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What do logicians mean when they refer to the notion of "semantics"? I don't find the definition "the connection between words and meaning of those words" to be that satisfactory here.

Informally, I have been taking it to mean "the formal apparatus of the particular logic of context" in my readings. By this mean if an author refers to a "semantics" of a logic, he means something like the set of formulas for a logic, the set of connectives, the set of interpretations (functions or relations from those formulas to truth and falsity if not more things), and so forth.

Am I correct in loosely interpreting the word "semantics" this way? If not, where am I going wrong?

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    A clear and exhaustive answer to this question should go into an eventual FAQ about terminology, similarly to this one - @NieldeBeaudrap I am looking at you! ;) – DBK Apr 4 '13 at 9:44
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    "I don't find the definition satisfactory" Why? What you look for? – Annotations Apr 4 '13 at 13:18
  • I've just realized that you asked two questions, one in the headline about semantics in general and one in the text about semantics of logic. I've only answered the second one. Natural language semantics may be taken to differ substantially from semantics of logical languages for various reasons, so another answer would be needed for the headline question. – Eric '3ToedSloth' Apr 5 '13 at 8:42
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You're interpreting the term too loosely in the second part of your question. Your first definition "the connection between words and meaning of those words" is acceptable for logic if you add "described in a precise, rule-governed way". Logicians speak of terms, formulas, operators, connectives, and quantifiers rather than mere words, though.

Unfortunately, when you step away from such informal definitions and take a closer look things become less clear. I've seen the term being used in at least three slightly different senses. The first one is the general case, whereas (2) and (3) are imprecise. However, when someone talks about "the semantics" of a particular formal system, he may mean any of (1)-(3):

(1) In a first, primordial sense, "semantics" is used for "the definition of possible interpretations" and "the definition of models for a formal language". The semantics of a logical system is a set of rules for interpreting well-defined expressions of a formal language, which is in turn defined by syntax rules (a grammar). A logical semantics usually leaves it open how extralogical expressions are interpreted and focuses on a particular subset of expressions of a language called logical expressions. These are for example terms (how to deal with them in general), operators, connectives, quantifiers, and formulas. For the extralogical expressions such as particular interpretations of terms, you may then provide different models within the range circumscribed by the general semantic rules (aka interpretation, evaluation rules).

(2) In a second, narrower sense based on (1), "semantics" sometimes abbreviates denotational semantics which is also sometimes called model-theoretic semantics. This involves the explicit mapping of expressions of the language to elements in sets and rules for counting and combining these elements expressed in the language of set theory or, in the simplest case, by providing truth tables. So here "semantics" is understood as (1)+a particular method of achieving (1).

(3) In a third sense closely related to (1), "semantics" is meant in various ways that may be precisely described as algebraic semantics, combinatorial semantics, category-theoretic semantics, and so forth, where the methods used aren't based on set theory and are generally more syntactic in nature. ("syntactic" is meant as symbol manipulation by means of rules here.) Again, this sense of "semantics" is (1)+ = a particular method of achieving (1). These kinds of semantics are not very far from proof-theoretic methods such as using axiom systems + deduction rules, natural deduction, tableaux which are commonly considered syntactic methods. As confusing as this might be, speaking of "semantics" despite a certain nearness to proof theory may be well-justified in such cases provided that the method is provably more powerful than any calculus could be.

The third use of "semantics" often conveys an overall proof-theoretic attitude of a logician, the view that the main business of logic (as opposed to mathematics) is to provide the meaning of expressions of logical languages by a well-behaved proof theory only. Logicians in this tradition will sometimes even speak of proof-theoretical semantics.

Things get really messy when you go beyond first-order logic to second-order logic with standard models, because second- and higher-order logics with standard models have no complete proof theory (axioms+symbol manipulation rules), and in that case which kind of semantics you choose can make a huge difference both conceptually and formally.

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From all your suggestions, only "the set of interpretations" is related to "semantics". For some given (formal) language, its "semantics" is what it can mean. There may be more than one way to interpret a given sentence (from a formal language). I'm mostly thinking about different compilers for a fixed programming language here, but the same is true in most other contexts as well.

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Language and sentences as its component have to different types of rules: syntactical and semantics rules.

Syntactical rules are the content of the grammar.

Semantics is the study of the meaning of a sentence or a word. Defining meaning is a much more difficult task than defining the syntax of a sentence. Different philosophical theories exist aiming at a definition of meaning, e.g. see Frege, Putnam http://mcps.umn.edu/assets/pdf/7.3_Putnam.pdf or Kripke. E.g. The meaning of a a word like "water* is to define in an abstract way, not by enumeration, to which objects this word refers to.

Besides syntacs and semantics of natural languages the same issue comes up for artificial languages, namely computer languages like C or Java. Here the compiler is able to check whether a piece of code, i.e. a set of sentences from the language, satisfies the syntactical rules of the language. What the program does during execution time of the code, that's the semantics of the code. In general, the semantics are the coded algorithms.

I have never heard speaking about the semantics of a logic. The term cannot denote the rules for the logic, because these rules are formal rules without semantics. Possibly it refers to the models of a given logic calculus.

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