What leads one down the garden path.

From my notes while reading Wittgenstein's Philosophical Investigations sec. 388-414.

Language L1 expresses the (set of) concepts C1.
“expresses a set of concepts”: all noncontradictory, coherent combinations of concepts are possible in grammatically acceptable sentences of Ln.
“noncontradictory combinations of concepts” are those which do not contain an impossibility of imagination or alternatively an implicit logical contradiction in any appropriate formalization of the terms of Ln.
Call such a formalization ƒ(Ln). [e.g. ‘x is a square’ becomes S(x) ]

Example of an unimaginable:
∃x(x is a square ^ x is a circle)

Example of an implicit logical contradiction (that is, a semantic one):
∀x(x is a circle only if x is not a square)
∀y((y is a square only if y is not a circle)
∴ ∀x∀y(x is a circle ^ y is a square iff x≠y) [steps omitted]

Second such example:
∀x(x is a number ↔ x has no color)
∴ ∼∃y(y = 1 ^ y is red)

Third (further) example
∀x∀ y(x is an elephant ^ y is a zebra iff x ≠ y)

Ln may express more, and more complex, concepts than necessary to express Cn. This includes concepts (e.g. cj) contradictory to concepts (e.g. ci) in Cn. E.g. ci = |square| and cj = |circle| we can express |square & circle|. Similarly for |elk| and |fox|, e.g. This leaves open the question just what a simple concept being “contradictory” to another might mean.

“coherent combinations of concepts” are complex or complexes of concepts that do violate the exclusivity constraint
It is impossible to express a concept cn two parts of which are such that:
being-F (= ci) and being-G (= cj) are mutually exclusive
[ ∀ x ∀ y (Fx.Gy only if ◻x≠y) ]
for some sort of combining function ƒ, cn = ƒ{ci, …, cj} or perhaps +ƒ〈…, ci, …, ck, …〉+.

Though note how the noncontradiction rule and the exclusivity constrain amount to the same demand for coherence or intelligibility. (“Making sense.”) Question: is it possible that there are concepts that it is impossible to express? "Expressions" qua literal, demonstrative utterances of a language probably cannot express every coherent noncontradictory concept of, say, Ce, the concept set expressed by my current (American English of the early 21st century) ideolect, Le.

NB: and Ln can be used to attempt to express some concept set Cm; I have used the alphanumeric symbology to indicate that, as in point 1, Ln expresses Cm iff n=m.
Poetry seems to allow the expression of complex concepts not expressable in Le. –A controversial claim. It is possible that poetry expresses cns which are inexpressible in demonstrative sentences by we limited beings—which is to say that it affords an alternative method for “expressing” some Cn. In a sense, poetry has a different meaning than prose, because of some feature of its structure, specifically that it associations of concepts available for ƒ [see above] that we cannot properly concatenate (before appreciating a poem). [Those are nonlinear or “nonlogical” associations… whatever that means.]

We have assumed that there are
Primitive concepts–some might call these “simple” concepts or ideas.
Complex concepts–the other ones.
These foregoing concerns commit us to the view that ∃x(x is a concept ^ x is structured).
So on this view I owe an account of how concepts are structured, and why (process).
Connectionism is an attractive move here...

One wonders what sort of vile epithets W. might have resorted to upon hearing such a claim.


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