What is the definition of a “deductive system”?












3














From what I can figure: A deductive system is a language $L$, a set of logical axioms $Delta_L$ that are formulas of the language $L$, and a set of ordered pairs of rules of inference $(Gamma, phi)$.



$text{Deductive System } = (L, Delta_L, {(Gamma, phi)})$



Is this the correct definition or am I misunderstanding something?






logic definition







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  • See proof system : "A set of axioms and a set of inference rules which are jointly used to deduce" theorems. See also Formal system.
    – Mauro ALLEGRANZA
    3 hours ago












  • See related posts : Axiom Systems and Formal Systems and Is a derivation a proof?.
    – Mauro ALLEGRANZA
    3 hours ago










  • See also Logics as consequence relations.
    – Mauro ALLEGRANZA
    3 hours ago
















3














From what I can figure: A deductive system is a language $L$, a set of logical axioms $Delta_L$ that are formulas of the language $L$, and a set of ordered pairs of rules of inference $(Gamma, phi)$.



$text{Deductive System } = (L, Delta_L, {(Gamma, phi)})$



Is this the correct definition or am I misunderstanding something?






logic definition







share|cite|improve this question






















  • See proof system : "A set of axioms and a set of inference rules which are jointly used to deduce" theorems. See also Formal system.
    – Mauro ALLEGRANZA
    3 hours ago












  • See related posts : Axiom Systems and Formal Systems and Is a derivation a proof?.
    – Mauro ALLEGRANZA
    3 hours ago










  • See also Logics as consequence relations.
    – Mauro ALLEGRANZA
    3 hours ago














3












3








3







From what I can figure: A deductive system is a language $L$, a set of logical axioms $Delta_L$ that are formulas of the language $L$, and a set of ordered pairs of rules of inference $(Gamma, phi)$.



$text{Deductive System } = (L, Delta_L, {(Gamma, phi)})$



Is this the correct definition or am I misunderstanding something?






logic definition







share|cite|improve this question













From what I can figure: A deductive system is a language $L$, a set of logical axioms $Delta_L$ that are formulas of the language $L$, and a set of ordered pairs of rules of inference $(Gamma, phi)$.



$text{Deductive System } = (L, Delta_L, {(Gamma, phi)})$



Is this the correct definition or am I misunderstanding something?






logic definition




logic definition






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asked 3 hours ago









Oliver G

1,6231529




1,6231529












  • See proof system : "A set of axioms and a set of inference rules which are jointly used to deduce" theorems. See also Formal system.
    – Mauro ALLEGRANZA
    3 hours ago












  • See related posts : Axiom Systems and Formal Systems and Is a derivation a proof?.
    – Mauro ALLEGRANZA
    3 hours ago










  • See also Logics as consequence relations.
    – Mauro ALLEGRANZA
    3 hours ago


















  • See proof system : "A set of axioms and a set of inference rules which are jointly used to deduce" theorems. See also Formal system.
    – Mauro ALLEGRANZA
    3 hours ago












  • See related posts : Axiom Systems and Formal Systems and Is a derivation a proof?.
    – Mauro ALLEGRANZA
    3 hours ago










  • See also Logics as consequence relations.
    – Mauro ALLEGRANZA
    3 hours ago
















See proof system : "A set of axioms and a set of inference rules which are jointly used to deduce" theorems. See also Formal system.
– Mauro ALLEGRANZA
3 hours ago






See proof system : "A set of axioms and a set of inference rules which are jointly used to deduce" theorems. See also Formal system.
– Mauro ALLEGRANZA
3 hours ago














See related posts : Axiom Systems and Formal Systems and Is a derivation a proof?.
– Mauro ALLEGRANZA
3 hours ago




See related posts : Axiom Systems and Formal Systems and Is a derivation a proof?.
– Mauro ALLEGRANZA
3 hours ago












See also Logics as consequence relations.
– Mauro ALLEGRANZA
3 hours ago




See also Logics as consequence relations.
– Mauro ALLEGRANZA
3 hours ago










2 Answers
2






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oldest

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2














There is not really a single agreed-on definition of "deductive system".



