Mixing
Formal Proof - premises and conclusions
0
$begingroup$
So I'm learning about formal proof and understand the beginning steps. However, after it gives an argument and conclusion. I then don'
t understand how to do the actual formal proofing. For example:
Premises: P ⇒ Q, P ∧ R.
Conclusion: Q.
Then the first two steps are the two premises, then step three says
P (2. Simplification) and step four says
Q (1,3. Modus Ponens).
What I don't understand is how you get the correct rule of inference and what the numbers are.
logic computer-science formal-proofs
asked Jan 28 at 21:10
JavaScrJavaScr
1
$endgroup$
|
show 2 more comments
0
$begingroup$
So I'm learning about formal proof and understand the beginning steps. However, after it gives an argument and conclusion. I then don'
t understand how to do the actual formal proofing. For example:
Premises: P ⇒ Q, P ∧ R.
Conclusion: Q.
Then the first two steps are the two premises, then step three says
P (2. Simplification) and step four says
Q (1,3. Modus Ponens).
What I don't understand is how you get the correct rule of inference and what the numbers are.
logic computer-science formal-proofs
asked Jan 28 at 21:10
JavaScrJavaScr
1
$endgroup$
2
$begingroup$
The numbers are telling you which steps to look at for the antecedents of the inference rule. How to find a proof is in general something you have to learn by experience.
$endgroup$
– Rob Arthan
Jan 28 at 21:19
$begingroup$
@RobArthan so for this example how do I know that the first is meant to be simplification. And then what does that 2. mean in regards to the simplification inference
$endgroup$
– JavaScr
Jan 28 at 21:26
$begingroup$
The proof tells you that the first step is by simplification. Knowing that that is the right thing to is something you have to learn by experience. The 2 means that you are simplifying the result of the second step in the proof, namely the premise $P land R$.
$endgroup$
– Rob Arthan
Jan 28 at 21:54
$begingroup$
Ok right thats the answer i seemed to come up with, but then got confused with the 1,3. And with another example the premise didnt match the rule of inference
$endgroup$
– JavaScr
Jan 28 at 22:06
$begingroup$
@JavaScr That the other example didn't match the rule of inference is possibly due to the rule being stated in terms of $P$'s and $Q$'s .... but obviously the $P$'s and $Q$'s could just as well have been $A$'s and $B$'s; as long as the example follows the abstract form of the rule, the rule can be used, which is why this is called formal proof. So, for example, Modus Ponens may be defined by your book as "you can infer $Q$ from $P rightarrow Q$ together with $P$", but that rule allows you also to infer $C lor D$ from $(A land B) rightarrow (C lor D)$ together with $A land B$
$endgroup$
– Bram28
Jan 28 at 23:35
|
show 2 more comments
0
0
0
$begingroup$
So I'm learning about formal proof and understand the beginning steps. However, after it gives an argument and conclusion. I then don'
t understand how to do the actual formal proofing. For example:
Premises: P ⇒ Q, P ∧ R.
Conclusion: Q.
Then the first two steps are the two premises, then step three says
P (2. Simplification) and step four says
Q (1,3. Modus Ponens).
What I don't understand is how you get the correct rule of inference and what the numbers are.
logic computer-science formal-proofs
asked Jan 28 at 21:10
JavaScrJavaScr
1
$endgroup$
So I'm learning about formal proof and understand the beginning steps. However, after it gives an argument and conclusion. I then don'
t understand how to do the actual formal proofing. For example:
Premises: P ⇒ Q, P ∧ R.
Conclusion: Q.
Then the first two steps are the two premises, then step three says
P (2. Simplification) and step four says
Q (1,3. Modus Ponens).
