[Paper Review] Semantic Information Measure with Two Types of Probability for Falsification and Confirmation

Cheguang Lu|arXiv (Cornell University)|Sep 26, 2016

Rough Sets and Fuzzy Logic3 citations

TL;DR

This paper proposes a Semantic Information Measure (SIM) that integrates Logical Probability (LP) and Statistical Probability (SP) to quantify semantic information for hypothesis confirmation and falsification. By combining LP for truth-functional evaluation with SP for sampling distributions, SIM enables a rigorous, information-theoretic criterion for Popperian falsification and computes Degree of Confirmation (DOC) with upper bounds, resolving paradoxes like the Raven Paradox and modeling diagnostic tests such as rapid HIV tests with precise DOC formulas.

ABSTRACT

Logical Probability (LP) is strictly distinguished from Statistical Probability (SP). To measure semantic information or confirm hypotheses, we need to use sampling distribution (conditional SP function) to test or confirm fuzzy truth function (conditional LP function). The Semantic Information Measure (SIM) proposed is compatible with Shannon's information theory and Fisher's likelihood method. It can ensure that the less the LP of a predicate is and the larger the true value of the proposition is, the more information there is. So the SIM can be used as Popper's information criterion for falsification or test. The SIM also allows us to optimize the true-value of counterexamples or degrees of disbelief in a hypothesis to get the optimized degree of belief, i. e. Degree of Confirmation (DOC). To explain confirmation, this paper 1) provides the calculation method of the DOC of universal hypotheses; 2) discusses how to resolve Raven Paradox with new DOC and its increment; 3) derives the DOC of rapid HIV tests: DOC of test-positive=1-(1-specificity)/sensitivity, which is similar to Likelihood Ratio (=sensitivity/(1-specificity)) but has the upper limit 1; 4) discusses negative DOC for excessive affirmations, wrong hypotheses, or lies; and 5) discusses the DOC of general hypotheses with GPS as example.

Motivation & Objective

• To develop a semantic information measure compatible with both Shannon's information theory and Fisher's likelihood principle.
• To distinguish Logical Probability (LP) for truth evaluation from Statistical Probability (SP) for sampling distributions in hypothesis testing.
• To provide a formal criterion for falsification based on semantic information, aligning with Popper's demarcation of science.
• To define and compute the Degree of Confirmation (DOC) for universal and general hypotheses, including counterexamples and false affirmations.
• To resolve classical paradoxes such as the Raven Paradox using a new DOC and its incremental change.

Proposed method

• The Semantic Information Measure (SIM) is defined as a function of Logical Probability (LP) and Statistical Probability (SP), where lower LP and higher truth value yield higher information.
• The method uses conditional SP (sampling distribution) to evaluate the likelihood of observed data under a hypothesis, while LP evaluates the truth function of the proposition.
• The Degree of Confirmation (DOC) is derived as 1 minus the ratio of (1 - specificity) to sensitivity for diagnostic tests, ensuring DOC ≤ 1.
• For universal hypotheses, DOC is computed based on the optimized true-value of counterexamples, minimizing disbelief.
• The framework applies to general hypotheses using GPS as a case study, showing how DOC evolves with evidence.
• The method incorporates negative DOC for excessive affirmations, false hypotheses, or lies, indicating information loss.

Experimental results

Research questions

• RQ1 How can semantic information be measured using two distinct types of probability—logical and statistical—for hypothesis confirmation?
• RQ2 Can the proposed Semantic Information Measure (SIM) serve as a formal criterion for falsification in the Popperian sense?
• RQ3 What is the mathematical form of the Degree of Confirmation (DOC) for universal and general hypotheses, and how does it handle counterexamples?
• RQ4 How does the new DOC resolve the Raven Paradox, and what is the role of DOC increment in evidence evaluation?
• RQ5 What is the DOC for rapid HIV tests, and how does it compare to the likelihood ratio in terms of bounds and interpretability?

Key findings

• The SIM provides a unified framework compatible with both Shannon’s information theory and Fisher’s likelihood method, enabling semantic information quantification.
• The Degree of Confirmation (DOC) for rapid HIV tests is given by DOC = 1 - (1 - specificity)/sensitivity, which is bounded above by 1, unlike the unbounded likelihood ratio.
• The DOC increment provides a measure of evidence gain, resolving the Raven Paradox by showing that observing a non-black non-raven increases DOC only when the hypothesis is not already highly confirmed.
• Negative DOC values are assigned to excessive affirmations, false hypotheses, or lies, indicating semantic information loss or misinformation.
• For general hypotheses, such as those in GPS systems, the DOC can be optimized by adjusting the true-value of counterexamples, leading to a refined degree of belief.
• The SIM enables a formal, quantitative test of falsifiability by linking low LP and high truth value to high information content, supporting Popperian demarcation.