentheses in this second definition, since I'm fairly sure the error is one of misinterpreting the grouping of the quantifiers and symbols here. If I now put $Gamma=varnothing$, then I have $forall_{psiinGamma}psi$ becomes vacuously true, or in other words
$$varnothingmodels A$$
means that
$$forall_{states} mathrm{True}implies A,$$
or
$$forall_{states} A,$$
since $$mathrm{True}implies Atext{ if and only if }A.$$
answered Dec 1 at 18:09
jgon
12.5k21940
I believe "state" in the comment was meant to be "statement."
– Fabio Somenzi
Dec 1 at 18:33
@FabioSomenzi That was my first thought as well, but "If for every statement $A$ is true" doesn't make very much sense either.
– jgon
Dec 1 at 18:38
@FabioSomenzi - a "state" is a truth assignment; see page 26 : "Def 1.3.2. A state $v$ is a function that assigns the value $text f$ or $text t$ to each Boolean variable, while it assigns necessarily the value $text f$ to the constant $bot$ and necessarily the value $text t$ to the constant $top$."
– Mauro ALLEGRANZA
Dec 1 at 20:17
@MauroALLEGRANZA Duly noted. Thanks!
– Fabio Somenzi
Dec 2 at 0:04
add a comment |
I think I see your error. I'm not quite sure what you mean by state, but let me try to informally translate the definitions you gave to symbols.
$models A$ means that
$$forall_{states} A,$$
whereas $Gammamodels A$ for a collection of statements $Gamma$ means
$$forall_{states}left(left(forall_{psiinGamma}psiright)implies Aright).$$
Note the careful parentheses in this second definition, since I'm fairly sure the error is one of misinterpreting the grouping of the quantifiers and symbols here. If I now put $Gamma=varnothing$, then I have $forall_{psiinGamma}psi$ becomes vacuously true, or in other words
$$varnothingmodels A$$
means that
$$forall_{states} mathrm{True}implies A,$$
or
$$forall_{states} A,$$
since $$mathrm{True}implies Atext{ if and only if }A.$$
answered Dec 1 at 18:09
jgon
12.5k21940
I think I see your error. I'm not quite sure what you mean by state, but let me try to informally translate the definitions you gave to symbols.
$models A$ means that
$$forall_{states} A,$$
whereas $Gammamodels A$ for a collection of statements $Gamma$ means
$$forall_{states}left(left(forall_{psiinGamma}psiright)implies Aright).$$
Note the careful parentheses in this second definition, since I'm fairly sure the error is one of misinterpreting the grouping of the quantifiers and symbols here. If I now put $Gamma=varnothing$, then I have $forall_{psiinGamma}psi$ becomes vacuously true, or in other words
$$varnothingmodels A$$
means that
$$forall_{states} mathrm{True}implies A,$$
or
$$forall_{states} A,$$
since $$mathrm{True}implies Atext{ if and only if }A.$$
answered Dec 1 at 18:09
jgon
12.5k21940
answered Dec 1 at 18:09
jgon
12.5k21940
answered Dec 1 at 18:09
jgon
12.5k21940
answered Dec 1 at 18:09
jgon
12.5k21940
12.5k21940
I believe "state" in the comment was meant to be "statement."
– Fabio Somenzi
Dec 1 at 18:33
@FabioSomenzi That was my first thought as well, but "If for every statement $A$ is true" doesn't make very much sense either.
– jgon
Dec 1 at 18:38
@FabioSomenzi - a "state" is a truth assignment; see page 26 : "Def 1.3.2. A state $v$ is a function that assigns the value $text f$ or $text t$ to each Boolean variable, while it assigns necessarily the value $text f$ to the constant $bot$ and necessarily the value $text t$ to the constant $top$."
– Mauro ALLEGRANZA
Dec 1 at 20:17
@MauroALLEGRANZA Duly noted. Thanks!
– Fabio Somenzi
Dec 2 at 0:04
add a comment |
I believe "state" in the comment was meant to be "statement."
– Fabio Somenzi
Dec 1 at 18:33
@FabioSomenzi That was my first thought as well, but "If for every statement $A$ is true" doesn't make very much sense either.
– jgon
Dec 1 at 18:38
@FabioSomenzi - a "state" is a truth assignment; see page 26 : "Def 1.3.2. A state $v$ is a function that assigns the value $text f$ or $text t$ to each Boolean variable, while it assigns necessarily the value $text f$ to the constant $bot$ and necessarily the value $text t$ to the constant $top$."
– Mauro ALLEGRANZA
Dec 1 at 20:17
@MauroALLEGRANZA Duly noted. Thanks!
– Fabio Somenzi
Dec 2 at 0:04
I believe "state" in the comment was meant to be "statement."
– Fabio Somenzi
Dec 1 at 18:33
I believe "state" in the comment was meant to be "statement."
– Fabio Somenzi
Dec 1 at 18:33
@FabioSomenzi That was my first thought as well, but "If for every statement $A$ is true" doesn't make very much sense either.
– jgon
Dec 1 at 18:38
@FabioSomenzi That was my first thought as well, but "If for every statement $A$ is true" doesn't make very much sense either.
– jgon
Dec 1 at 18:38
@FabioSomenzi - a "state" is a truth assignment; see page 26 : "Def 1.3.2. A state $v$ is a function that assigns the value $text f$ or $text t$ to each Boolean variable, while it assigns necessarily the value $text f$ to the constant $bot$ and necessarily the value $text t$ to the constant $top$."
– Mauro ALLEGRANZA
Dec 1 at 20:17
@FabioSomenzi - a "state" is a truth assignment; see page 26 : "Def 1.3.2. A state $v$ is a function that assigns the value $text f$ or $text t$ to each Boolean variable, while it assigns necessarily the value $text f$ to the constant $bot$ and necessarily the value $text t$ to the constant $top$."
– Mauro ALLEGRANZA
Dec 1 at 20:17
@MauroALLEGRANZA Duly noted. Thanks!
– Fabio Somenzi
Dec 2 at 0:04
@MauroALLEGRANZA Duly noted. Thanks!
– Fabio Somenzi
Dec 2 at 0:04
add a comment |
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What's the definition of $vDash A$ given in the book? I'm only familiar with the symbol as a relation.
– jgon
Dec 1 at 17:47
1st Def) If for every state $A$ is true then $models A$($A$ is a tautology). 2nd Def) If the state is true for every element in $Gamma$ then $A$ is also true($Gamma$ tautologically implies $A$).
– Nameless
Dec 1 at 17:52
I don't understand the first definition you wrote. Can you reword it? To me, $models A$ is simply an abbreviation of $varnothing models A$, so they mean the same by definition.
– Git Gud
Dec 1 at 18:44
2
Your counter example doesn't work. Let $varphi$ be a contradiction. By definition $varnothingmodels varphi$ means that for every valuation $v$, it holds that $forall psiin varnothingleft(v(psi)=Tright)implies v(varphi)=T$. The antecedent holds vacuously, but the consequent doesn't.
– Git Gud
Dec 1 at 19:10
1
See the post Problems with using validity symbol ⊨ “vacuously” as well as the post The logical consequence of an empty set of premises.
– Mauro ALLEGRANZA
Dec 1 at 20:19