Reformulating the Second Law

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Paul Patton

IHPST, University of Toronto [email protected]

Nicholas Overgaard

Hakob Barseghyan

theoretical scientonomy, the second law, employed method, theory assessment, assessment outcomes, theory acceptance, method satisfaction, method, theory

The current formulation of the second law is flawed since it does not specify the causal relations between the outcomes of theory assessment and the actual acceptance/unacceptance of a theory; it merely tells us that a theory was assessed by the method employed at the time. We propose a new formulation of the

second law: “If a theory satisfies the acceptance criteria of the method actually employed at the time, then

it becomes accepted into the mosaic; if it does not, it remains unaccepted; if it is inconclusive whether the theory satisfies the method, the theory can be accepted or not accepted.” This new formulation makes the

causal connection between theory assessment outcomes and cases of theory acceptance/unacceptance explicit. Also, this new formulation is not a tautology because it forbids certain logically possible scenarios, such as a theory satisfying the method of the time yet remaining unaccepted. Finally, we outline what inferences an observational scientonomist can make regarding theory assessment outcomes from the record of accepted theories.

Patton, P; Overgaard, N. & Barseghyan, H. (2017). Scientonomy 1, 29-39 https://www.scientowiki.com/Patton, Overgaard, and Barseghyan (2017)

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30 Patton, Overgaard, & Barseghyan

Thesecond law of scientific change, also known as thelaw of theory acceptance, is a core element of the current scientonomic theory (Barseghyan, 2015). The law is supposed to specify the conditions under which a contender theory is accepted into a scientific mosaic. As currently formulated, it states that “in order to become accepted into the mosaic, a theory is assessed by the method actually employed at the time” (Barseghyan, 2015, p. 129). However, this formulation does not explicitly specify the relationship between the acceptance of a theory and the outcome of its assessment. While the law says that a theory is assessed by the method actually employed at the time, it does not specify under which conditions it becomes accepted or remains unaccepted. Thus, the current formulation of the law is flawed.

The problem becomes deeper because in later deductions a more specific formulation of the law is tacitly assumed. In particular, the deductions of the necessary and possible mosaic split theorems assume that the second law provides a clear account of the conditions under which theories become accepted or remain unaccepted (Barseghyan, 2015, pp. 202-208). Consider the following diagram from the section Mosaic Split and Mosaic Merge (Barseghyan, 2015, p. 205):

The tacit assumption here is that the content of this diagram somehow follows from the second law. But this is not actually the case; the current law only stipulates that theory assessment has to do with the method of the time. It says nothing about specific outcomes of assessment and how exactly they determine theory acceptance or unacceptance.

In this paper, we propose a new formulation of the second law that corrects this flaw by clearly stating the effect that each assessment outcome has on theories in a mosaic. We also show that this new formulation of the second law is not a tautology. Finally, we consider its role in inferring assessment outcomes from the actual changes in a mosaic.

The process of theory change as posited by the current scientonomic theory can be broken down into two logically separable parts. The assessment outcome of a theory is one part, while the theory’s acceptance or unacceptance is the other.

Let us first consider assessment outcomes. When a theory is assessed by a method, one of the three following outcomes can obtain:

Theory Assessment

Outcomes

Accept

Not accept

Inconclusive

Actual Courses of Events

Theory remains unaccepted

Theory becomes accepted

Impossible Necessary

Necessary Impossible

Possible Possible

Mosaic splits

Impossible

Possible

Outcome: Inconclusive ≡

It is unclear whether or not the requirements of the method employed at the time

are met.

Outcome: Not Satisfied ≡

The theory is deemed to conclusively not meet the requirements of the method

employed at the time.

Outcome: Satisfied ≡

The theory is deemed to conclusively meet the requirements of the method

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www.scientojournal.com • Reformulating the Second Law 31

In other words, the theory can either conclusively satisfy the requirements of the method, or it can conclusively fail to satisfy the requirements of the method, or the outcome of the assessment can be inconclusive.

As for theory acceptance/unacceptance, when confronted with a contender theory, it is possible, in principle, for a scientific community to act in three different ways:

• Acceptance: the contender theory is accepted into the mosaic. • Unacceptance: the contender theory remains unaccepted.

• Mosaic Split: the mosaic splits into two mosaics; the contender theory becomes accepted into one mosaic and remains unaccepted in the other.

These are the three possible courses of events that can obtain during the process of scientific change.

It is important not to confuse assessment outcomes with theory acceptance/unacceptance. In a sense, an assessment outcome is what the community thinks about the merits of the theory, while acceptance or unacceptance is what the community actually does with the theory. Thus, saying that a theory satisfies the requirements of the time is one thing; saying that it is actually accepted by the community of the time is another. Similarly, the fact that a theory does not satisfy the method of the time does not logically imply that a theory necessarily remains unaccepted.

The assessment outcomes and acceptance/unacceptance of a theory are summarized in the table below:

Now, it is the task of the second law to specify what happens to a mosaic when a specific assessment outcome obtains, i.e. to causally connect assessment outcomes with theory acceptance/unacceptance. Although the posited relationships between assessment outcomes and theory acceptance/unacceptance are implied in the tables on pages 205-207 of The Laws of Scientific Change, it is safe to say that the current second law does not distinguish the phenomenon of assessment outcome from that of theory acceptance/unacceptance. In fact, the definitions of the three assessment outcomes on page 199 of The Laws of Scientific Change result from an unfortunate conflation of assessment outcomes with theory acceptance/unacceptance. It is not clear whether they refer to the possible outcomes of a theory’s assessment by a method, or whether they refer to what happens to the theory in terms of its acceptance/unacceptance.

