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I am trying to gain a deeper understanding of logic, especially proof theory in its many forms, and I am curious about the common use of the term "Logic". It is often confusing what the precise semantics of "logic" are in any isolated reading of the many descriptions of the topic area. For instance, the term "Propositional Calculus" and "Propositional Logic" are sometimes used interchangeably. Confusing things more, "Propositional Logic" and "Classical Logic" are used interchangeably.

From what I can intuit, it seems that there are a few popular logics; Classical, Intuitionistic, and Linear. Each with differing definitions of the semantics of truth and falsehood. Then there are algebras for logic; propositional, relational, first order, and second order. Lastly (at least for the distance i am currently willing to travel) there are proof calculi; sequent calculus, natural deduction, and axiomatic (i.e. Hilbert style).

Am I way off here? Are there any resources for clarifying this (please don't reply with wikipedia because that is causing some of the difficuties)?

Thanks in advance.

Chuck

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    $\begingroup$ There are a few popular apples; Golden Delicious, Granny Smith, Fuji, Jonagold, Pink Lady, ...., each with differing taste and coloration. Does this make the use of the term, "apple", confusing to you? $\endgroup$ – Gerry Myerson Aug 9 '16 at 13:19
  • $\begingroup$ Good question...I had to think about that analogy a while. I'm thinking the analogy you propose does not quite capture the confusion I am experiencing. It seems as if the analogy would need to be extended to apple pie or sauce which is what one can do with apples. Also, apple peeling or throwing. Sometimes an apple is described as apple sauce and sometimes it is a gala apple...I guess the analogy breaks down in there. The point I am driving towards is sometimes classical logic is described as propositional logic, but can intuitionistic logic also use the construct of propositional logic? $\endgroup$ – Chuck Winters Aug 9 '16 at 16:39
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    $\begingroup$ Where have you seen "propositional" and "classical" used interchangeably? This is plain wrong. $\endgroup$ – Rob Arthan Aug 10 '16 at 15:41
  • $\begingroup$ @GerryMyerson: By the way, thanks for all the apples! $\endgroup$ – user21820 Aug 12 '16 at 9:06
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I think you mean "predicate logic" (which has quantifiers and equality) rather than "propositional logic" (which has only boolean connectives). The reason "classical logic" is often called "predicate logic" is because the ones using the term are not even talking about non-classical logic so there is no need to specifically say "classical predicate logic". And Intuitionistic logic uses exactly the same language as classical logic but with different inference rules.

But what you need is to start with learning one logic first otherwise you're going to get confused very quickly, just like you'll get a stomachache if you try all the kinds of apples in the world all at one go. Naturally, the first logic to study should be classical first-order logic. I suggest the first two references in this post, which cover the intuition behind first-order logic, as well as common deductive systems (also called proof calculii) including natural deduction systems, Hilbert-style systems and the tableaux method (which is surprisingly simple).

While you are at it, always keep in mind that to study any formal system you always need to work in a meta-system. So take note what exactly is carried out within the formal system and what is carried out outside in the meta-system. I mention this because many students do not distinguish carefully between the two and get very confused. By the way, most logicians will take ZFC as their meta-system, but usually you do not need that much. You will see as you go just how much you need.

After the basics are there, it is not too hard to learn about higher-order logics and non-classical logics on your own, and you would be able to ask specific questions about those on Math SE if you cannot figure out something. Sorry for citing Wikipedia, but it is indeed a reasonable starting point once you have sufficient background knowledge. I agree it is quite useless if you don't, which is why one should go to proper texts for introductory material.

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  • $\begingroup$ Good point. Predicate logic is often used. That point coupled with trying to answer the question by @Rob Arthan I think I realized some of my confusions. The use of the term predicate logic is a contributing factor. The other was propositional classical logic. Also the use of the term "first order logic" must imply either classical, intuitionistic, or something. That is a source of confusion. I guess the way I am thinking about it now (which seems to help unless I find out it is incredibly incorrect) is that there are languages of logic, semantics for logic, and proof systems for logic. $\endgroup$ – Chuck Winters Aug 11 '16 at 17:26
  • $\begingroup$ So languages would be propositional, relational, first order. Semantics would be classical, intuitionistic, linear, etc. Proof systems would be natural deduction, tableux, trees, axiomatic, etc. Now I believe a propositional language can be interpreted with intuitionistic or classical interpretation as well as the First order language of logic. Am I way off in this thinking? $\endgroup$ – Chuck Winters Aug 11 '16 at 17:31
  • $\begingroup$ @ChuckWinters: So far so good. Your rough idea is about right. There is only a slight catch. Semantics is tightly related to the formal system itself, not directly (because the formal system exists even without any semantics) but indirectly because the intended semantics drives the design of the formal system. For example, classical logic will have LEM (law of excluded middle) as an axiom or some other inference rules that give the same, simply because we want it to be so. $\endgroup$ – user21820 Aug 12 '16 at 9:05
  • $\begingroup$ @ChuckWinters: So yes the same sentence in first-order logic can be interpreted classically or intuitionistically, and may be provable in classical but not intuitionistic logic. And it turns out that there is a semantics for intuitionistic logic under which semantic entailment is exactly provability (see math.stackexchange.com/a/1804379). Thus you can say that intuitionistic logic is complete with respect to Kripke semantics, just like classical logic is complete with respect to classical semantics. $\endgroup$ – user21820 Aug 12 '16 at 9:10

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