What I don't understand is how you get the correct rule of inference and what the numbers are.
logic computer-science formal-proofs
logic computer-science formal-proofs
asked Jan 28 at 21:10
JavaScrJavaScr
1
asked Jan 28 at 21:10
JavaScrJavaScr
1
asked Jan 28 at 21:10
JavaScrJavaScr
1
asked Jan 28 at 21:10
JavaScrJavaScr
1
asked Jan 28 at 21:10
JavaScrJavaScr
1
1
2
$begingroup$
The numbers are telling you which steps to look at for the antecedents of the inference rule. How to find a proof is in general something you have to learn by experience.
$endgroup$
– Rob Arthan
Jan 28 at 21:19
$begingroup$
@RobArthan so for this example how do I know that the first is meant to be simplification. And then what does that 2. mean in regards to the simplification inference
$endgroup$
– JavaScr
Jan 28 at 21:26
$begingroup$
The proof tells you that the first step is by simplification. Knowing that that is the right thing to is something you have to learn by experience. The 2 means that you are simplifying the result of the second step in the proof, namely the premise $P land R$.
$endgroup$
– Rob Arthan
Jan 28 at 21:54
$begingroup$
Ok right thats the answer i seemed to come up with, but then got confused with the 1,3. And with another example the premise didnt match the rule of inference
$endgroup$
– JavaScr
Jan 28 at 22:06
$begingroup$
@JavaScr That the other example didn't match the rule of inference is possibly due to the rule being stated in terms of $P$'s and $Q$'s .... but obviously the $P$'s and $Q$'s could just as well have been $A$'s and $B$'s; as long as the example follows the abstract form of the rule, the rule can be used, which is why this is called formal proof. So, for example, Modus Ponens may be defined by your book as "you can infer $Q$ from $P rightarrow Q$ together with $P$", but that rule allows you also to infer $C lor D$ from $(A land B) rightarrow (C lor D)$ together with $A land B$
$endgroup$
– Bram28
Jan 28 at 23:35
|
show 2 more comments
2
$begingroup$
The numbers are telling you which steps to look at for the antecedents of the inference rule. How to find a proof is in general something you have to learn by experience.
$endgroup$
– Rob Arthan
Jan 28 at 21:19
$begingroup$
@RobArthan so for this example how do I know that the first is meant to be simplification. And then what does that 2. mean in regards to the simplification inference
$endgroup$
– JavaScr
Jan 28 at 21:26
$begingroup$
The proof tells you that the first step is by simplification. Knowing that that is the right thing to is something you have to learn by experience. The 2 means that you are simplifying the result of the second step in the proof, namely the premise $P land R$.
$endgroup$
– Rob Arthan
Jan 28 at 21:54
$begingroup$
Ok right thats the answer i seemed to come up with, but then got confused with the 1,3. And with another example the premise didnt match the rule of inference
$endgroup$
– JavaScr
Jan 28 at 22:06
$begingroup$
@JavaScr That the other example didn't match the rule of inference is possibly due to the rule being stated in terms of $P$'s and $Q$'s .... but obviously the $P$'s and $Q$'s could just as well have been $A$'s and $B$'s; as long as the example follows the abstract form of the rule, the rule can be used, which is why this is called formal proof. So, for example, Modus Ponens may be defined by your book as "you can infer $Q$ from $P rightarrow Q$ together with $P$", but that rule allows you also to infer $C lor D$ from $(A land B) rightarrow (C lor D)$ together with $A land B$
$endgroup$
– Bram28
Jan 28 at 23:35
2
2
$begingroup$
The numbers are telling you which steps to look at for the antecedents of the inference rule. How to find a proof is in general something you have to learn by experience.
$endgroup$
– Rob Arthan
Jan 28 at 21:19
$begingroup$
The numbers are telling you which steps to look at for the antecedents of the inference rule. How to find a proof is in general something you have to learn by experience.