It's more of a stepping-stone system that each author will set up with the details they need for being able to define the actual systems they want to teach about. Other authors who want to speak about other actual systems may need to choose different details for their definition.



This is one of the few places in mathematics where the generalization is much less important than the concrete instances.



Everybody agrees quite well on what the entailment relations of, say, intuitionistic propositional logic or classical first-order logic are. But the precise systems that realize these entailment relations vary from author to author, so -- if they bother to define a general concept of "deductive system" at all -- their helper concepts are slightly different too.



What you're suggesting is close enough to the general kind of abstractions people tend to call "deductive system", that it ought to enable you to follow their train of thought when they use the term. Just as long as you don't take the details too seriously.






share|cite|improve this answer





























    1














    To expand on Henning's Answer:



    There are many 'deductive systems'.



    Your definition will be able to 'fit' a good number of deductive systems, but there are also some that do not allow such a nice generalization:



    Some systems define subproofs (see, for example, Fitch systems) and an inference is based on whole subproofs, rather than just statements



    Truth trees (sometimes called semantic tableaux) work quite differently yet.



    Of course, some may argue that those that don't fit your definition are not 'deductive systems'. But others construe them more broadly. I really don't think there is any universally agreed upon nice and clean definition. The same seems to be true for 'systems of natural deduction': some have a more narrow definition for those than others (I am sure to get comments on this :) )



    Point is: there is a whole taxonomy of these kinds of systems. The best thing to do is just to learn how they all work (indeed, you can really gain some deeper insights into logic if you look at different systems, rather than sticking than just one).






    share|cite|improve this answer





















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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2














      There is not really a single agreed-on definition of "deductive system".



      It's more of a stepping-stone system that each author will set up with the details they need for being able to define the actual systems they want to teach about. Other authors who want to speak about other actual systems may need to choose different details for their definition.



      This is one of the few places in mathematics where the generalization is much less important than the concrete instances.



      Everybody agrees quite well on what the entailment relations of, say, intuitionistic propositional logic or classical first-order logic are. But the precise systems that realize these entailment relations vary from author to author, so -- if they bother to define a general concept of "deductive system" at all -- their helper concepts are slightly different too.



      What you're suggesting is close enough to the general kind of abstractions people tend to call "deductive system", that it ought to enable you to follow their train of thought when they use the term. Just as long as you don't take the details too seriously.






      share|cite|improve this answer


























        2














        There is not really a single agreed-on definition of "deductive system".



        It's more of a stepping-stone system that each author will set up with the details they need for being able to define the actual systems they want to teach about. Other authors who want to speak about other actual systems may need to choose different details for their definition.



        This is one of the few places in mathematics where the generalization is much less important than the concrete instances.



        Everybody agrees quite well on what the entailment relations of, say, intuitionistic propositional logic or classical first-order logic are. But the precise systems that realize these entailment relations vary from author to author, so -- if they bother to define a general concept of "deductive system" at all -- their helper concepts are slightly different too.



        What you're suggesting is close enough to the general kind of abstractions people tend to call "deductive system", that it ought to enable you to follow their train of thought when they use the term. Just as long as you don't take the details too seriously.






        share|cite|improve this answer
























          2












          2








          2






          There is not really a single agreed-on definition of "deductive system".



          It's more of a stepping-stone system that each author will set up with the details they need for being able to define the actual systems they want to teach about. Other authors who want to speak about other actual systems may need to choose different details for their definition.



          This is one of the few places in mathematics where the generalization is much less important than the concrete instances.



          Everybody agrees quite well on what the entailment relations of, say, intuitionistic propositional logic or classical first-order logic are. But the precise systems that realize these entailment relations vary from author to author, so -- if they bother to define a general concept of "deductive system" at all -- their helper concepts are slightly different too.



          What you're suggesting is close enough to the general kind of abstractions people tend to call "deductive system", that it ought to enable you to follow their train of thought when they use the term. Just as long as you don't take the details too seriously.






          share|cite|improve this answer












          There is not really a single agreed-on definition of "deductive system".



          It's more of a stepping-stone system that each author will set up with the details they need for being able to define the actual systems they want to teach about. Other authors who want to speak about other actual systems may need to choose different details for their definition.



