Answers.Body: <p>From your examples it seems you treat (UN)KNOWLEDGE as some predicate which rightfully can always be negated or double-negated as predicate of some semantically appropriate subject to continue to form well syntactically, however, unlike the real predicate (UN)KNOW, it's not the case at least in the theoretical classic logic sense of natural language. And unsurprisingly some non-empty positive (noun) element of our language's domain of discourse such as KNOWLEDGE may not always have its &quot;negative&quot; counterpart, and actually it makes no sense in logic for the notion of antonyms. Thus from rule-base logic expert system point of view there would better be logical constraints imposed on the possible forms of prompt.</p>
  <p><a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#First_incompleteness_theorem" rel="nofollow noreferrer">Gödel's incompleteness theorems</a> are only applicable to certain recursively axiomatizable formal system which can perform (equivalent) arithmetic additions <em>and</em> multiplications such as PA and ZFC involving leading universal quantifier(s), thus not applicable to systems of any natural language which cannot be formalized and your concerned sentence &quot;UNKNOWLEDGE is in the dictionary&quot; is quantifier free.</p>
  
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Answers.CreationDate: 2022-12-25T03:31:50.547Z
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Body: <p>Truthfulness is a quality that Sam Altman mentioned to be improvable on ChatGPT. This question considers a small example to try to understand the problem.</p>
  <p>A large language model includes a large set of true sentences, as well as a mechanism to obtain new sentences from known sentences. For convenience, let us map this to a simpler problem with a similar or equivalent form.</p>
  <p>A dialog system includes a set of valid words (lexicon), and a mechanism to obtain new words from known words (composition)</p>
  <p>For instance, a training set (or knowledge base), may look like this.</p>
  <p>KNOW is in the dictionary</p>
  <p>UNKNOW is in the dictionary</p>
  <p>KNOWN is in the dictionary</p>
  <p>UNKNOWN is in the dictionary</p>
  <p>Now, a system capable of subword composing from these sentences is prompted:  &quot;KNOWLEDGEABLE is in the dictionary&quot;. It will likely continue with &quot;UNKNOWLEDGEABLE is in the dictionary&quot;, which happens to be true by chance.</p>
  <p>However, when prompted &quot;KNOWLEDGE is in the dictionary&quot;, the likely continuation is &quot;UNKNOWLEDGE is in the dictionary&quot;, which is a well-formed and likely sentence, but false for some reason.</p>
  <p>The truth value of these two continuations cannot be derived from the four sentences in the training set, it would take an actual dictionary look-up to check.  In other types of sentences, such as those true by definition (+5 is a positive number, -5 is a negative number), it can actually be computed, as the validity of the sentence comes from the symbolic structure of the subject. It is true that -(digits) is a negative number, while it is only possible that UN(word_in_lexicon) is a word in the lexicon.</p>
  <p>Therefore, better continuations for both prompts would have been &quot;Possibly UN(X) is in the dictionary&quot;, but this is impossible to get from the training set, unless we preappended &quot;Possibly&quot; to the second and fourth training sentences.</p>
  <p>Is this impossibility to check for the validity of a new sentence simply a consequence of first Gödel's incompleteness theorem?</p>
  
Tags: <logic><chatgpt>
Title: Can the truth value of unobserved sentences be computed from known sentences only?
UserDisplayName: Jaume Oliver Lafont
ThreadText: # Can the truth value of unobserved sentences be computed from known sentences only?
  
  (Asked by: username_0 on 2022-12-23)
  
  Truthfulness is a quality that Sam Altman mentioned to be improvable on ChatGPT. This question considers a small example to try to understand the problem.
  
  A large language model includes a large set of true sentences, as well as a mechanism to obtain new sentences from known sentences. For convenience, let us map this to a simpler problem with a similar or equivalent form.
  
  A dialog system includes a set of valid words (lexicon), and a mechanism to obtain new words from known words (composition)
  
  For instance, a training set (or knowledge base), may look like this.
  
  KNOW is in the dictionary
  
  UNKNOW is in the dictionary
  
  KNOWN is in the dictionary
  
  UNKNOWN is in the dictionary
  
  Now, a system capable of subword composing from these sentences is prompted: "KNOWLEDGEABLE is in the dictionary". It will likely continue with "UNKNOWLEDGEABLE is in the dictionary", which happens to be true by chance.
  
  However, when prompted "KNOWLEDGE is in the dictionary", the likely continuation is "UNKNOWLEDGE is in the dictionary", which is a well-formed and likely sentence, but false for some reason.
  
  The truth value of these two continuations cannot be derived from the four sentences in the training set, it would take an actual dictionary look-up to check. In other types of sentences, such as those true by definition (+5 is a positive number, -5 is a negative number), it can actually be computed, as the validity of the sentence comes from the symbolic structure of the subject. It is true that -(digits) is a negative number, while it is only possible that UN(word\_in\_lexicon) is a word in the lexicon.
  
  Therefore, better continuations for both prompts would have been "Possibly UN(X) is in the dictionary", but this is impossible to get from the training set, unless we preappended "Possibly" to the second and fourth training sentences.
  
  Is this impossibility to check for the validity of a new sentence simply a consequence of first Gödel's incompleteness theorem?
  
  **Tags:** logic, chatgpt
  
  
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  ## 1 answers
  
  ### Answer by username_1 on 2022-12-25. Score: 1 | Accepted answer
  
  From your examples it seems you treat (UN)KNOWLEDGE as some predicate which rightfully can always be negated or double-negated as predicate of some semantically appropriate subject to continue to form well syntactically, however, unlike the real predicate (UN)KNOW, it's not the case at least in the theoretical classic logic sense of natural language. And unsurprisingly some non-empty positive (noun) element of our language's domain of discourse such as KNOWLEDGE may not always have its "negative" counterpart, and actually it makes no sense in logic for the notion of antonyms. Thus from rule-base logic expert system point of view there would better be logical constraints imposed on the possible forms of prompt.
  
  [Gödel's incompleteness theorems](https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#First_incompleteness_theorem) are only applicable to certain recursively axiomatizable formal system which can perform (equivalent) arithmetic additions *and* multiplications such as PA and ZFC involving leading universal quantifier(s), thus not applicable to systems of any natural language which cannot be formalized and your concerned sentence "UNKNOWLEDGE is in the dictionary" is quantifier free.
  
  
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