Cohen suggests that the standard of proof should be understood entirely as a matter of weight of evidence, which he argues is a matter of how many tests or challenges a factual hypothesis faces in court. It accounts for the legal conclusion of fact in relation to a representation of inductive probability inspired by the works of writers such as Francis Bacon and J.S. Mill. Inductive probability works differently from classical probability theory. It is based on an inductive support for the generalization of common sense, which makes it possible to draw the relevant conclusion. The inductive support of generalization is graduated according to the number of tests it has passed, or in other words, according to its degree of resistance to falsification by relevant variables. The inductive probability of an argument is equal to the degree of reliability of the inductive support for the generalization that covers the argument. One way to avoid the paradox of conjunction is to take the position that it should not be enough for each element to cross the probabilistic threshold. The applicant (or prosecutor) should only win if the likelihood of the applicant`s case (or the prosecutor`s office) as a whole exceeds the applicable probability threshold. In our example, the applicant should lose because the overall probability is less than 0.5. But the proposed solution is not satisfactory.
The overall likelihood required would then depend on the number of elements of the civil suit or criminal charge. The higher the number of elements, the greater the probability up to which, on average, each of them must be proven. This is considered arbitrary and therefore reprehensible. As two commentators have noted, the legal definition of theft contains more elements than murder. Criminal law is not the same in all countries. We can take the following as a convenient approximation of what the law is in some countries: murder is (1) an act that caused the death of a person, (2) that was committed with intent to cause death, and to constitute theft, there must be (1) an intent to take property, (2) dishonesty in the removal of property, (3) removal of property from the possession of another person, and (4) lack of consent of that person. Since the offence of theft contains twice as many elements as murder, the individual elements of theft should be proven with a much higher probability (so that the probability of their connection exceeds the overall threshold) than the individual elements for the much more serious crime of murder (Allen and Leiter 2001: 1504-5). This is intuitively unacceptable. In Section 3.1, we focused on the weight of evidence in terms of the weight of evidence of individual evidence. The concept of weight may also apply to all evidence presented at trial; Here, „weight“ is commonly referred to as „sufficiency of evidence.“  The law allocates the legal burden of proof between the disputing parties.
For example, in criminal proceedings, the presumption of innocence of the accused applies and the burden of proof is on the prosecution to prove guilt. In order to obtain a conviction, the evidence presented at trial must be sufficient to meet the standard of proof. As a general rule, a judgment is in favour of the party bearing the legal burden of proof only if the factual intermediary is satisfied, after taking into account all the evidence, that the applicable standard of proof is met. The standard of proof has been interpreted differently. The second definition is found in the United States Federal Rule of Evidence 401, which (as revised) states that evidence is relevant if it „tends to make a fact more or less likely than it would be without the evidence“ (emphasis added). The word „likely“ in these and other standard definitions is sometimes interpreted as carrying the mathematical meaning of probability.  In an editorial, Lempert cited this example to show how relevance affects the probability ratio. The prosecution adduces evidence that the author`s blood found at the scene is type A. The accused has the same blood type. Suppose fifty percent of the suspect population has type A blood.
If the defendant is indeed guilty, the probability that the blood found at the scene is type A is 1.0. But if he is indeed innocent, the probability of finding type A blood at the crime scene is 0.5 – that is, it corresponds to the background probability of type A blood of the suspect population. The probability ratio is the ratio of the first probability to the second – 1.0:0.5 or more simply 2:1. Evidence is considered relevant as long as the probability ratio is not 1:1 (Lempert 1977). If the ratio is 1:1, it means that the probability of obtaining evidence is the same as if the accused is guilty or innocent. The notion of relevance plays a central role in the legal determination of the facts. Thayer (1898:266, 530) articulates its importance in relation to two fundamental principles of the law of evidence: first, without exception, nothing irrelevant can be accepted as evidence by the court, and second, subject to numerous exceptions and limitations, what is relevant can be accepted by the court as evidence. Thayer`s views have been influential and find expression in legal sources, for example in Rule 402 of the Federal Rules of Evidence in the United States.  Thayer argues, and is now widely accepted, that relevance is a „logical“ rather than a legal concept; In section 2.1.3, we will examine this allegation and the contradiction expressed by Wigmore. In addition to the dissenting opinion for the time being, we will first look at possible conceptions of relevance in the traditional sense of logical relevance. Fifth, according to Haack, this is wrong in Bayesian theory.
What matters is not the strength of the investigator`s own faith. Rather, the standard of proof should be understood in terms of what the investigator may reasonably believe in light of the evidence presented, and the extent to which the belief is justified by the evidence. The evidence is legally sufficient if it substantiates the factual allegation complained of to the extent prescribed by law. Whether a factual allegation is supported by the evidence depends on the strength with which the evidence supports the allegation, the independence of the evidence, and the amount of relevant evidence available to the investigator (i.e., the completeness of the evidence – see additional explanation in section 3.3 below). Haack is against identifying mandate notes with mathematical probabilities. Mandate notes do not correspond to the axioms of standard probability theory. For example, if the evidence is weak, neither p nor non-p can be justified; In contrast, the probability of p and the probability of non-P must total 1. If the probability of p and the probability of q are both less than 1, the probability of p and q, which is the product of the probability of p and the probability of q, is less than the probability of both. On the other hand, the degree of the mandate for the connection of p and q may be higher than the mandate for both.  (Cf. Haack 2004, 2008a,b, 2012, 2014 on the legal application of their general theory of epistemology.
On his general theory of epistemology, see Haack 1993: chap. 4; Haack 2009: Kap. 4; Haack 2003: Kap. 3.) Note that different types of cases and legal proceedings have different burdens of proof. Clear and convincing evidence is a higher standard of proof than the balance of the standard of proof. We have just examined the first condition of the requirement, namely relevance. The fact that fact A is relevant to fact B is not sufficient to prove fact A in court. In addition, B must be a „substantial“ fact.
The relative importance of the facts in a particular case is determined by the law applicable to that case. In a criminal prosecution, this depends on the law that defines the crime with which the accused is charged and, in civil proceedings, the law that determines the elements of the prosecution against the accused (Wigmore 1983a, 15-19; Montrose 1954: 536–537). Given that the relevant utility depends on individual circumstances, such as the seriousness of the offence and the severity of the sentence, the adjudicative theoretical presentation of the standard of proof in both the simple and amended versions seems to lead to the conclusion that the probabilistic threshold should vary from case to case (Lillquist, 2002; Bartels, 1981; Laudan and Saunders, 2009; Ribeiro, 2019). In other words, the standard of proof should be flexible or floating. This view is seen as problematic. Even if the theory is correct, it does not necessarily follow that the exclusion rules should be abolished once the jury system is abolished.