Modus Ponens - Implication vs Disjunction The Next CEO of Stack OverflowShowing logical argument is validAgain about McGee objections to modus ponensWhy isn't Modus Ponens valid hereIs the modus ponens is an axiom in formal logic?Proof using deductive system and modus ponensDemonstrate the next propositional excersise( The Constructive Dilema)How do I find the contradition in this indirect proof?Deduction of “Disjunction elimination”Is modus ponens a tautology?Modus ponens as a rule of inference vs. tautology

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Modus Ponens - Implication vs Disjunction



The Next CEO of Stack OverflowShowing logical argument is validAgain about McGee objections to modus ponensWhy isn't Modus Ponens valid hereIs the modus ponens is an axiom in formal logic?Proof using deductive system and modus ponensDemonstrate the next propositional excersise( The Constructive Dilema)How do I find the contradition in this indirect proof?Deduction of “Disjunction elimination”Is modus ponens a tautology?Modus ponens as a rule of inference vs. tautology










2












$begingroup$


The Modus Ponens inference rule is generally expressed as:



$$ beginarrayrl
& Prightarrow Q \
& P \
hline
therefore & Qendarray $$




Is the below rule also considered to be Modus Ponens?



$$ beginarrayrl
& P lor lnot Q \
& Q \
hline
therefore & Pendarray $$










share|cite|improve this question











$endgroup$







  • 3




    $begingroup$
    Translate $P lor lnot Q$ into $lnot Q lor P$ (by commutativity of $lor$) and then to an implication
    $endgroup$
    – Bernard Massé
    Mar 18 at 20:16










  • $begingroup$
    Ah! I don't know why I didn't think of that! Thanks!
    $endgroup$
    – Abhilash k
    Mar 18 at 20:27






  • 1




    $begingroup$
    You didn't think of it because you are new to the game. By practice and by asking questions when you are stuck, you'll become an expert or at least a good player in the game of logic.
    $endgroup$
    – Bernard Massé
    Mar 18 at 20:34










  • $begingroup$
    Yes; it is a particular case of the Resolution rule.
    $endgroup$
    – Mauro ALLEGRANZA
    Mar 19 at 8:10















2












$begingroup$


The Modus Ponens inference rule is generally expressed as:



$$ beginarrayrl
& Prightarrow Q \
& P \
hline
therefore & Qendarray $$




Is the below rule also considered to be Modus Ponens?



$$ beginarrayrl
& P lor lnot Q \
& Q \
hline
therefore & Pendarray $$










share|cite|improve this question











$endgroup$







  • 3




    $begingroup$
    Translate $P lor lnot Q$ into $lnot Q lor P$ (by commutativity of $lor$) and then to an implication
    $endgroup$
    – Bernard Massé
    Mar 18 at 20:16










  • $begingroup$
    Ah! I don't know why I didn't think of that! Thanks!
    $endgroup$
    – Abhilash k
    Mar 18 at 20:27






  • 1




    $begingroup$
    You didn't think of it because you are new to the game. By practice and by asking questions when you are stuck, you'll become an expert or at least a good player in the game of logic.
    $endgroup$
    – Bernard Massé
    Mar 18 at 20:34










  • $begingroup$
    Yes; it is a particular case of the Resolution rule.
    $endgroup$
    – Mauro ALLEGRANZA
    Mar 19 at 8:10













2












2








2





$begingroup$


The Modus Ponens inference rule is generally expressed as:



$$ beginarrayrl
& Prightarrow Q \
& P \
hline
therefore & Qendarray $$




Is the below rule also considered to be Modus Ponens?



$$ beginarrayrl
& P lor lnot Q \
& Q \
hline
therefore & Pendarray $$










share|cite|improve this question











$endgroup$




The Modus Ponens inference rule is generally expressed as:



$$ beginarrayrl
& Prightarrow Q \
& P \
hline
therefore & Qendarray $$




Is the below rule also considered to be Modus Ponens?



$$ beginarrayrl
& P lor lnot Q \
& Q \
hline
therefore & Pendarray $$







logic propositional-calculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 18 at 20:18