          This is one of the few places in mathematics where the generalization is much less important than the concrete instances.



          Everybody agrees quite well on what the entailment relations of, say, intuitionistic propositional logic or classical first-order logic are. But the precise systems that realize these entailment relations vary from author to author, so -- if they bother to define a general concept of "deductive system" at all -- their helper concepts are slightly different too.



          What you're suggesting is close enough to the general kind of abstractions people tend to call "deductive system", that it ought to enable you to follow their train of thought when they use the term. Just as long as you don't take the details too seriously.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 3 hours ago









          Henning Makholm

          238k16303537




          238k16303537























              1














              To expand on Henning's Answer:



              There are many 'deductive systems'.



              Your definition will be able to 'fit' a good number of deductive systems, but there are also some that do not allow such a nice generalization:



              Some systems define subproofs (see, for example, Fitch systems) and an inference is based on whole subproofs, rather than just statements



              Truth trees (sometimes called semantic tableaux) work quite differently yet.



              Of course, some may argue that those that don't fit your definition are not 'deductive systems'. But others construe them more broadly. I really don't think there is any universally agreed upon nice and clean definition. The same seems to be true for 'systems of natural deduction': some have a more narrow definition for those than others (I am sure to get comments on this :) )



              Point is: there is a whole taxonomy of these kinds of systems. The best thing to do is just to learn how they all work (indeed, you can really gain some deeper insights into logic if you look at different systems, rather than sticking than just one).






              share|cite|improve this answer


























                1














                To expand on Henning's Answer:



                There are many 'deductive systems'.



                Your definition will be able to 'fit' a good number of deductive systems, but there are also some that do not allow such a nice generalization:



                Some systems define subproofs (see, for example, Fitch systems) and an inference is based on whole subproofs, rather than just statements



                Truth trees (sometimes called semantic tableaux) work quite differently yet.



                Of course, some may argue that those that don't fit your definition are not 'deductive systems'. But others construe them more broadly. I really don't think there is any universally agreed upon nice and clean definition. The same seems to be true for 'systems of natural deduction': some have a more narrow definition for those than others (I am sure to get comments on this :) )



                Point is: there is a whole taxonomy of these kinds of systems. The best thing to do is just to learn how they all work (indeed, you can really gain some deeper insights into logic if you look at different systems, rather than sticking than just one).






                share|cite|improve this answer
























                  1












                  1








                  1






                  To expand on Henning's Answer:



                  There are many 'deductive systems'.



                  Your definition will be able to 'fit' a good number of deductive systems, but there are also some that do not allow such a nice generalization:



                  Some systems define subproofs (see, for example, Fitch systems) and an inference is based on whole subproofs, rather than just statements



                  Truth trees (sometimes called semantic tableaux) work quite differently yet.



                  Of course, some may argue that those that don't fit your definition are not 'deductive systems'. But others construe them more broadly. I really don't think there is any universally agreed upon nice and clean definition. The same seems to be true for 'systems of natural deduction': some have a more narrow definition for those than others (I am sure to get comments on this :) )



                  Point is: there is a whole taxonomy of these kinds of systems. The best thing to do is just to learn how they all work (indeed, you can really gain some deeper insights into logic if you look at different systems, rather than sticking than just one).






                  share|cite|improve this answer












                  To expand on Henning's Answer:



                  There are many 'deductive systems'.



                  Your definition will be able to 'fit' a good number of deductive systems, but there are also some that do not allow such a nice generalization:



                  Some systems define subproofs (see, for example, Fitch systems) and an inference is based on whole subproofs, rather than just statements



                  Truth trees (sometimes called semantic tableaux) work quite differently yet.



                  Of course, some may argue that those that don't fit your definition are not 'deductive systems'. But others construe them more broadly. I really don't think there is any universally agreed upon nice and clean definition. The same seems to be true for 'systems of natural deduction': some have a more narrow definition for those than others (I am sure to get comments on this :) )



                  Point is: there is a whole taxonomy of these kinds of systems. The best thing to do is just to learn how they all work (indeed, you can really gain some deeper insights into logic if you look at different systems, rather than sticking than just one).







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 3 hours ago









                  Bram28

                  60.2k44590




                  60.2k44590






























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