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Carnap from 1950 sets out his exposition on defining a system purely on inductive logic that he attributes with epistemic qualities from probability.

This journey begins with his 'degree of confirmation', which requires evidence and a hypothesis to explicate a confirmed 'scientific sentence'. He details this further with modal logic and various syntactical deductions to define it by ordering finitely-many signs in a language system L, and assigning individual constants (events) to a number of predicates in that system. These signs prove the L-determinate class of sentences and so whether they're True or False.

Applying his method of induction to his own system - we enquire into the relevance of proving a system of reasoning purely under inductive logic. Such that, joining theorems of probability and modal logic to ultimately define probabilities by each sentence and a subset of proposition contained in any set of sentences.

In reference to the original question, how would the degree of confirmation actually apply to these methods of induction which seem psychological. For example: -

A clear sky on a sunny-day at the beach; the perception of colour attributed to the sky remains different to the oceans perceived colour.

If I take a finite number of sentences on a compact set; so a sequence of sentences with a finite subsequence. Assuming the limit point is the colour blue, I may find that the skies colour fits within a neighbourhood around the limit point whereas the oceans colour looks as if it's at some distance. However, I may also conclude the colour of the ocean also converges within that neighbourhood if I take a delta big enough.

How reliable is the evidence in this case - the colour of the sky and the ocean determined under sight and experience. Whereas the hypothesis seems more valid in terms of reliability and evidence because it avoids those sentences defined mainly on the weakness of identifying whether the colour is blue due to defaults in perception.

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    Reichenbach's inductive straight rule conjecturing the observed relative frequency will converge to the true limiting relative frequency, but it's not the only rule with this property so its justification relative to the goal of converging to limiting relative frequencies is at least incomplete. Also Goodman showed unlike deductive logic whether a hypothesis is confirmed by evidence depends on features other than mere syntactical rules. Later Bayesian theory replaced Carnap’s logical interpretation of probability with a subjective interpretation as degree of belief qua fair betting ratio... – Double Knot Feb 20 '22 at 02:31
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    In Lenz's paper [Carnap on Defining "Degree of Confirmation"](https://www.journals.uchicago.edu/doi/abs/10.1086/287488?journalCode=phos#:~:text=Carnap%2C%20as%20we%20have%20noted,be%20seen%20to%20be%20high.): *the central problem of the logic theory is...to find such a definition (of degree of confirmation)... is an insurmountable one...without either presupposing the frequency view or else committing oneself to the a priori...* – Double Knot Feb 20 '22 at 04:19
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    @DoubleKnot Thank you for these comments! After having a quick read on the article you have kindly provided; I see Reichenbachs comment on the c function as a degree of confirmation with a view as having error. For example, If we define a language system L, with a finite number of sentence and correspondingly a finite number of c-functions. Let's assume a modular arithmetic is applied to L whenever the belief reached by a sentence reaches a threshold on the c-function. This would at some sense reduce the frequency view but apply bounds on the total belief acquired as a conclusion to DOC. – Stackcans Feb 20 '22 at 11:30
  • @DoubleKnot This invites a stopping criterion similar to machine learning algorithms. Proving a closer relation to the bayesian view. Perhaps the DOC provides an alternative measure towards L-determinate assertions when a finite number of signs are involved. This measure opens up the beliefs possible for further investigation however there are limits as discussed previously. – Stackcans Feb 20 '22 at 11:32

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Here's an attempt to add further details on the choice of 'degree of confirmation'.

According to Bar-hillel (1955), Carnap defined the degree of confirmation in complete treatment by applying symbolic methods of logic and mathematical inequalities. This incurred the idea of a relevant criterion for a hypothesis, or evidence to apply towards the degree of confirmation. The following article is in objection to Karl poppers view which he coined in 1945, however Hillel maintains Carnap had supported this idea from 1936.

Here's my note on the approach Hillel, Carnap and Popper take on the language use revolving the variables x,y.

  • to qualify a sentence which happens to combine both x and y to a mode of communication. i.e. "the power of y to support x"; "a measure of the increase or decrease due to y, in the probability of "x.

Practice to define the degree of confirmation largely involves comparing between variables with methods of mathematical analysis; the finite methods used to suggest whether a sentence imposes a higher quality of success than another. Given this, Popper (1954)says "that all probability functions are inadequate to serve as a degree of confirmation". Proving a different conclusion to Carnap (I have yet to read up on Carnaps conclusion.)

Popper (1955), provides keywords to justify his choices for an explicandum that Hillel subjects as having large similarity existing in Carnaps Ideas. Popper does not deny this, but as previously mentioned his conclusions drawn from these examples are different.

It largely seems that explanations to deduce those inequalities that describe the degree of confirmation is circular in terms of practicality. Because evidence always implies an increment by time on some measure of what’s available i.e. resources, or even creativity.

References:

Bar-Hillel, Y. 1995. Comments on 'Degree of Confirmation' by Professor K. R. Popper. The British Journal for the Philosophy of Science. Vol. 6, No. 22: 155-157

Popper, K.R. 1955. ‘Content’ and ‘Degree of Confirmation’: A Reply to Dr Bar-Hillel. The British Journal for the Philosophy of Science. Vol. 6, No. 22: 157-163,

Popper, K.R. 1954. Degree of Confirmation. The British Journal for the Philosophy of Science. Vol. 5, No. 18: 143-149,

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