Abhilash k

















asked Mar 18 at 20:10









Abhilash kAbhilash k

1314




1314







  • 3




    $begingroup$
    Translate $P lor lnot Q$ into $lnot Q lor P$ (by commutativity of $lor$) and then to an implication
    $endgroup$
    – Bernard Massé
    Mar 18 at 20:16










  • $begingroup$
    Ah! I don't know why I didn't think of that! Thanks!
    $endgroup$
    – Abhilash k
    Mar 18 at 20:27






  • 1




    $begingroup$
    You didn't think of it because you are new to the game. By practice and by asking questions when you are stuck, you'll become an expert or at least a good player in the game of logic.
    $endgroup$
    – Bernard Massé
    Mar 18 at 20:34










  • $begingroup$
    Yes; it is a particular case of the Resolution rule.
    $endgroup$
    – Mauro ALLEGRANZA
    Mar 19 at 8:10












  • 3




    $begingroup$
    Translate $P lor lnot Q$ into $lnot Q lor P$ (by commutativity of $lor$) and then to an implication
    $endgroup$
    – Bernard Massé
    Mar 18 at 20:16










  • $begingroup$
    Ah! I don't know why I didn't think of that! Thanks!
    $endgroup$
    – Abhilash k
    Mar 18 at 20:27






  • 1




    $begingroup$
    You didn't think of it because you are new to the game. By practice and by asking questions when you are stuck, you'll become an expert or at least a good player in the game of logic.
    $endgroup$
    – Bernard Massé
    Mar 18 at 20:34










  • $begingroup$
    Yes; it is a particular case of the Resolution rule.
    $endgroup$
    – Mauro ALLEGRANZA
    Mar 19 at 8:10







3




3




$begingroup$
Translate $P lor lnot Q$ into $lnot Q lor P$ (by commutativity of $lor$) and then to an implication
$endgroup$
– Bernard Massé
Mar 18 at 20:16




$begingroup$
Translate $P lor lnot Q$ into $lnot Q lor P$ (by commutativity of $lor$) and then to an implication
$endgroup$
– Bernard Massé
Mar 18 at 20:16












$begingroup$
Ah! I don't know why I didn't think of that! Thanks!
$endgroup$
– Abhilash k
Mar 18 at 20:27




$begingroup$
Ah! I don't know why I didn't think of that! Thanks!
$endgroup$
– Abhilash k
Mar 18 at 20:27




1




1




$begingroup$
You didn't think of it because you are new to the game. By practice and by asking questions when you are stuck, you'll become an expert or at least a good player in the game of logic.
$endgroup$
– Bernard Massé
Mar 18 at 20:34




$begingroup$
You didn't think of it because you are new to the game. By practice and by asking questions when you are stuck, you'll become an expert or at least a good player in the game of logic.
$endgroup$
– Bernard Massé
Mar 18 at 20:34












$begingroup$
Yes; it is a particular case of the Resolution rule.
$endgroup$
– Mauro ALLEGRANZA
Mar 19 at 8:10




$begingroup$
Yes; it is a particular case of the Resolution rule.
$endgroup$
– Mauro ALLEGRANZA
Mar 19 at 8:10










2 Answers
2






active

oldest

votes


















2












$begingroup$

Yes.



Thanks to @Bernard Massé for pointing me in the right direction.



Here's the proof:



  1. $(P lor lnot Q)$ can be written as $ (lnot Q lor P) $ - Commutative Property


  2. $ (lnot Q lor P) $ can be written as $ (Q rightarrow P) $ - Material Implication


  3. By Modus Ponens :


$$ beginarrayrl
& Qrightarrow P \
& Q \
hline
therefore & Pendarray $$

This is equivalent to



$$ beginarrayrl
& P lor lnot Q \
& Q \
hline
therefore & Pendarray $$






share|cite|improve this answer









$endgroup$








  • 3




    $begingroup$
    This is an answer to a slightly different question than what you asked which is whether the latter rule would be considered Modus Ponens. I, personally, would consider it (a special case of) resolution. For me to call it Modus Ponens, I would have had to define $Pto Q$ as a syntactic abbrevation for $neg Plor Q$. Part of the reason I have this view is that the "Material Implication" equivalence is not derivable in constructive logic while both of the rules you list are, so these rules are talking about different things.
    $endgroup$
    – Derek Elkins
    Mar 19 at 0:03


















1












$begingroup$

I would consider it an application of Disjunctive Syllogism, which is typically stated as:



$P lor Q$



$neg P$



$therefore Q$



Of course, that is not exactly the same pattern, but the basic idea of Disjunctive Syllogism is that you have two options ... but it isn't one of them, and therefore you are left with the other one. Your argument is like that too: it is either P, or $neg Q$, but given $Q$ it is not $neg Q$, and so you are left with $P$






share|cite|improve this answer









$endgroup$













    Your Answer





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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    Yes.



