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Can the word belief be replaced with memo, short for MEntal MOdel?

The word I am looking for means "cognitive content that an agent categorizes as true".

Consider a statement X that could potentially be a fact, e.g. "Alice was born in NYC" or "Quebec City is Canada's capital" or "The earth is over 4 billion years old". Which of the following are synonymous? Which are logically connected (so that the truth of one implies the truth of another)?

# X.
# X is true.
# Bob thinks that X.
# Bob believes X.
# Bob considers X to be true.
# Bob holds X to be true.
# Bob categorizes X as true.
# Bob knows X.
# Bob would bet on X.
# Bob would bet $100 on X.
# Bob would bet his house on X.
# Bob would bet his life on X.
# Bob claims that X.
# Bob would say X is true if asked.
# Bob would say X is true if asked and he had no reason to lie.
# Bob behaves as if X is true.
# Bob infers X from other facts he thinks are true.
# Bob would infer X from other facts he thinks are true.
# Bob could infer X from other facts he thinks are true.
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For each statement three factors, all [fuzzy], and thus in the range of -1 (false) through 0 (unassertainable) through 1 (true) need to be addressed, viz The external falsifiable X, the process of arriving at the conclusion (1 rational (supported by evidence), 0 indeterminate and -1 irrational (contradicts evidence)) and the internal model of X held by Bob.

Examining the degenerate cases where the values are known, it is immediately apparent that there are  only three "useful" (i.e. non-delusional) states, and that these comprise two positions providing knowledge (high utility) and one which should result in a search for further information). All of the other possible states require the assumption of the unproveable or the rejection of evidence and thus holding such states as true is not helpful. It can also be seen that when the '[fuzzy values]' representing the external environment and Bob's internal representation are out of kilter, that the results will deviate from the useful toward the non-useful. The same naturally happens when confidence levels are low. Also looking at Table 1, it can be seen that where the end result is unknown, the process is non-evaluatable (irrespective of the truth of the source). Where the end internal map matches the external state, there is a high correlation with a rational process having been used to reach the conclusion. Finally, where the conclusion differs from the external, there is a reasonable presumption of irrational processing.

''Table 1: Utility Mapping''

|External|Bob's Perspective|Usefulness|Process
|True|True|High|Rational
|True|Unknown|None|Indeterminate
|True|False|None|Irrational
|Unknown|True|None|Irrational
|Unknown|Unknown|Moderate|Rational
|Unknown|False|None|Irrational
|False|True|None|Irrational
|False|None|Indeterminate|Unknown
|False|False|High|Rational

We can reverse this process too, in that by examining the inputs and the process, the input and output, or the output and the process, simple logic allows us to determine the probable missing values.

In addition to assigning a truth value to any assertion, we can assign an estimated certainty. Where possible, for each value determined, we will also assign a probable accuracy as reflected in Table 2, in the range of 0 to 1, where 0 is unknown, and is certainty. This will be used to scale the results. Notice that when we are certain that a value cannot be assigned to a factor that 1 is still used for probability.

''Table 2: Probability Map''

|0|Unlikely
|0.2|Possibly
|0.4|Maybe
|0.6|Likely
|0.8|Probably
|1|Certainty

Now let us reevaluate the above examples in terms of this logical approach.

''Table 3: Multicomponent Analysis''

