The logic of the probabilistic language

Article by Gianni Frisardi · Riccardo Azzali · Flavio Frisardi

Abstract

The text deals with the logic of probabilistic language applied to the medical field, highlighting how uncertainty is an intrinsic part of scientific practice. Through probabilistic and statistical concepts, efforts are made to manage and understand the uncertainties associated with medical theory and practice. The role of probability in the relationship between theory and observation is emphasized, distinguishing between subjective uncertainty and randomness. Subjective uncertainty concerns individuals' state of knowledge and belief, while randomness refers to the lack of a certain connection between cause and effect. In the medical approach, the importance of understanding and distinguishing between subjective and objective probability is discussed. Subjective probability reflects individual belief, while objective probability is based on data and empirical evidence. The concept of probabilistic-causal analysis is then further explored, which seeks to quantify the relationship between events and random processes in clinical diagnosis. A detailed exposition is presented on how conditional probabilities can be interpreted and how causal relevance partitioning can be used to formulate a differential diagnosis. Finally, the theme of interdisciplinarity in scientific research is addressed, highlighting the importance of an interdisciplinary approach to tackling complex problems. Fuzzy logic is also mentioned as a possible tool for managing uncertainty in medical contexts.

Subjective Uncertainty and Causality

This section examines the internal uncertainties that individuals may experience when faced with a diagnosis, using Mary Poppins as a fictional reference. The concepts of subjective and objective uncertainties are explored:

Subjective Uncertainty: This type of uncertainty is dependent on an individual's knowledge and beliefs. It is highlighted that objectivity in science is often a shared consensus of subjective views, termed as intersubjectivity.
Causality: The relationship between causality and uncertainty is discussed in the medical context. Using mathematical expressions, it is shown how medical phenomena do not always follow a deterministic pattern but are rather influenced by probabilistic factors.

Subjective and Objective Probability

This chapter revisits the discussions from Kazem Sadegh-Zadeh's work on the logic of medical language and applies it to the clinical case of Mary Poppins. It elaborates on how events are categorized as probable based on their randomness and subjective uncertainty:

Subjective Probability: Involves individual belief and varies according to the information available to the person making the assessment.
Objective Probability: In contrast, objective probability is based on empirical evidence and statistical data, reflecting a more quantifiable aspect of probability.

Probabilistic-Causal Analysis

Probabilistic-causal analysis is used to describe how clinical diagnoses are formulated. This involves assessing the likelihood of various conditions based on observed probabilities:

The text details the process of causal relevance and how it can be used to infer potential diagnoses from given data sets.
A specific focus is placed on the interpretation of conditional probabilities and how they can aid in the differential diagnosis.

Interdisciplinarity in Scientific Research

The importance of interdisciplinary approaches in scientific research is underscored, particularly in complex fields like medicine. The use of fuzzy logic as a method for handling uncertainty in medical contexts is proposed as an innovative approach to problem-solving across different disciplines.

Introdduction

Every scientific idea—whether in medicine, architecture, engineering, chemistry, or any other field—when implemented, is prone to small errors and uncertainties. Mathematics, through the lens of probability theory and statistical inference, aids in precisely managing and thereby mitigating these uncertainties. It must always be considered that in all practical scenarios, "the outcomes also depend on many other external factors to the theory," be they initial and environmental conditions, experimental errors, or others.

The uncertainties surrounding these factors render the theory-observation relationship probabilistic. In medical practice, two types of uncertainty predominantly impact diagnoses: subjective uncertainty and causality.^[1]^[2] Therefore, in this context, it becomes crucial to differentiate between these two uncertainties and to demonstrate that the concept of probability assumes different meanings in these contexts. We will endeavor to elucidate these concepts by connecting each critical step to the clinical approach that has been documented in previous chapters, particularly focusing on the dental and neurological domains in vying for diagnostic supremacy for our dear Mary Poppins.

Subjective Uncertainty and Causality

Let's imagine asking Mary Poppins which of the two medical colleagues—the dentist or the neurologist—is correct.

The question would generate a kind of agitation based on internal uncertainty; therefore, the notions of certainty and uncertainty refer to the subjective epistemic states of human beings, and not to states of the external world, because there is no certainty or uncertainty in that world. In this sense, as we have mentioned, there are an inner world and an external world that both do not adhere to canons of uncertainty, but rather to probability.

Mary Poppins may be subjectively certain or uncertain as to whether she is suffering from TMDs or a neuropathic or neuromuscular form of OP. This is because "uncertainty" is a subjective, epistemic state below the threshold of knowledge and belief; hence the term.

Subjective Uncertainty

Without a doubt, the term ‘subjective’ alarms many, especially those who aim to practice science by pursuing the noble ideal of ‘objectivity,’ as this term is perceived by common sense. Therefore, it is appropriate to make some clarifications on the use of this term in this context:

‘Subjective’ indicates that the probability assessment depends on the information status of the individual who performs it.

‘Objective’ does not mean arbitrary.

The so-called ‘objectivity,’ as perceived by those outside scientific research, is achieved when a community of rational beings shares the same state of information. But even in this case, one should more properly speak of ‘intersubjectivity’ (i.e., the sharing of subjective opinions by a group).

In clinical cases—precisely because patients rarely possess advanced notions of medicine—subjective uncertainty must be considered. Living with uncertainty requires us to adopt a probabilistic approach.

Causality

Causality indicates the lack of a certain connection between cause and effect. The uncertainty of a close union between the source and the phenomenon is among the most challenging problems in determining a diagnosis.

In a clinical case, a phenomenon $A(x)$ (such as a malocclusion, a crossbite, an openbite, etc.) is randomly associated with another phenomenon $B(x)$ (such as TMJ bone degeneration); when there are exceptions for which the logical proposition $A(x)\rightarrow B(x)$ is not always true (but it is most of the time), we will say that the relation $A(x)\rightarrow B(x)$ ......................