    Thanks to @Bernard Massé for pointing me in the right direction.



    Here's the proof:



    1. $(P lor lnot Q)$ can be written as $ (lnot Q lor P) $ - Commutative Property


    2. $ (lnot Q lor P) $ can be written as $ (Q rightarrow P) $ - Material Implication


    3. By Modus Ponens :


    $$ beginarrayrl
    & Qrightarrow P \
    & Q \
    hline
    therefore & Pendarray $$

    This is equivalent to



    $$ beginarrayrl
    & P lor lnot Q \
    & Q \
    hline
    therefore & Pendarray $$






    share|cite|improve this answer









    $endgroup$








    • 3




      $begingroup$
      This is an answer to a slightly different question than what you asked which is whether the latter rule would be considered Modus Ponens. I, personally, would consider it (a special case of) resolution. For me to call it Modus Ponens, I would have had to define $Pto Q$ as a syntactic abbrevation for $neg Plor Q$. Part of the reason I have this view is that the "Material Implication" equivalence is not derivable in constructive logic while both of the rules you list are, so these rules are talking about different things.
      $endgroup$
      – Derek Elkins
      Mar 19 at 0:03















    2












    $begingroup$

    Yes.



    Thanks to @Bernard Massé for pointing me in the right direction.



    Here's the proof:



    1. $(P lor lnot Q)$ can be written as $ (lnot Q lor P) $ - Commutative Property


    2. $ (lnot Q lor P) $ can be written as $ (Q rightarrow P) $ - Material Implication


    3. By Modus Ponens :


    $$ beginarrayrl
    & Qrightarrow P \
    & Q \
    hline
    therefore & Pendarray $$

    This is equivalent to



    $$ beginarrayrl
    & P lor lnot Q \
    & Q \
    hline
    therefore & Pendarray $$






    share|cite|improve this answer









    $endgroup$








    • 3




      $begingroup$
      This is an answer to a slightly different question than what you asked which is whether the latter rule would be considered Modus Ponens. I, personally, would consider it (a special case of) resolution. For me to call it Modus Ponens, I would have had to define $Pto Q$ as a syntactic abbrevation for $neg Plor Q$. Part of the reason I have this view is that the "Material Implication" equivalence is not derivable in constructive logic while both of the rules you list are, so these rules are talking about different things.
      $endgroup$
      – Derek Elkins
      Mar 19 at 0:03













    2












    2








    2





    $begingroup$

    Yes.



    Thanks to @Bernard Massé for pointing me in the right direction.



    Here's the proof:



    1. $(P lor lnot Q)$ can be written as $ (lnot Q lor P) $ - Commutative Property


    2. $ (lnot Q lor P) $ can be written as $ (Q rightarrow P) $ - Material Implication


    3. By Modus Ponens :


    $$ beginarrayrl
    & Qrightarrow P \
    & Q \
    hline
    therefore & Pendarray $$

    This is equivalent to



    $$ beginarrayrl
    & P lor lnot Q \
    & Q \
    hline
    therefore & Pendarray $$






    share|cite|improve this answer









    $endgroup$



    Yes.



    Thanks to @Bernard Massé for pointing me in the right direction.



