Frege
Frege (1884/1950:§§5,88) and others noted a number of problems with Kant’s “containment” metaphor.
ΕΛΕΥΘΕΡΩΣΗ ΑΠΟ ΤΟΝ ΨΥΧΟΛΟΓΙΣΜΟ: ΤΙΣ ΤΥΧΑΙΕΣ ΝΟΗΤΙΚΕΣ ΔΙΕΡΓΑΣΙΕΣ ΤΩΝ ΣΚΕΠΤΟΜΕΝΩΝ ΚΑΙ ΑΝΑΖΗΤΗΣΗ ΤΗΣ ΑΛΗΘΕΙΑΣ ΤΟΥ ΙΔΙΟΥ ΤΟΥ ΠΡΑΓΜΑΤΟΣ.
In the first place, as Kant himself probably would have agreed, the criterion would need to be freed of “psychologistic” suggestions, or claims about merely the accidental thought processes of thinkers, as opposed to claims about truth and justification that are presumably at issue with the analytic.
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Even were Kant to have solved these problems, it isn’t clear how his notion of “containment” would cover all the cases of what seem to many to be as “analytic” as any of set II. Thus, consider:
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II. (cont.)
(11) If Bob is married to Sue, then Sue is married to Bob.
(12) Anyone who’s an ancestor of an ancestor of Bob is an ancestor of Bob.
(13) If x is bigger than y, and y is bigger than z, then x is bigger than z.
(14) If something is red, then it’s colored.
The symmetry of the marriage relation, or the transitivity of “ancestor” and “bigger than” are not obviously “contained in” the corresponding thoughts in the way that the idea of extension is plausibly “contained in” the notion of body, or male in the notion of bachelor. (14) has seemed particularly troublesome: what else besides “colored” could be included in the analysis? Red is colored and what else? It is hard to see what else to “add”—except red itself! (See §3.4 below for further discussion.)
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Frege attempted to remedy the situation by completely rethinking the foundations of logic, developing what we now think of as modern symbolic logic .
Ο FREGE ΟΡΙΣΕ ΜΙΑ ΠΛΗΡΩΣ ΑΚΡΙΒΗ “ΤΥΠΙΚΗ” ΓΛΩΣΣΑ, δηλ., ΜΙΑ ΓΛΩΣΣΑ ΠΟΥ ΘΕΜΑΤΟΠΟΙΕΙ ΤΗ “ΜΟΡΦΗ” ΤΩΝ ΕΚΦΡΑΣΕΩΝ ΤΗΣ, ΚΑΙ ΠΡΟΣΕΚΤΙΚΑ ΘΕΤΕΙ ΜΙΑ ΘΕΩΡΙΑ ΤΗΣ ΣΥΝΤΑΞΗΣ ΚΑΙ ΤΗΣ ΣΗΜΑΝΤΙΚΗΣ ΑΥΤΩΝ ΠΟΥ ΟΝΟΜΑΖΟΝΤΑΙ “ΛΟΓΙΚΕΣ ΣΤΑΘΕΡΕΣ,” ΟΠΩΣ “ΚΑΙ”, “ΕΙΤΕ”, “ΟΧΙ”, “ΟΛΑ” και “ΚΑΠΟΙΑ”, ΔΕΙΧΝΟΝΤΑΣ ΠΩΣ ΝΑ ΣΥΛΛΑΒΟΥΜΕ ΜΙΑ ΠΟΛΥ ΕΥΡΕΙΑ ΤΑΞΗ ΕΓΚΥΡΩΝ ΣΥΜΠΕΡΑΣΜΩΝ.
(He defined a perfectly precise “formal” language, i.e., a language characterized by the “form” –standardly, the shape– of its expressions, and he carefully set out an account of the syntax and semantics of what are called the “logical constants,” such as “and”, “or”, “not”, “all” and “some”, showing how to capture a very wide class of valid inferences.)
——ΠΩΣ ΝΑ ΕΠΙΛΕΞΕΤΕ ΛΟΓΙΚΕΣ ΣΤΑΘΕΡΕΣ (those parts of language that don’t “point” or “function referentially”)———-
Just how these constants are selected is a matter of some dispute (see Logical Constants), but intuitively, the constants can be thought of as those parts of language that don’t “point” or “function referentially”, aiming to refer to something in the world, in the way that ordinary nouns, verbs and adjectives seem to do: “dogs” refers to dogs, “clever” to clever and/or clever things, and even “Zeus” aims to refer to a Greek god; but words like “or” and “all” don’t seem to function referentially at all: it doesn’t seem to make sense to think of there being “or”s in the world, along with the dogs and their properties.
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This distinction between referring expressions and logical constants allows us to define a logical truth as a sentence that is true no matter what referring expressions occur in it.
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Frege was mostly interested in formalizing arithmetic, and so considered the logical forms of a relative minority of natural language sentences in a deliberately spare formalism. Work on the logical (or syntactic) structure of the full range of sentences of natural language has blossomed since then, initially in the work of Russell (1905), in his famous theory of definite descriptions, where the criterion is applied to whole phrases in context, but then especially in the work of Chomsky and other “generative” linguists (see §4.3 below). Whether Frege’s criterion of analyticity will work for the rest of II and other analyticities depends upon the details of those proposals (see, e.g., Katz 1972, Montague 1974, Hornstein 1984 and Pietroski 2005).
eidetisch 