Can existential quantifiers be used in modal logic to demonstrate the necessity of existence?
$begingroup$
Let E = existence, or 'at least one thing exists'
Let ~E = total non-existence, or 'nothing exists'
P1 // If and only if total nonexistence is possible, then it is possible there exists some x such that total nonexistence is true of x.
◊~E <=> ◊(ꓱx~Ex)
P2 // It is not possible that there exists some x such that total nonexistence is true of x.
~◊(ꓱx~Ex)
C1 //Therefore, existence is necessary
.'. □E
Defending P1:
This premise could be stated more generally as, "If and only if a proposition is possible, then it is possible there exists some x such that the proposition is true of x," and formalized as follows: ◊P <=> ◊(ꓱxPx).
Now, I'm not sure if that runs into any problems, formally, since I've just started trying to teach myself formal logic. I wouldn't be surprised if this premise simply doesn't work, but it seems to make sense in my head. In order to say that a thing is 'possible,' are you not granting that, in some possible world, 'ꓱxPx' would be a true statement? And if there is no possible world in which 'ꓱxPx' could be true, then is that not logically equivalent to '~◊P'?
Defending Premise 2:
Given that ~E = total non-existence, it seems prima facie obvious to me that no x could 'exist' to give truth to ~E. Any x which 'existed' would contradict ~E, and as such, it is impossible that some x could exist to make ~E true.
logic modal-logic
asked Jan 18 at 17:32
zjameszjames
1
$endgroup$
add a comment |
$begingroup$
Let E = existence, or 'at least one thing exists'
Let ~E = total non-existence, or 'nothing exists'
P1 // If and only if total nonexistence is possible, then it is possible there exists some x such that total nonexistence is true of x.
◊~E <=> ◊(ꓱx~Ex)
P2 // It is not possible that there exists some x such that total nonexistence is true of x.
~◊(ꓱx~Ex)
C1 //Therefore, existence is necessary
.'. □E
Defending P1:
This premise could be stated more generally as, "If and only if a proposition is possible, then it is possible there exists some x such that the proposition is true of x," and formalized as follows: ◊P <=> ◊(ꓱxPx).
Now, I'm not sure if that runs into any problems, formally, since I've just started trying to teach myself formal logic. I wouldn't be surprised if this premise simply doesn't work, but it seems to make sense in my head. In order to say that a thing is 'possible,' are you not granting that, in some possible world, 'ꓱxPx' would be a true statement? And if there is no possible world in which 'ꓱxPx' could be true, then is that not logically equivalent to '~◊P'?
Defending Premise 2:
Given that ~E = total non-existence, it seems prima facie obvious to me that no x could 'exist' to give truth to ~E. Any x which 'existed' would contradict ~E, and as such, it is impossible that some x could exist to make ~E true.
logic modal-logic
asked Jan 18 at 17:32
zjameszjames
1
$endgroup$
$begingroup$
What resources are you using to teach yourself formal logic? In formal logic we identify a definite system of rules for carrying out proofs. There are many different logical systems that we may consider and these may give different answers to things like "the necessity of existence", so your question has no one answer. The whole point of formal logic is to avoid having to appeal to informal arguments like "that makes sense to me" or "that's obvious".
$endgroup$
– Rob Arthan
Jan 18 at 21:00
add a comment |
$begingroup$
Let E = existence, or 'at least one thing exists'
Let ~E = total non-existence, or 'nothing exists'
P1 // If and only if total nonexistence is possible, then it is possible there exists some x such that total nonexistence is true of x.
◊~E <=> ◊(ꓱx~Ex)
P2 // It is not possible that there exists some x such that total nonexistence is true of x.
~◊(ꓱx~Ex)
C1 //Therefore, existence is necessary
.'. □E
Defending P1:
This premise could be stated more generally as, "If and only if a proposition is possible, then it is possible there exists some x such that the proposition is true of x," and formalized as follows: ◊P <=> ◊(ꓱxPx).
Now, I'm not sure if that runs into any problems, formally, since I've just started trying to teach myself formal logic. I wouldn't be surprised if this premise simply doesn't work, but it seems to make sense in my head. In order to say that a thing is 'possible,' are you not granting that, in some possible world, 'ꓱxPx' would be a true statement? And if there is no possible world in which 'ꓱxPx' could be true, then is that not logically equivalent to '~◊P'?