    Here's the proof:



    1. $(P lor lnot Q)$ can be written as $ (lnot Q lor P) $ - Commutative Property


    2. $ (lnot Q lor P) $ can be written as $ (Q rightarrow P) $ - Material Implication


    3. By Modus Ponens :


    $$ beginarrayrl
    & Qrightarrow P \
    & Q \
    hline
    therefore & Pendarray $$

    This is equivalent to



    $$ beginarrayrl
    & P lor lnot Q \
    & Q \
    hline
    therefore & Pendarray $$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Mar 18 at 20:36









    Abhilash kAbhilash k

    1314




    1314







    • 3




      $begingroup$
      This is an answer to a slightly different question than what you asked which is whether the latter rule would be considered Modus Ponens. I, personally, would consider it (a special case of) resolution. For me to call it Modus Ponens, I would have had to define $Pto Q$ as a syntactic abbrevation for $neg Plor Q$. Part of the reason I have this view is that the "Material Implication" equivalence is not derivable in constructive logic while both of the rules you list are, so these rules are talking about different things.
      $endgroup$
      – Derek Elkins
      Mar 19 at 0:03












    • 3




      $begingroup$
      This is an answer to a slightly different question than what you asked which is whether the latter rule would be considered Modus Ponens. I, personally, would consider it (a special case of) resolution. For me to call it Modus Ponens, I would have had to define $Pto Q$ as a syntactic abbrevation for $neg Plor Q$. Part of the reason I have this view is that the "Material Implication" equivalence is not derivable in constructive logic while both of the rules you list are, so these rules are talking about different things.
      $endgroup$
      – Derek Elkins
      Mar 19 at 0:03







    3




    3




    $begingroup$
    This is an answer to a slightly different question than what you asked which is whether the latter rule would be considered Modus Ponens. I, personally, would consider it (a special case of) resolution. For me to call it Modus Ponens, I would have had to define $Pto Q$ as a syntactic abbrevation for $neg Plor Q$. Part of the reason I have this view is that the "Material Implication" equivalence is not derivable in constructive logic while both of the rules you list are, so these rules are talking about different things.
    $endgroup$
    – Derek Elkins
    Mar 19 at 0:03




    $begingroup$
    This is an answer to a slightly different question than what you asked which is whether the latter rule would be considered Modus Ponens. I, personally, would consider it (a special case of) resolution. For me to call it Modus Ponens, I would have had to define $Pto Q$ as a syntactic abbrevation for $neg Plor Q$. Part of the reason I have this view is that the "Material Implication" equivalence is not derivable in constructive logic while both of the rules you list are, so these rules are talking about different things.
    $endgroup$
    – Derek Elkins
    Mar 19 at 0:03











    1












    $begingroup$

    I would consider it an application of Disjunctive Syllogism, which is typically stated as:



    $P lor Q$



    $neg P$



    $therefore Q$



    Of course, that is not exactly the same pattern, but the basic idea of Disjunctive Syllogism is that you have two options ... but it isn't one of them, and therefore you are left with the other one. Your argument is like that too: it is either P, or $neg Q$, but given $Q$ it is not $neg Q$, and so you are left with $P$






    share|cite|improve this answer









    $endgroup$

















      1












      $begingroup$

      I would consider it an application of Disjunctive Syllogism, which is typically stated as:



      $P lor Q$



      $neg P$



      $therefore Q$



      Of course, that is not exactly the same pattern, but the basic idea of Disjunctive Syllogism is that you have two options ... but it isn't one of them, and therefore you are left with the other one. Your argument is like that too: it is either P, or $neg Q$, but given $Q$ it is not $neg Q$, and so you are left with $P$






      share|cite|improve this answer









      $endgroup$















        1












        1








        1





        $begingroup$

        I would consider it an application of Disjunctive Syllogism, which is typically stated as:



        $P lor Q$



        $neg P$



        $therefore Q$



        Of course, that is not exactly the same pattern, but the basic idea of Disjunctive Syllogism is that you have two options ... but it isn't one of them, and therefore you are left with the other one. Your argument is like that too: it is either P, or $neg Q$, but given $Q$ it is not $neg Q$, and so you are left with $P$






        share|cite|improve this answer









        $endgroup$



        I would consider it an application of Disjunctive Syllogism, which is typically stated as:



        $P lor Q$



        $neg P$



        $therefore Q$



        Of course, that is not exactly the same pattern, but the basic idea of Disjunctive Syllogism is that you have two options ... but it isn't one of them, and therefore you are left with the other one. Your argument is like that too: it is either P, or $neg Q$, but given $Q$ it is not $neg Q$, and so you are left with $P$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Mar 21 at 1:50









        Bram28Bram28

        63.9k44793




        63.9k44793



























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