Using Semantic Arguments in Proving Validity
up vote
0
down vote
favorite
I have to show that $forall x (phi to psi) to (forall x phi to forall x psi)$ is valid using semantic arguments. However, what does "semantic arguments" mean? Do I use rules of inference?
logic first-order-logic
asked Nov 18 at 3:12
numericalorange
1,669311
add a comment |
up vote
0
down vote
favorite
I have to show that $forall x (phi to psi) to (forall x phi to forall x psi)$ is valid using semantic arguments. However, what does "semantic arguments" mean? Do I use rules of inference?
logic first-order-logic
asked Nov 18 at 3:12
numericalorange
1,669311
2
semantic arguments refer to models
– Gödel
Nov 18 at 4:21
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have to show that $forall x (phi to psi) to (forall x phi to forall x psi)$ is valid using semantic arguments. However, what does "semantic arguments" mean? Do I use rules of inference?
logic first-order-logic
asked Nov 18 at 3:12
numericalorange
1,669311
I have to show that $forall x (phi to psi) to (forall x phi to forall x psi)$ is valid using semantic arguments. However, what does "semantic arguments" mean? Do I use rules of inference?
logic first-order-logic
logic first-order-logic
asked Nov 18 at 3:12
numericalorange
1,669311
asked Nov 18 at 3:12
numericalorange
1,669311
asked Nov 18 at 3:12
numericalorange
1,669311
asked Nov 18 at 3:12
numericalorange
1,669311
asked Nov 18 at 3:12
numericalorange
1,669311
1,669311
2
semantic arguments refer to models
– Gödel
Nov 18 at 4:21
add a comment |
2
semantic arguments refer to models
– Gödel
Nov 18 at 4:21
2
2
semantic arguments refer to models
– Gödel
Nov 18 at 4:21
semantic arguments refer to models
– Gödel
Nov 18 at 4:21
add a comment |
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
When having a logical formula, there're usually two methods to prove validity.
A syntactic one where we use rules of inference only by looking at the syntactic structure of the formula. For instance if we have $P$ and $P rightarrow Q$ then we can syntactically produce Q with a rule called modus ponens $frac{Pquad Prightarrow Q}{Q}$.
A semantic one which consists of interpreting/evaluating the meaning of the formula with respect to a subjective conception of meaning. For instance if we have the propositional formula $phi = P land Q$ its interpreation will be $[![ A land B ]!]_sigma = AND([![ A ]!]_sigma, [![ B ]!]_sigma$) where $sigma$ associates a truth value in ${0, 1}$ to each variables and $AND : {0, 1} times {0, 1} longrightarrow {0, 1}$ is a function reproducing the truth table of $land$. To be valid is to be evaluated to the truth value $1$ for any interpretation.
In first-order logic, we have to extend the method to models to interpret entities in a particular universe. When we write $forall x. P(x)$, the $x$ doesn't refer to a truth value but to an entity (e.g a number). Then we seek an universe making the formula true.
The theorem of soundness and completude relates these two methods by saying that they are equivalents.
answered Nov 18 at 10:54
Boris E.
388113
add a comment |
up vote
1
down vote
By proving that a formula is valid by semantic arguments one usually means to prove that it is logically valid, that is that it is true in every possible interpretation.
So what you are asked is to prove that the formula
$$forall x (varphi to psi) to (forall x varphi to forall x psi)$$
is true in every interpretation.
In order to do that you have to use the basic definition of truth for a first-order formula.
I suggest you try to continue from here and in case you find any difficulty fell free to ask here for some hints.
I hope this helps.
answered Nov 18 at 16:04
Giorgio Mossa
13.7k11748
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
When having a logical formula, there're usually two methods to prove validity.
A syntactic one where we use rules of inference only by looking at the syntactic structure of the formula. For instance if we have $P$ and $P rightarrow Q$ then we can syntactically produce Q with a rule called modus ponens $frac{Pquad Prightarrow Q}{Q}$.
A semantic one which consists of interpreting/evaluating the meaning of the formula with respect to a subjective conception of meaning. For instance if we have the propositional formula $phi = P land Q$ its interpreation will be $[![ A land B ]!]_sigma = AND([![ A ]!]_sigma, [![ B ]!]_sigma$) where $sigma$ associates a truth value in ${0, 1}$ to each variables and $AND : {0, 1} times {0, 1} longrightarrow {0, 1}$ is a function reproducing the truth table of $land$. To be valid is to be evaluated to the truth value $1$ for any interpretation.
In first-order logic, we have to extend the method to models to interpret entities in a particular universe. When we write $forall x. P(x)$, the $x$ doesn't refer to a truth value but to an entity (e.g a number). Then we seek an universe making the formula true.
The theorem of soundness and completude relates these two methods by saying that they are equivalents.
answered Nov 18 at 10:54
Boris E.
388113
add a comment |
up vote
1
down vote
accepted
When having a logical formula, there're usually two methods to prove validity.
A syntactic one where we use rules of inference only by looking at the syntactic structure of the formula. For instance if we have $P$ and $P rightarrow Q$ then we can syntactically produce Q with a rule called modus ponens $frac{Pquad Prightarrow Q}{Q}$.
A semantic one which consists of interpreting/evaluating the meaning of the formula with respect to a subjective conception of meaning. For instance if we have the propositional formula $phi = P land Q$ its interpreation will be $[![ A land B ]!]_sigma = AND([![ A ]!]_sigma, [![ B ]!]_sigma$) where $sigma$ associates a truth value in ${0, 1}$ to each variables and $AND : {0, 1} times {0, 1} longrightarrow {0, 1}$ is a function reproducing the truth table of $land$. To be valid is to be evaluated to the truth value $1$ for any interpretation.
