Does a formal system having inference rules imply that it is a logic system?

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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– Arturo Magidin Feb 16 '12 at 21:30
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. – Arturo Magidin Feb 16 '12 at 21:40
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". – Arturo Magidin Feb 16 '12 at 21:43
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. – Arturo Magidin Feb 16 '12 at 21:50
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. – André Nicolas Feb 16 '12 at 21:54

As already noted in the comments, this depends essentially on what you choose to take as your definition of "a logic".

However, you would need an extremely expansive definition of "logic" in order to have the term cover all formal systems. In computer science, formal systems according to the description you quote crop up everywhere: they are the framework for discussing generative grammars, a diverse selection of models of computation, type systems, most styles of operational semantics for programming languages, and much more.

The general formalism and vocabulary of formal systems are deeply influenced by logic -- logics are in some sense paradigmatic examples of formal system, but that doesn't mean that every formal system is logic. Most of the examples I've listed here lack the central semantic characteristic of logic, namely that formulas can be thought of as being true or false (or at least have properties that are generalizations of truth and falsehood), and that the truth value of a formula depends in some way on the truth value of its components.

For example, the strings derived by a generative grammar usually don't represent any claims that can be true or false -- except in the rather vacuous sense that one can consider a string of letters to represent the proposition that this particular string of letters can be generated by the grammar. But the grammar can still be studied as an example of a formal system.

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As I think already pointed out, the formal system also needs a particular semantics to qualify as a logical system. Nothing in the definition of a formal system says anything about a semantics. Sometimes in logic it comes as just as useful, if not more useful, to notice what a definition does NOT say than to notice what it does say.

Additionally, the semantics of any arbitrarily given logical system, even if happening in a "truth-value" interpretation, need not happen on the truth set {true, false}. Mulit-valued logics, such as a three-valued logic where we have truth-values of {true, undecided, false} do get studied. Also, if we let "1" represent "truth" (or "absolute truth"), "0" falsity, and any number between 0 and 1 represent a degree of truth, there exist formal systems which have [0, 1] as its truth set.

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