Saul Kripke:
NAMING AND NECESSITY (Harvard University Press, 1980)

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These three lectures were first published in 1972, and expanded in 1980, but the ideas had already been formulated in 1963, starting with "Semantical Analysis of Modal Logic". The first lecture begins with an attack against the Frege-Russell theory of reference.

John Stuart Mill introduced the concepts of "connotation" and "denotation". The expression "the president of Russia" denotes Vladimir Putin but it can be understood even by those who don't know who the current president of Russia is (i.e., it has a meaning regardless of whether you know that the president of Russia is Putin or not). Mill then divided names into general and singular (the latter including proper names). He then argued that general terms denote and connote, whereas singular terms merely denote. The word "Piero" simply refers to me, but it means nothing. Proper names have no meaning, general names have connotation (meaning). Frege and Russell interpreted proper names differently, as abbreviations of descriptions, and then the description is the sense (meaning) of the name ("sense" being Frege's term for connotation, and "reference" being Frege's term for denotation). In this view all names have connotation/sense. Kripke objected that Frege's "sense" is both the meaning of a name and the way its reference is determined. Kripke distinguishes the meaning of a designator and the way its reference is determined (which are both "sense" in Frege). In Kripke's view, names are rigid designators (their meaning is the same in every possible world); definite descriptions such as "the president of Russia", on the other hand, may have a different meaning in different worlds.

Kripke replaces Frege's description with a casual chain of communication Initially, the reference of a name is fixed by some operation (e.g., by description or by baptism), then the name is passed from person to person, from generation to generation. A name is not identified by a set of unique properties satisfied by the referent (Frege's description): the speaker may have erronous beliefs about those properties or they may not be unique. A property cannot determine the reference as the object might not have that property in all worlds. Instead, the name is passed to the speaker by tradition from link to link. And names for natural kinds behave in a similar way to proper names. Natural kinds and proper names are not as different as traditionally assumed. Names are linked to their referents through a casual chain. A term applies directly to an object via a connection that was set in place by the initial naming of the object. A speaker is always a member of a community of speakers who use that name. New discoveries do not change the meaning of names: if we discovered that we have always been wrong about the chemical composition of gold, the name "gold" would still mean gold.

Neither proper nor common nouns are associated with properties that serve to select their referents. Names are just "rigid designators". Both proper and common names have a referent, but not a Fregean sense.

Kripke developed a model-theoretic interpretation of various axiom sets for modal logic. Modality can be represented by recurring to the notion of possible worlds. A property is necessary if it is true in all worlds, a property is possible if there is at least a world in which it is true.
The extensional analysis of language cannot account for sentences that are very common such as those that employ opaque contexts (to know, to believe, to think) and those that employ modal operators (all words that can be reduced to "it is possible that" and "it is necessary that"). These senteces are not extensional, meaning that they do not satisfy Leibniz's law. These sentences can be interpreted in Kripke's model-theoretic semantics. A statement that is false in this universe can be true in another universe. The truth values of a sentence are always relative to a particular world.
Tarski's theory is purely extensional (for each model the truth of a predicate is determined by the list of objects for which it is true), Kripke's modal logic is intensional. An extensional definition would actually be impossible, as the set of objects is infinite.

Mental states cannot be identical to physical states because both are rigid designators and they might designate different objects in different worlds.