In first-order logic, we have to extend the method to models to interpret entities in a particular universe. When we write $forall x. P(x)$, the $x$ doesn't refer to a truth value but to an entity (e.g a number). Then we seek an universe making the formula true.
The theorem of soundness and completude relates these two methods by saying that they are equivalents.
answered Nov 18 at 10:54
Boris E.
388113
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
When having a logical formula, there're usually two methods to prove validity.
A syntactic one where we use rules of inference only by looking at the syntactic structure of the formula. For instance if we have $P$ and $P rightarrow Q$ then we can syntactically produce Q with a rule called modus ponens $frac{Pquad Prightarrow Q}{Q}$.
A semantic one which consists of interpreting/evaluating the meaning of the formula with respect to a subjective conception of meaning. For instance if we have the propositional formula $phi = P land Q$ its interpreation will be $[![ A land B ]!]_sigma = AND([![ A ]!]_sigma, [![ B ]!]_sigma$) where $sigma$ associates a truth value in ${0, 1}$ to each variables and $AND : {0, 1} times {0, 1} longrightarrow {0, 1}$ is a function reproducing the truth table of $land$. To be valid is to be evaluated to the truth value $1$ for any interpretation.
In first-order logic, we have to extend the method to models to interpret entities in a particular universe. When we write $forall x. P(x)$, the $x$ doesn't refer to a truth value but to an entity (e.g a number). Then we seek an universe making the formula true.
The theorem of soundness and completude relates these two methods by saying that they are equivalents.
answered Nov 18 at 10:54
Boris E.
388113
When having a logical formula, there're usually two methods to prove validity.
A syntactic one where we use rules of inference only by looking at the syntactic structure of the formula. For instance if we have $P$ and $P rightarrow Q$ then we can syntactically produce Q with a rule called modus ponens $frac{Pquad Prightarrow Q}{Q}$.
A semantic one which consists of interpreting/evaluating the meaning of the formula with respect to a subjective conception of meaning. For instance if we have the propositional formula $phi = P land Q$ its interpreation will be $[![ A land B ]!]_sigma = AND([![ A ]!]_sigma, [![ B ]!]_sigma$) where $sigma$ associates a truth value in ${0, 1}$ to each variables and $AND : {0, 1} times {0, 1} longrightarrow {0, 1}$ is a function reproducing the truth table of $land$. To be valid is to be evaluated to the truth value $1$ for any interpretation.
In first-order logic, we have to extend the method to models to interpret entities in a particular universe. When we write $forall x. P(x)$, the $x$ doesn't refer to a truth value but to an entity (e.g a number). Then we seek an universe making the formula true.
The theorem of soundness and completude relates these two methods by saying that they are equivalents.
answered Nov 18 at 10:54
Boris E.
388113
answered Nov 18 at 10:54
Boris E.
388113
answered Nov 18 at 10:54
Boris E.
388113
answered Nov 18 at 10:54
Boris E.
388113
388113
add a comment |
add a comment |
up vote
1
down vote
By proving that a formula is valid by semantic arguments one usually means to prove that it is logically valid, that is that it is true in every possible interpretation.
So what you are asked is to prove that the formula
$$forall x (varphi to psi) to (forall x varphi to forall x psi)$$
is true in every interpretation.
In order to do that you have to use the basic definition of truth for a first-order formula.
I suggest you try to continue from here and in case you find any difficulty fell free to ask here for some hints.
I hope this helps.
answered Nov 18 at 16:04
Giorgio Mossa
13.7k11748
add a comment |
up vote
1
down vote
By proving that a formula is valid by semantic arguments one usually means to prove that it is logically valid, that is that it is true in every possible interpretation.
So what you are asked is to prove that the formula
$$forall x (varphi to psi) to (forall x varphi to forall x psi)$$
is true in every interpretation.
In order to do that you have to use the basic definition of truth for a first-order formula.
I suggest you try to continue from here and in case you find any difficulty fell free to ask here for some hints.
I hope this helps.
answered Nov 18 at 16:04
Giorgio Mossa
13.7k11748
add a comment |
up vote
1
down vote
up vote
1
down vote
By proving that a formula is valid by semantic arguments one usually means to prove that it is logically valid, that is that it is true in every possible interpretation.
So what you are asked is to prove that the formula
$$forall x (varphi to psi) to (forall x varphi to forall x psi)$$
is true in every interpretation.
In order to do that you have to use the basic definition of truth for a first-order formula.
I suggest you try to continue from here and in case you find any difficulty fell free to ask here for some hints.
I hope this helps.
answered Nov 18 at 16:04
Giorgio Mossa
13.7k11748
By proving that a formula is valid by semantic arguments one usually means to prove that it is logically valid, that is that it is true in every possible interpretation.
So what you are asked is to prove that the formula
$$forall x (varphi to psi) to (forall x varphi to forall x psi)$$
is true in every interpretation.
In order to do that you have to use the basic definition of truth for a first-order formula.
I suggest you try to continue from here and in case you find any difficulty fell free to ask here for some hints.
I hope this helps.
answered Nov 18 at 16:04
Giorgio Mossa
13.7k11748
answered Nov 18 at 16:04
Giorgio Mossa
13.7k11748
answered Nov 18 at 16:04
Giorgio Mossa
13.7k11748
answered Nov 18 at 16:04
Giorgio Mossa
13.7k11748
13.7k11748
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003098%2fusing-semantic-arguments-in-proving-validity%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
2
semantic arguments refer to models
– Gödel
Nov 18 at 4:21