Piero Scaruffi(Copyright © 2013 Piero Scaruffi | Legal restrictions )
These are excerpts and elaborations from my book "The Nature of Consciousness"
Possible World Semantics
In the 1960s the US philosopher Saul Kripke expanded Tarski’s model-theoretic interpretation to Modal Logic. Modal Logic is a logic that adds two more truth values, "possible" and "necessary" (also known as “modal” values) to the two traditional ones, "true" and "false".
Kripke defined modality through the notion of possible worlds: a property is necessary if it is true in all worlds, a property is possible if it is true in at least one world. Thanks to these two operators, it is possible to discriminate between sentences that are false but have different intension. In classical Logic, sentences such as “Piero Scaruffi is the author of the Divine Comedy” and “Piero Scaruffi is a billionaire” have the same extension, because they are both false. In Modal Logic they have different extensions, because the former is impossible (because I was not alive at the time), whereas the latter is also false but could be true. Also, Modal Logic avoids paradoxes that classical Logic cannot deal with. For example, the sentence “all mermaids are male” is intuitively false, but classical Logic would consider it true (because the sentence “all mermaids are male” translates into a logical formula of the negation of something that is not true, i.e. that is always true). In Modal Logic this sentence is false in the world where mermaids do exist.
The advantage of Kripke's semantics is that it can interpret sentences that are not extensional (that do not satisfy Leibniz's law), such as those that employ opaque contexts (to know, to believe, to think) and those that employ modal operators. Put bluntly, Kripke's semantics can interpret all sentences that can be reduced to "it is possible that" and "it is necessary that". The trick is that in his 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. A proposition does not have a truth value, but a set of truth values, one for each possible 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), whereas Kripke's modal logic is intensional. An extensional definition would actually be impossible, since the set of objects is infinite.
Kripke’s semantics can explain how we can refer to a thing by its name, even when we do not know the properties of that thing.
Proper names and definite descriptions are “designators”. A non-rigid designator is a term that changes its referent across possible worlds. But proper names are “rigid” designators, i.e. in every possible world they designate the same object (“Piero Scaruffi” is always the same person). So are natural kings: gold is always gold, and water is always water. Kripke (unlike Frege) carefully distinguished the meaning of a designator and the way its reference is determined (which are both "sense" in Frege). If it turned out that water is not H2O, I would still recognize water as water. The term “water” still designates water in a world in which water is not made of H2O.
Kripke’s explanation is his “causal theory of naming": names are linked to their referents through a causal chain. A term applies directly to an object via a connection that was set in place by the initial naming of the object. Initially, the reference of a name is fixed by some operation (e.g., by description), and then the name is passed from speaker to speaker basically by tradition. A name is not identified by a set of unique properties satisfied by the referent: the speaker may have erroneous beliefs about those properties or they may not be unique.
Kripke rejects the view that either proper or common nouns are associated with properties that serve to select their referents. Names are just "rigid designators". Both proper names and names of natural kinds have a referent, but not a Fregean sense. The property cannot determine the reference as the object might not have that property in all worlds. For example, gold may not be precious in all worlds.
Analogously, Jerry Fodor argued in favor of two types of meaning: one is the "narrow content" of a mental representation, which is a semantic representation and is purely mental and does not depend on anything else; and the other is the "broad content", a function that yields the referent in every possible world, and depends on the external world. Narrow content is a conceptual role. Meaning needs both narrow and broad contents.
In the 1980s the US mathematicians Jon Barwise and John Perry proposed “situation semantics”, a relation theory of meaning: the meaning of a sentence provides a constraint between the utterance and the described situation. Sentences stand for situations rather than for truth values. Properties and relations are primitive entities. Situations turn out to be more flexible than Kripke's possible worlds because they don't need to be coherent and don't need to be maximal. Just like mental states.
The US philosopher David Lewis even argued that possible worlds should be assumed to be real ("modal realism"). Things that may have been are no less real to him than things that actually are.
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