[FOM] Paradox and cognitive mechanism

Sat Aug 5 13:11:40 EDT 2006

  In my argument for the unknowability of AI I argued that a certain sentence (G’) would have a truth value if AI were true. Now I am convinced that the sentence at issue expresses no proposition (or, at least, not all utterances do). So it seems to me I can argue against AI itself. As you can see, the argument is essentially the same as before though I have introduced minor modifications.

  I add some remarks as to what the ultimate reason for the result could be.

  My apologies for the abuse of abbreviations; my excuse is that it makes easier the hard task of writing in English.

  The argument is as follows.

  DEFINITION 1: consistent bivalent logic (CBL) is the claim that 

  A. whenever, according to some linguistic code L, a sentence or a sentence-token depicts a well defined state of affairs, then, according to L, it expresses a proposition 

  and 

  B. every proposition has exactly one truth value, namely one of the following two: true, false, 

  and

  C. no proposition and its negation have the same truth value,

  and

  D. if CLST is a (maybe infinite) class of well defined states of affairs, then “each element of CLST is the case” depicts a well defined state of affairs.

  A, B, C and D fit the classical conception of logic as opposed, for instance, to multivalued, intuitionistic or paraconsistent logics. 

  PROPOSITION 1: CBL implies the existence of pairs of tokens of the same sentence-type, both of which would express the same proposition, according to some linguistic code L, if each of them expressed a proposition, yet one of which expresses no proposition according to L.

  JUSTIFICATION

  Assume CBL. I present an example. Call “(1)” the following sentence-token:

                       (1) expresses no true proposition

  So we include in our linguistic code L that (1) is a name for the sentence-token above. A double reductio shows that (1) is neither true nor false; so (1) has no truth value and consequently expresses no true proposition. This conclusion entitles us to state:

                      (1) expresses no true proposition

  which we call “(2)”.  Sentence-token (2) has to be true while (1), though a token of the same sentence-type, has no truth value. We will say that (1) and (2) are different cases of a same sentence.

  Obviously, according to L, (2) says that (1) expresses no true proposition but, according to L, this cannot be said by (1) itself. So, the fact that (1) has no truth value is not a well defined state of affairs for (the one uttering) (1).

  DEFINITION 2: a cognitive behavior is a sequence of acts which contains acts of assertion concerning some particular propositions that get asserted on certain grounds, and some times also acts of derivation of a proposition from some other propositions on certain grounds; all this can be reduced to sequences of acts of assertion concerning propositions that can be expressed by sentences or sentence-tokens of the form:  A because B where A and B are propositions. 

  Note this definition does not imply that every cognitive behavior can be reduced to the generation of a certain sequence of sentences since this could not always guarantee that the corresponding assertions have been really accomplished and accomplished on the corresponding grounds.

  DEFINITION 3: let (G) be the following sentence:

  “appended to its own quotation yields no sentence some correct cognitive behavior proves” appended to its own quotation yields no sentence some correct cognitive behavior proves

  This is an informal Quinean way of constructing a sentence (G) such that, if (G) expressed a proposition, (G) would say that no correct cognitive behavior can prove (G). Of course, that is so for (G) only according to the linguistic code LC we use here to interpret it.

  DEFINITION 4: a correct cognitive behavior is a cognitive behavior which accomplishes no incorrect acts of assertion and which interprets (G) according to the same linguistic code LC we employ here to interpret it. 

  Therefore we need to assume there is at least one linguistic code LC permitting to interpret (G) the way we do; we need not assume, however, that LC is recursively axiomatizable. The existence of LC is empirically evident but it is no logical or mathematical evidence; consequently those of my coming results which assume the existence of correct cognitive behaviors or rely on reasoning about (G) will be conditional on the existence of such LC.

  PROPOSITION 2: if LC exists, then CBL implies that it is always a well defined state of affairs whether a particular cognitive behavior is a correct cognitive behavior (even if that problem is not algorithmically decidable).

  JUSTIFICATION: the correctness of a cognitive behavior is always equivalent to the truth of each proposition of a certain set of propositions, one of them asserting that the concerned cognitive behavior applies LC to the intrepretation of (G), i. e. interprets (G) the same way we have interpreted it; thus the principle of bivalence guarantees it is always an objective state of affairs whether a cognitive behavior is correct.

  PROPOSITION 3: if LC exists, then CBL implies that, according to LC, (G) has a case expressing no proposition.

  JUSTIFICATION

  We assume CBL and the existence of LC. We show we can reproduce for (G) essentially the same kind of reasoning we have presented for (1).