Defending Premise 2:
Given that ~E = total non-existence, it seems prima facie obvious to me that no x could 'exist' to give truth to ~E. Any x which 'existed' would contradict ~E, and as such, it is impossible that some x could exist to make ~E true.
logic modal-logic
asked Jan 18 at 17:32
zjameszjames
1
$endgroup$
Let E = existence, or 'at least one thing exists'
Let ~E = total non-existence, or 'nothing exists'
P1 // If and only if total nonexistence is possible, then it is possible there exists some x such that total nonexistence is true of x.
◊~E <=> ◊(ꓱx~Ex)
P2 // It is not possible that there exists some x such that total nonexistence is true of x.
~◊(ꓱx~Ex)
C1 //Therefore, existence is necessary
.'. □E
Defending P1:
This premise could be stated more generally as, "If and only if a proposition is possible, then it is possible there exists some x such that the proposition is true of x," and formalized as follows: ◊P <=> ◊(ꓱxPx).
Now, I'm not sure if that runs into any problems, formally, since I've just started trying to teach myself formal logic. I wouldn't be surprised if this premise simply doesn't work, but it seems to make sense in my head. In order to say that a thing is 'possible,' are you not granting that, in some possible world, 'ꓱxPx' would be a true statement? And if there is no possible world in which 'ꓱxPx' could be true, then is that not logically equivalent to '~◊P'?
Defending Premise 2:
Given that ~E = total non-existence, it seems prima facie obvious to me that no x could 'exist' to give truth to ~E. Any x which 'existed' would contradict ~E, and as such, it is impossible that some x could exist to make ~E true.
logic modal-logic
logic modal-logic
asked Jan 18 at 17:32
zjameszjames
1
asked Jan 18 at 17:32
zjameszjames
1
asked Jan 18 at 17:32
zjameszjames
1
asked Jan 18 at 17:32
zjameszjames
1
asked Jan 18 at 17:32
zjameszjames
1
1
$begingroup$
What resources are you using to teach yourself formal logic? In formal logic we identify a definite system of rules for carrying out proofs. There are many different logical systems that we may consider and these may give different answers to things like "the necessity of existence", so your question has no one answer. The whole point of formal logic is to avoid having to appeal to informal arguments like "that makes sense to me" or "that's obvious".
$endgroup$
– Rob Arthan
Jan 18 at 21:00
add a comment |
$begingroup$
What resources are you using to teach yourself formal logic? In formal logic we identify a definite system of rules for carrying out proofs. There are many different logical systems that we may consider and these may give different answers to things like "the necessity of existence", so your question has no one answer. The whole point of formal logic is to avoid having to appeal to informal arguments like "that makes sense to me" or "that's obvious".
$endgroup$
– Rob Arthan
Jan 18 at 21:00
$begingroup$
What resources are you using to teach yourself formal logic? In formal logic we identify a definite system of rules for carrying out proofs. There are many different logical systems that we may consider and these may give different answers to things like "the necessity of existence", so your question has no one answer. The whole point of formal logic is to avoid having to appeal to informal arguments like "that makes sense to me" or "that's obvious".
$endgroup$
– Rob Arthan
Jan 18 at 21:00
$begingroup$
What resources are you using to teach yourself formal logic? In formal logic we identify a definite system of rules for carrying out proofs. There are many different logical systems that we may consider and these may give different answers to things like "the necessity of existence", so your question has no one answer. The whole point of formal logic is to avoid having to appeal to informal arguments like "that makes sense to me" or "that's obvious".
$endgroup$
– Rob Arthan
Jan 18 at 21:00
add a comment |
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$begingroup$
What resources are you using to teach yourself formal logic? In formal logic we identify a definite system of rules for carrying out proofs. There are many different logical systems that we may consider and these may give different answers to things like "the necessity of existence", so your question has no one answer. The whole point of formal logic is to avoid having to appeal to informal arguments like "that makes sense to me" or "that's obvious".
$endgroup$
– Rob Arthan
Jan 18 at 21:00