|Case|Assertion|Discussion|Ext Truth|Ext Certainty|Process|Process Certainty|Int Perception|Int Certainty
|1|X|This is merely an assertion that X exists. No internal or external truth value is assigned to X, and no process is defined.|0|1|0|1|0|1
|2|X is true.|This statement asserts that X exists and is true, but makes no assertion about Bob's internal truth perception or the process applied|1|1|0|1|0|1
|3|Bob thinks that X.|A process of evaluation (thinking) leads Bob to assign an internal truth value to X. Thinking takes into account the fact that results are probable, not absolute.|1|0.6|1|0.6|1|0.6
|4|Bob believes X.|No process of evaluation has occured, so no knowledge of X is available. We know that the process used did not involve evaluation (or the result would not require belief), so we know that it was not rational, but do not know that it was irrational (although it may have been). Belief need not account for the limits of truth|0|1|0|1|1|1
|5|Bob considers X to be true.|A process of evaluation (consideration) leads Bob to assign a truth value to X. Consideration should take into account the fact that results are probable, not absolute.|1|0.6|1|0.6|1|0.6
|6|Bob holds X to be true.|No process of evaluation is asserted, so no knowledge of X is available. The fact that the value is "held" true, possibly irrespective of evidence, means that the process is probably irrational.|0|0|-1|0.6|1|1
|7|Bob categorizes X as true.| Categorization, while similar to cognition, implies a much greater degree of knowledge about the subject of study. It also implies a formal process of creating internal maps, giving higher certainty to the likelihood that the intrnal map will be accurate.|1|0.8|1|0.8|1|0.8
|8|Bob knows X.|A process of evaluation (knowledge development) leads Bob to assign a truth value to X. Similar to categorization, but not speaking to the process involved.|1|0.8|1|0.6|1|0.8
|9|Bob would bet on X.|No assertion is made about X or the process through which Bob reached his position, nor even that Bob considers X true or false. Assuming Bob expects to win his bet.|0|0.2|0|0|1|0.2b
|10|Bob would bet $100 on X.|No assertion is made about X or the process through which Bob reached his position - just the iimplication that Bob feels more positive about winning than in 9.|0|0.2|0|0|1|0.4
|11|Bob would bet his house on X.|No assertion is made about X or the process through which Bob reached his position - just the iimplication that Bob feels more positive about winning than in 9 or 10.|0|0.2|0|0|1|0.6
|12|Bob would bet his life on X.|No assertion is made about X or the process through which Bob reached his position - just the iimplication that Bob feels more positive about winning than in 11, 10 or 9.|0|0.2|0|0|1|0.8
|13|Bob claims that X.|No assertion is made about X or the process through which Bob reached his position.|0|0|0|0|1|1
|14|Bob would say X is true if asked.|No assertion is made about X or the process through which Bob reached his position.(although I think that the implication may be that this is less certain than 13).|0|0|0|0|1|0.8
|15|Bob would say X is true if asked and he had no reason to lie.|No assertion is made about X or the process through which Bob reached his position (although I think that the implication may be that this is less certain than 14 or 13)|0|0|0|0|1|0.6
|16|Bob behaves as if X is true.|No process of evaluation is asserted, so no knowledge of X or process is available.|0|0|0|0|1|1
|17|Bob infers X from other facts he thinks are true.|X is evaluated against other evaluated things|1|0.8|0|0|1|0.8
|18|Bob would infer X from other facts he thinks are true.|No process of evaluation of X is asserted, even though it might be infered from other evaluated things. The process and conclusions sound weaker than in 17.|1|0.6|0|0|1|0.6
|19|Bob could infer X from other facts he thinks are true.|No process of evaluation of X is asserted, even though it might be infered from other evaluated things. This sounds weaker than 18 or 17.|1|0.4|0|0|1|0.4

Now multiplying the conclusions by the probabilities we get:

''Table 4: Conclusions''

|External Truth|Process|Internal Truth|Case|Assertion
|0|0|0|1|X.
|1|0|0|2|X is true.
|0.6|0.6|0.6|3|Bob thinks that X.
|0|0|1|4|Bob believes X.
|0.6|0.6|0.6|5|Bob considers X to be true.
|0|-0.6|1|6|Bob holds X to be true.
|0.8|0.8|0.8|7|Bob categorizes X as true.
|0.8|0.6|0.8|8|Bob knows X.
|0.2|0|0.2|9|Bob would bet on X.
|0.2|0|0.4|10|Bob would bet $100 on X.
|0.2|0|0.6|11|Bob would bet his house on X.
|0.2|0|0.8|12|Bob would bet his life on X.
|0|0|1|13|Bob claims that X.
|0|0|0.8|14|Bob would say X is true if asked.
|0|0|0.6|15|Bob would say X is true if asked and he had no reason to lie.
|0|0|1|16|Bob behaves as if X is true.
|0.8|0|0.8|17|Bob infers X from other facts he thinks are true.
|0.6|0|0.4|18|Bob would infer X from other facts he thinks are true.
|0.4|0|0.4|19|Bob could infer X from other facts he thinks are true.

