What is the difference between validity and satisfiability?
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As the title says, What is the difference between validity and satisfiability?
Suppose I have a sentence If the sun is made of blue cheese, then cats fly.
How do I tell if its valid or satisfiable?
logic
asked Sep 9 at 6:53
Helosy
183
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up vote
2
down vote
favorite
As the title says, What is the difference between validity and satisfiability?
Suppose I have a sentence If the sun is made of blue cheese, then cats fly.
How do I tell if its valid or satisfiable?
logic
asked Sep 9 at 6:53
Helosy
183
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
As the title says, What is the difference between validity and satisfiability?
Suppose I have a sentence If the sun is made of blue cheese, then cats fly.
How do I tell if its valid or satisfiable?
logic
asked Sep 9 at 6:53
Helosy
183
As the title says, What is the difference between validity and satisfiability?
Suppose I have a sentence If the sun is made of blue cheese, then cats fly.
How do I tell if its valid or satisfiable?
logic
logic
asked Sep 9 at 6:53
Helosy
183
asked Sep 9 at 6:53
Helosy
183
asked Sep 9 at 6:53
Helosy
183
asked Sep 9 at 6:53
Helosy
183
asked Sep 9 at 6:53
Helosy
183
183
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
In mathematical logic, satisfiability and validity are elementary concepts of semantics.
A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true.
A formula is valid if all interpretations make the formula true.
A formula $varphi$ is valid iff its negation : $lnot varphi$ is unsatisfiable.
Examples from propositional logic :
$lnot p lor p$ is valid;
$p lor q$ and $p to q$ are examples of satisfiable formulas (but not valid);
$p land lnot p$ is unsatisfiable.
The above concepts appy to formulas; a single statement of natural language is either true or false.
Specifically, the statement :
"If the sun is made of blue cheese, then cats fly",
if we read the connective "if..., then ..." in the truth-functional way (i.e. as the material conditiona), is true, because the antecedent : "the sun is made of blue cheese" is false.
The sentence is an instance of the formula $p to q$, that is satisfiable (but not valid).
answered Sep 9 at 7:56
Mauro ALLEGRANZA
61.5k446105
I think the line "The above concepts appy to formulas; a single statement of natural language is either true or false." is misleading, since it's easy to read the emphasis as being on formulas vs. sentences (and of course a sentence is not true or false in the absence of an interpretation either); in fact, even the reference to natural language is misleading, since the real point is the role of "the world" as an intended interpretation.
– Noah Schweber
Sep 9 at 15:56
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
In mathematical logic, satisfiability and validity are elementary concepts of semantics.
A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true.
A formula is valid if all interpretations make the formula true.
A formula $varphi$ is valid iff its negation : $lnot varphi$ is unsatisfiable.
Examples from propositional logic :
$lnot p lor p$ is valid;
$p lor q$ and $p to q$ are examples of satisfiable formulas (but not valid);
$p land lnot p$ is unsatisfiable.
The above concepts appy to formulas; a single statement of natural language is either true or false.
Specifically, the statement :
"If the sun is made of blue cheese, then cats fly",
if we read the connective "if..., then ..." in the truth-functional way (i.e. as the material conditiona), is true, because the antecedent : "the sun is made of blue cheese" is false.
The sentence is an instance of the formula $p to q$, that is satisfiable (but not valid).
answered Sep 9 at 7:56
Mauro ALLEGRANZA
61.5k446105
I think the line "The above concepts appy to formulas; a single statement of natural language is either true or false." is misleading, since it's easy to read the emphasis as being on formulas vs. sentences (and of course a sentence is not true or false in the absence of an interpretation either); in fact, even the reference to natural language is misleading, since the real point is the role of "the world" as an intended interpretation.
– Noah Schweber
Sep 9 at 15:56
add a comment |Â
up vote
2
down vote
accepted
In mathematical logic, satisfiability and validity are elementary concepts of semantics.
A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true.
A formula is valid if all interpretations make the formula true.
A formula $varphi$ is valid iff its negation : $lnot varphi$ is unsatisfiable.
Examples from propositional logic :
$lnot p lor p$ is valid;
$p lor q$ and $p to q$ are examples of satisfiable formulas (but not valid);
$p land lnot p$ is unsatisfiable.
The above concepts appy to formulas; a single statement of natural language is either true or false.