$endgroup$
– Rob Arthan
Jan 28 at 21:19
$begingroup$
@RobArthan so for this example how do I know that the first is meant to be simplification. And then what does that 2. mean in regards to the simplification inference
$endgroup$
– JavaScr
Jan 28 at 21:26
$begingroup$
@RobArthan so for this example how do I know that the first is meant to be simplification. And then what does that 2. mean in regards to the simplification inference
$endgroup$
– JavaScr
Jan 28 at 21:26
$begingroup$
The proof tells you that the first step is by simplification. Knowing that that is the right thing to is something you have to learn by experience. The 2 means that you are simplifying the result of the second step in the proof, namely the premise $P land R$.
$endgroup$
– Rob Arthan
Jan 28 at 21:54
$begingroup$
The proof tells you that the first step is by simplification. Knowing that that is the right thing to is something you have to learn by experience. The 2 means that you are simplifying the result of the second step in the proof, namely the premise $P land R$.
$endgroup$
– Rob Arthan
Jan 28 at 21:54
$begingroup$
Ok right thats the answer i seemed to come up with, but then got confused with the 1,3. And with another example the premise didnt match the rule of inference
$endgroup$
– JavaScr
Jan 28 at 22:06
$begingroup$
Ok right thats the answer i seemed to come up with, but then got confused with the 1,3. And with another example the premise didnt match the rule of inference
$endgroup$
– JavaScr
Jan 28 at 22:06
$begingroup$
@JavaScr That the other example didn't match the rule of inference is possibly due to the rule being stated in terms of $P$'s and $Q$'s .... but obviously the $P$'s and $Q$'s could just as well have been $A$'s and $B$'s; as long as the example follows the abstract form of the rule, the rule can be used, which is why this is called formal proof. So, for example, Modus Ponens may be defined by your book as "you can infer $Q$ from $P rightarrow Q$ together with $P$", but that rule allows you also to infer $C lor D$ from $(A land B) rightarrow (C lor D)$ together with $A land B$
$endgroup$
– Bram28
Jan 28 at 23:35
$begingroup$
@JavaScr That the other example didn't match the rule of inference is possibly due to the rule being stated in terms of $P$'s and $Q$'s .... but obviously the $P$'s and $Q$'s could just as well have been $A$'s and $B$'s; as long as the example follows the abstract form of the rule, the rule can be used, which is why this is called formal proof. So, for example, Modus Ponens may be defined by your book as "you can infer $Q$ from $P rightarrow Q$ together with $P$", but that rule allows you also to infer $C lor D$ from $(A land B) rightarrow (C lor D)$ together with $A land B$
$endgroup$
– Bram28
Jan 28 at 23:35
|
show 2 more comments
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2
$begingroup$
The numbers are telling you which steps to look at for the antecedents of the inference rule. How to find a proof is in general something you have to learn by experience.
$endgroup$
– Rob Arthan
Jan 28 at 21:19
$begingroup$
@RobArthan so for this example how do I know that the first is meant to be simplification. And then what does that 2. mean in regards to the simplification inference
$endgroup$
– JavaScr
Jan 28 at 21:26
$begingroup$
The proof tells you that the first step is by simplification. Knowing that that is the right thing to is something you have to learn by experience. The 2 means that you are simplifying the result of the second step in the proof, namely the premise $P land R$.
$endgroup$
– Rob Arthan
Jan 28 at 21:54
$begingroup$
Ok right thats the answer i seemed to come up with, but then got confused with the 1,3. And with another example the premise didnt match the rule of inference
$endgroup$
– JavaScr
Jan 28 at 22:06
$begingroup$
@JavaScr That the other example didn't match the rule of inference is possibly due to the rule being stated in terms of $P$'s and $Q$'s .... but obviously the $P$'s and $Q$'s could just as well have been $A$'s and $B$'s; as long as the example follows the abstract form of the rule, the rule can be used, which is why this is called formal proof. So, for example, Modus Ponens may be defined by your book as "you can infer $Q$ from $P rightarrow Q$ together with $P$", but that rule allows you also to infer $C lor D$ from $(A land B) rightarrow (C lor D)$ together with $A land B$
$endgroup$
– Bram28
Jan 28 at 23:35