And grouping these:

''Table 5: Ordered by Knowledge of X''

|External Truth|Process|Internal Truth|Case|Assertion
|0|0|0|1|X.
|0|0|0.8|14|Bob would say X is true if asked.
|0|0|0.6|15|Bob would say X is true if asked and he had no reason to lie.
|0|-0.6|1|6|Bob holds X to be true.
|0|0|1|4|Bob believes X.
|0|0|1|13|Bob claims that X.
|0|0|1|16|Bob behaves as if X is true.
|0.2|0|0.2|9|Bob would bet on X.
|0.2|0|0.4|10|Bob would bet $100 on X.
|0.2|0|0.6|11|Bob would bet his house on X.
|0.2|0|0.8|12|Bob would bet his life on X.
|0.4|0|0.4|19|Bob could infer X from other facts he thinks are true.
|0.6|0|0.4|18|Bob would infer X from other facts he thinks are true.
|0.6|0.6|0.6|3|Bob thinks that X.
|0.6|0.6|0.6|5|Bob considers X to be true.
|0.8|0.8|0.8|7|Bob categorizes X as true.
|0.8|0.6|0.8|8|Bob knows X.
|0.8|0|0.8|17|Bob infers X from other facts he thinks are true.
|1|0|0|2|X is true.

''Table 6: Ordered by Perception of X''

|External Truth|Process|Internal Truth|Case|Assertion
|0|0|0|1|X.
|1|0|0|2|X is true.
|0.6|0|0.4|18|Bob would infer X from other facts he thinks are true.
|0.4|0|0.4|19|Bob could infer X from other facts he thinks are true.
|0|0|0.6|15|Bob would say X is true if asked and he had no reason to lie.
|0.2|0|0.2|9|Bob would bet on X.
|0.2|0|0.4|10|Bob would bet $100 on X.
|0.2|0|0.6|11|Bob would bet his house on X.
|0.6|0.6|0.6|3|Bob thinks that X.
|0.6|0.6|0.6|5|Bob considers X to be true.
|0|0|0.8|14|Bob would say X is true if asked.
|0.2|0|0.8|12|Bob would bet his life on X.
|0.8|0.8|0.8|7|Bob categorizes X as true.
|0.8|0.6|0.8|8|Bob knows X.
|0.8|0|0.8|17|Bob infers X from other facts he thinks are true.
|0|-0.6|1|6|Bob holds X to be true.
|0|0|1|4|Bob believes X.
|0|0|1|13|Bob claims that X.
|0|0|1|16|Bob behaves as if X is true.

''Table 7: Ordered by Usefulness of Perception''

|External Truth|Process|Internal Truth|Case|Assertion
|0.8|0.8|0.8|7|Bob categorizes X as true.
|0.8|0.6|0.8|8|Bob knows X.
|0.8|0|0.8|17|Bob infers X from other facts he thinks are true.
|0.6|0.6|0.6|5|Bob considers X to be true.
|0.6|0.6|0.6|3|Bob thinks that X.
|0.6|0|0.4|18|Bob would infer X from other facts he thinks are true.
|0.4|0|0.4|19|Bob could infer X from other facts he thinks are true.
|0.2|0|0.8|12|Bob would bet his life on X.
|0.2|0|0.6|11|Bob would bet his house on X.
|0.2|0|0.4|10|Bob would bet $100 on X.
|0.2|0|0.2|9|Bob would bet on X.
|0|0|0.6|15|Bob would say X is true if asked and he had no reason to lie.
|0|0|0.8|14|Bob would say X is true if asked.
|0|0|1|16|Bob behaves as if X is true.
|0|0|1|13|Bob claims that X.
|0|0|1|4|Bob believes X.
|0|-0.6|1|6|Bob holds X to be true.
|1|0|0|2|X is true.
|0|0|0|1|X.


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From my perspective VectorHermit has taken my question a long way in the wrong direction.

Though the three factors that he mentions are interesting, only one needs to be addressed and that is the internal model that is held by Bob. The external truth of X is needed to evaluate just three statements:

1. X

2. X is true

and

8. Bob knows X.

Saying that someone knows X implies that both Bob categorizes X as true __and__ X *is* true. In other words, given that Bob know X, you can infer that X is true and that Bob thinks X is true. And given that X is true and Bob thinks X is true, you can infer that Bob knows X is true.

The external truth of X cannot be inferred by any of the other statements, as is easily seen by substituting something blatantly false like "the earth is flat" or "1+1=3" for X. I'm not sure where that leaves VectorHermit's analysis but I'm not hopeful.

I also think Hermit is reading way too much into the statements, inferring process where none is implied. Only statement 17 "Bob infers X from other facts he thinks are true" mentions how he came to categorize, hold, think, or believe X is true (and I see absolutely no difference between those, not even connotative).
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