  No correct cognitive behavior C proves (G), because if some C proved (G), then (G) would be true and then no C would prove (G). 

  Therefore we can say: “appended to its own quotation yields no sentence some correct cognitive behavior proves” appended to its own quotation yields no sentence some correct cognitive behavior proves.

  Assume now (G) always expresses a proposition; then (G) is true, because what it says is true, and false, because we have correctly proved it above. 

  So, if CBL holds, there is a case of (G) that, according to LC, expresses no proposition and a different case (in bold face above) that is true according to the same code. Consequently, if CBL holds, the state of affairs (G) would depict if (G) expressed a proposition (namely, that (G) is not provable) is not a well defined state of affairs for (the one uttering) the non propositional case of (G).

  DEFINITION 5: let cognitive mechanism (CM) be the claim that any possible cognitive behavior is an algorithm.

  PROPOSITION 4: if CM and CBL hold and there is a code LC, then every correct cognitive behavior is an algorithm.

  JUSTIFICATION: from definition 5 and proposition 2.

  PROPOSITION 5: if CM and CBL hold and there is a code LC, then, according to LC, there is no case of (G) expressing no proposition.

  JUSTIFICATION

  Assume CM, CBL and the existence of LC. 

  Then any correct cognitive behavior C is an algorithm (proposition 4) and the class CL of all C is a well defined class of algorithms (from proposition 2). The behavior of a particular algorithm is always a well defined physical and arithmetical state of affairs. Thus, for each element C of CL, whether C proves a particular linguistic object (whatever it is) is a physical and arithmetical state of affairs st. Let CLST be the class of all st. CLST is a class of well defined states of affairs. According to definition 1 C., “each st is the case” always depicts a well defined state of affairs ST. 

  So, the ST corresponding to the linguistic object (G) is always a well defined state of affairs and, according to LC, this is the state of affairs that (G) would depict if it depicted any; so, according to definition 1 A. and LC, (G) depicts in any case a well defined state of affairs. Then according to definition 1 B. and LC, (G) always expresses a proposition.

  PROPOSITION 6: if there is a LC, then if CBL is, CM is false.

  JUSTIFICATION: from proposition 3 and proposition 5.

  REMARKS

  The argument immediately relies on the fact that sometimes a state of affairs about an utterance U is not available as such for the one who utters U, while a physical (or arithmetical) state of affairs is always available as such. As a consequence, there can be paradoxes regarding the former but not regarding the latter.

  But, as I see it, the ultimate root of the proposed incompatibility might be the fact that, according to CBL, some references to mental intentional acts could be essential (i. e. endowed with a functional role) for the correctness of some cognitive behaviors; for instance, we can only accept (2) as logically other than (1) because we realize that (1) cannot express the thought that (2) expresses; mechanical devices are purely syntactical objects, so that such references cannot be functionally relevant for them.

  This can be couched as the existence of an irreducible distinction between sentences and propositions more concretely, as the impossibility of constructing a language L allowing for 1-1 mapping between sentences and propositions. For instance, it is easy to show that there is no L containing only disambiguated sentences and able to express all of the adjective expressions that can be expressed in natural languages: we can use an “heterological” defined over L to diagonalize out of the set of all adjective expressions in L, so that if L contains that “heterological”, then L contains a paradoxical “heterological” and then “ ‘heterological’ is heterological” is not disambiguated in L.

  The impossibility of retrieving propositions from sentences unambiguously in any occasion renders the very notion of “mechanical cognitive behavior” non well defined.

  But even if it were so, the notion of “correct mechanical cognitive behavior” would not be. All purely syntactical devices able to use a language as strong as natural language will generate sentences whose truth value is indefinite; therefore, whether a cognitive mechanical behavior is correct or incorrect, is not always a well defined state of affairs. This implies that the class CLA of all correct mechanical cognitive behaviors is not well defined. Therefore, according to CBL the class CL of all correct cognitive behaviors is well defined while CLA is not. But according to CM, CLA and CL are the same class; therefore CBL and CM are incompatible.

  The sentence

      no correct mechanical cognitive behavior proves this sentence

  and similar others (including the result of substituting “correct mechanical cognitive behavior” for “correct cognitive behavior” in (G)) are easily shown to be paradoxical according to CBL. This is not surprising, since CLA is not well defined; but if we assume CM we can prove as well that CLA is well defined so that those sentences cannot be paradoxical; so CBL and CM are mutually inconsistent.

  Regards,

  Laureano Luna Cabañero

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