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From Wikipedia

Formal systems in mathematics consist of the following elements:

  • A finite set of symbols (i.e. the alphabet), that can be used for constructing formulas (i.e. finite strings of symbols).
  • A grammar, which tells how well-formed formulas (abbreviated wff) are constructed out of the symbols in the alphabet. It is usually
    required that there be a decision procedure for deciding whether a
    formula is well formed or not.
  • A set of axioms or axiom schemata: each axiom must be a wff.
  • A set of inference rules.

I was wondering if "inference rules" of a formal system means inference rules of a logic system? If yes, is a formal system therefore also a logic system?

But a logic system is just an example or a model of a formal system by an interpretation mapping to $\{ true, false\}$, isn't it? From the last link

A logical system or, for short, logic, is a formal system together with a form of semantics, usually in the form of model-theoretic interpretation, which assigns truth values to sentences of the formal language, that is, formulae that contain no free variables. A logic is sound if all sentences that can be derived are true in the interpretation, and complete if, conversely, all true sentences can be derived.

Thanks and regards!

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    http://en.wikipedia.org/wiki/Inference_rule.2012-02-16
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    @ArturoMagidin: Thanks! I had my questions after reading that.2012-02-16
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    Your question makes no sense in the absence of a *definition* of "logic system" and "inference rules of a logic system." As such, your confusion seems to be self-inflicted. An inference rule is merely a mechanism/function/rule that takes wffs as inputs, and produces a wff as an output; we call the inputs "premises" and the output "conclusion" or "inference", exactly as described in the wikipedia page.2012-02-16
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    PS Your title is ungrammatical. What is the "it" to which "is" refers? Not "inference rules"; so it must be "inference rules of a formal system". But then they aren't an "it is", they are a "they are".2012-02-16
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    @ArturoMagidin: Thanks! Please see my update.2012-02-16
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    So, logic systems are a special case of formal systems, and you are asking if a formal system is a special case of a logic system? No. Note, moreover, that "rules of inference" appear nowhere explicitly in the definition of "logic system", so why would you assume that the definition of "formal system" somehow requires "rules of inference" to be defined in terms of "logic systems"? Rules of inference have nothing to do with *semantics*; they are purely syntactical.2012-02-16
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    @ArturoMagidin: (1) Re your first sentence, yes, that is what I perceive as a contradiction. (2) Do you mean rules of inference can exist outside of any logic system? Are there some examples?2012-02-16
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    Even though it seems to have a Wikipedia entry, the general specification of what one might mean by a "logic system," and the term itself, are not useful.2012-02-16
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    @Tim: As I said: A rule of inference is a function/mechanism/rule that takes wffs as inputs and produces wffs as outputs. They can be purely syntactical, with absolutely no semantic meaning. Hofstadter gives examples in "Goedel, Escher, Bach". Here's a perfectly valid rule of inference: a wff is a finite sequence of $0$s and $1$s. The rule of inference takes a wff, and produces the wff that is obtained from the input by replacing every `0` by `01`, and every `1` by `11`.2012-02-16

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