remove quantifiers from a Formula - Propositional Logic
up vote
0
down vote
favorite
I am trying to present a graph formula using propositional logic without any quantifiers $forall$ and $exists$. Is this possible?
The formula is:
$$∃v_1, . . . , ∃v_n forall u exists v(adj(u, v) vee u = v) land (v = v_1 vee · · · vee v = v_n))$$
logic propositional-calculus quantifiers
asked Nov 21 at 8:51
Courtney Mill
527
add a comment |
up vote
0
down vote
favorite
I am trying to present a graph formula using propositional logic without any quantifiers $forall$ and $exists$. Is this possible?
The formula is:
$$∃v_1, . . . , ∃v_n forall u exists v(adj(u, v) vee u = v) land (v = v_1 vee · · · vee v = v_n))$$
logic propositional-calculus quantifiers
asked Nov 21 at 8:51
Courtney Mill
527
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to present a graph formula using propositional logic without any quantifiers $forall$ and $exists$. Is this possible?
The formula is:
$$∃v_1, . . . , ∃v_n forall u exists v(adj(u, v) vee u = v) land (v = v_1 vee · · · vee v = v_n))$$
logic propositional-calculus quantifiers
asked Nov 21 at 8:51
Courtney Mill
527
I am trying to present a graph formula using propositional logic without any quantifiers $forall$ and $exists$. Is this possible?
The formula is:
$$∃v_1, . . . , ∃v_n forall u exists v(adj(u, v) vee u = v) land (v = v_1 vee · · · vee v = v_n))$$
logic propositional-calculus quantifiers
logic propositional-calculus quantifiers
asked Nov 21 at 8:51
Courtney Mill
527
asked Nov 21 at 8:51
Courtney Mill
527
asked Nov 21 at 8:51
Courtney Mill
527
asked Nov 21 at 8:51
Courtney Mill
527
asked Nov 21 at 8:51
Courtney Mill
527
527
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
Suppose you have an $n$-object finite domain, where every object has a name $a_1, a_2, ldots, a_n$. Then there is a sense in which $forall xvarphi x$ could be traded in for $varphi a_1 land varphi a_2 land ldots land varphi a_n$; and $exists xvarphi x$ could be traded in $varphi a_1 lor varphi a_2 lor ldots lor varphi a_n$. (Though even here we have to be careful -- e.g. we can't say the two wffs in each pair are logically equivalent.)
But finite-object domains are the exception, not the rule, in maths. And so in general we can't trade in quantified claims for long conjunctions/disjunctions. The use of the quantifiers is ineliminable.
answered Nov 21 at 9:34
Peter Smith
40.4k339118
Assuming that infinity consists of a legitimate mathematical concept, of course. Though, of course, such a denial these days seems unusual.
– Doug Spoonwood
Nov 21 at 14:12
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Suppose you have an $n$-object finite domain, where every object has a name $a_1, a_2, ldots, a_n$. Then there is a sense in which $forall xvarphi x$ could be traded in for $varphi a_1 land varphi a_2 land ldots land varphi a_n$; and $exists xvarphi x$ could be traded in $varphi a_1 lor varphi a_2 lor ldots lor varphi a_n$. (Though even here we have to be careful -- e.g. we can't say the two wffs in each pair are logically equivalent.)
But finite-object domains are the exception, not the rule, in maths. And so in general we can't trade in quantified claims for long conjunctions/disjunctions. The use of the quantifiers is ineliminable.
answered Nov 21 at 9:34
Peter Smith
40.4k339118
Assuming that infinity consists of a legitimate mathematical concept, of course. Though, of course, such a denial these days seems unusual.
– Doug Spoonwood
Nov 21 at 14:12
add a comment |
up vote
1
down vote
Suppose you have an $n$-object finite domain, where every object has a name $a_1, a_2, ldots, a_n$. Then there is a sense in which $forall xvarphi x$ could be traded in for $varphi a_1 land varphi a_2 land ldots land varphi a_n$; and $exists xvarphi x$ could be traded in $varphi a_1 lor varphi a_2 lor ldots lor varphi a_n$. (Though even here we have to be careful -- e.g. we can't say the two wffs in each pair are logically equivalent.)
But finite-object domains are the exception, not the rule, in maths. And so in general we can't trade in quantified claims for long conjunctions/disjunctions. The use of the quantifiers is ineliminable.
answered Nov 21 at 9:34
Peter Smith
40.4k339118
Assuming that infinity consists of a legitimate mathematical concept, of course. Though, of course, such a denial these days seems unusual.
– Doug Spoonwood
Nov 21 at 14:12
add a comment |
up vote
1
down vote
up vote
1
down vote
Suppose you have an $n$-object finite domain, where every object has a name $a_1, a_2, ldots, a_n$. Then there is a sense in which $forall xvarphi x$ could be traded in for $varphi a_1 land varphi a_2 land ldots land varphi a_n$; and $exists xvarphi x$ could be traded in $varphi a_1 lor varphi a_2 lor ldots lor varphi a_n$. (Though even here we have to be careful -- e.g. we can't say the two wffs in each pair are logically equivalent.)
But finite-object domains are the exception, not the rule, in maths. And so in general we can't trade in quantified claims for long conjunctions/disjunctions. The use of the quantifiers is ineliminable.
answered Nov 21 at 9:34
Peter Smith
40.4k339118
Suppose you have an $n$-object finite domain, where every object has a name $a_1, a_2, ldots, a_n$. Then there is a sense in which $forall xvarphi x$ could be traded in for $varphi a_1 land varphi a_2 land ldots land varphi a_n$; and $exists xvarphi x$ could be traded in $varphi a_1 lor varphi a_2 lor ldots lor varphi a_n$. (Though even here we have to be careful -- e.g. we can't say the two wffs in each pair are logically equivalent.)
But finite-object domains are the exception, not the rule, in maths. And so in general we can't trade in quantified claims for long conjunctions/disjunctions. The use of the quantifiers is ineliminable.
answered Nov 21 at 9:34
Peter Smith
40.4k339118
answered Nov 21 at 9:34
Peter Smith
40.4k339118
answered Nov 21 at 9:34
Peter Smith
40.4k339118
answered Nov 21 at 9:34
Peter Smith
40.4k339118
40.4k339118
Assuming that infinity consists of a legitimate mathematical concept, of course. Though, of course, such a denial these days seems unusual.
– Doug Spoonwood
Nov 21 at 14:12
add a comment |
Assuming that infinity consists of a legitimate mathematical concept, of course. Though, of course, such a denial these days seems unusual.
– Doug Spoonwood
Nov 21 at 14:12
Assuming that infinity consists of a legitimate mathematical concept, of course. Though, of course, such a denial these days seems unusual.
– Doug Spoonwood
Nov 21 at 14:12
Assuming that infinity consists of a legitimate mathematical concept, of course. Though, of course, such a denial these days seems unusual.
– Doug Spoonwood
Nov 21 at 14:12
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%2f3007444%2fremove-quantifiers-from-a-formula-propositional-logic%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