Specifically, the statement :
"If the sun is made of blue cheese, then cats fly",
if we read the connective "if..., then ..." in the truth-functional way (i.e. as the material conditiona), is true, because the antecedent : "the sun is made of blue cheese" is false.
The sentence is an instance of the formula $p to q$, that is satisfiable (but not valid).
answered Sep 9 at 7:56
Mauro ALLEGRANZA
61.5k446105
I think the line "The above concepts appy to formulas; a single statement of natural language is either true or false." is misleading, since it's easy to read the emphasis as being on formulas vs. sentences (and of course a sentence is not true or false in the absence of an interpretation either); in fact, even the reference to natural language is misleading, since the real point is the role of "the world" as an intended interpretation.
– Noah Schweber
Sep 9 at 15:56
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
In mathematical logic, satisfiability and validity are elementary concepts of semantics.
A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true.
A formula is valid if all interpretations make the formula true.
A formula $varphi$ is valid iff its negation : $lnot varphi$ is unsatisfiable.
Examples from propositional logic :
$lnot p lor p$ is valid;
$p lor q$ and $p to q$ are examples of satisfiable formulas (but not valid);
$p land lnot p$ is unsatisfiable.
The above concepts appy to formulas; a single statement of natural language is either true or false.
Specifically, the statement :
"If the sun is made of blue cheese, then cats fly",
if we read the connective "if..., then ..." in the truth-functional way (i.e. as the material conditiona), is true, because the antecedent : "the sun is made of blue cheese" is false.
The sentence is an instance of the formula $p to q$, that is satisfiable (but not valid).
answered Sep 9 at 7:56
Mauro ALLEGRANZA
61.5k446105
In mathematical logic, satisfiability and validity are elementary concepts of semantics.
A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true.
A formula is valid if all interpretations make the formula true.
A formula $varphi$ is valid iff its negation : $lnot varphi$ is unsatisfiable.
Examples from propositional logic :
$lnot p lor p$ is valid;
$p lor q$ and $p to q$ are examples of satisfiable formulas (but not valid);
$p land lnot p$ is unsatisfiable.
The above concepts appy to formulas; a single statement of natural language is either true or false.
Specifically, the statement :
"If the sun is made of blue cheese, then cats fly",
if we read the connective "if..., then ..." in the truth-functional way (i.e. as the material conditiona), is true, because the antecedent : "the sun is made of blue cheese" is false.
The sentence is an instance of the formula $p to q$, that is satisfiable (but not valid).
answered Sep 9 at 7:56
Mauro ALLEGRANZA
61.5k446105
edited Sep 12 at 8:13
answered Sep 9 at 7:56
Mauro ALLEGRANZA
61.5k446105
answered Sep 9 at 7:56
Mauro ALLEGRANZA
61.5k446105
answered Sep 9 at 7:56
Mauro ALLEGRANZA
61.5k446105
61.5k446105
I think the line "The above concepts appy to formulas; a single statement of natural language is either true or false." is misleading, since it's easy to read the emphasis as being on formulas vs. sentences (and of course a sentence is not true or false in the absence of an interpretation either); in fact, even the reference to natural language is misleading, since the real point is the role of "the world" as an intended interpretation.
– Noah Schweber
Sep 9 at 15:56
add a comment |Â
I think the line "The above concepts appy to formulas; a single statement of natural language is either true or false." is misleading, since it's easy to read the emphasis as being on formulas vs. sentences (and of course a sentence is not true or false in the absence of an interpretation either); in fact, even the reference to natural language is misleading, since the real point is the role of "the world" as an intended interpretation.
– Noah Schweber
Sep 9 at 15:56
I think the line "The above concepts appy to formulas; a single statement of natural language is either true or false." is misleading, since it's easy to read the emphasis as being on formulas vs. sentences (and of course a sentence is not true or false in the absence of an interpretation either); in fact, even the reference to natural language is misleading, since the real point is the role of "the world" as an intended interpretation.
– Noah Schweber
Sep 9 at 15:56
I think the line "The above concepts appy to formulas; a single statement of natural language is either true or false." is misleading, since it's easy to read the emphasis as being on formulas vs. sentences (and of course a sentence is not true or false in the absence of an interpretation either); in fact, even the reference to natural language is misleading, since the real point is the role of "the world" as an intended interpretation.
– Noah Schweber
Sep 9 at 15:56
add a comment |Â
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