Thus, it is not surprising that the current formulation of the second law fails to state explicitly what exactly happens to a theory in terms of its acceptance/unacceptance when a specific assessment outcome obtains. Consider the current formulation of the second law (Barseghyan, 2015, p. 129):

What is wrong with the current formulation of the second law is that it does not actually specify the effect each assessment outcome has on the state of a mosaic. In particular, it does not say that the satisfaction of the method

Theory Acceptance/Unacceptance Assessment Outcomes

Satisfied

Not Satisfied

Inconclusive

Acceptance

Unacceptance

Mosaic Split

2nd Law: Theory Acceptance

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of the time leads to a theory’s acceptance. Similarly, it does not say that when a theory does not satisfy the requirements of the time it remains unaccepted. A proper formulation of the second law must specify the relationship between assessment outcomes and the actual changes to a mosaic, so that the acceptance or non-acceptance of a theory is caused by a certain assessment outcome. Otherwise, the current second law merely expresses two disconnected events: that a theory was assessed (the outcome of that assessment remaining unstated) and that it was accepted and/or not accepted. Crucially, both assessment outcomes and the acceptance/unacceptance of a theory are historical events, and as such, can be subject to historical analysis. In some cases, these events are separable in time; in others, method satisfaction and theory acceptance occur almost simultaneously. What is important for our sake is that assessment outcomes and theory acceptance/unacceptance are logically separable, and that the current second law makes no such logical distinction.

The table below illustrates that the causal relations between assessment outcomes and theory acceptance/unacceptance have not been specified:

Those are exactly the relations that any proper law of theory acceptance should specify. In the following section, a more precise formulation of the second law is proposed, and the relationship between assessment outcomes and theory acceptance/unacceptance is reanalyzed.

Let us first consider an obvious reformulation: “A theory is accepted into the mosaic if and only if it satisfies the requirements of the method actually employed at the time”. Albeit succinct, this formulation is only workable for cases in which the outcome of theory assessment is conclusive. It is, however, clearly problematic for inconclusive assessment outcomes. Barseghyan defines an inconclusive assessment outcome as one which “allows for a theory to be accepted, but doesn’t dictate so” (Barseghyan, 2015, p. 199). These are cases where a theory becomes accepted not because it conclusively met the requirements of the time, but because the assessment outcome was not clear to the community itself. The reformulation does not allow for such a scenario. In fact, it clearly denies that a theory can become accepted while its assessment outcome is inconclusive, since it says a theory is accepted if and only if it satisfies the requirements of the method.

However, the lengthy discussion of scientific underdetermination and mosaic split (Barseghyan, 2015, pp. 196-216) makes it clear that inconclusive assessment outcomes are an important aspect of the process of scientific change. Inconclusive assessments may arise in cases when the method of assessment is vague. Barseghyan argues convincingly that such cases might exist. For example, the Aristotelian-medieval method required that theories be formulated in axiomatic deductive form and that the axioms be intuitively true propositions (Barseghyan, 2015,

Satisfied

Not Satisfied

Inconclusive

Acceptance

Unacceptance

Mosaic Split

The causal relations between assessment outcomes and theory acceptance/unacceptance are not specified by the current second law.

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p. 213). The notion of intuitive truth is likely to yield a great deal of difference of opinion, as different people have different intuitions. The seeming proliferation of mosaics among Ancient Greek schools of thought may have also been due to the vagueness of such methods (Barseghyan, 2015, p. 237). Therefore, inconclusiveness must be accommodated by a proper formulation of the second law; a more complex formulation is needed.

Here is the formulation that we suggest:

The reader can easily verify that the new formulation of the second law, unlike its predecessor, is capable of underwriting the causal relationships between assessment outcomes and the effects these outcomes have on a mosaic. It explicitly connects the three possible assessment outcomes with the three possible courses of events. In particular, it stipulates that the conclusive “satisfied” outcome necessitates the theory’s acceptance. Similarly, the conclusive “not satisfied” outcome always results in the theory’s remaining unaccepted. And, finally, an inconclusive assessment outcome can lead to the theory’s acceptance or unacceptance. The relationships between assessment outcomes and the actual course of events specified by the new formulation are as follows:

This diagram suggests that the new second law is not a tautology, as it clearly forbids certain logically possible relations. In The Laws of Scientific Change, Barseghyan writes that “the second law is not a law in the traditional sense; for normally a law is supposed to have some empirical content, i.e. its opposite should be conceivable, at least in principle. Obviously, the [current] second law is a tautology, since it follows from the definition of employed method. My only excuse for presenting it as law is that it is too important to be lost in the thickets of definitions” (Barseghyan, 2015, p. 129, footnote 18). Leaving the question of whether the current second law is or is not a tautology aside, the important point is that the new formulation is clearly not a tautology.

A tautology is a statement that is true in all logically possible worlds. If we can coherently imagine a logically possible scenario forbidden by the second law, then we can conclude that the second law is not a tautology. Now, it is clear from the new formulation that the second law forbids certain logically possible scenarios. One scenario forbidden by this formulation is the case when a theory satisfies the method of the time and yet remains unaccepted. The new second law says that such a situation is impossible. Similarly, the new formulation forbids

2nd_{Law: Theory Acceptance}

If a theory satisfies the acceptance criteria of the method employed at the time, it becomes accepted into the mosaic;

if it does not, it remains unaccepted; if assessment is inconclusive, the theory can be accepted or not accepted.

Satisfied

Not Satisfied

The new second law specifies the causal relationships between assessment outcomes and theory acceptance/unacceptance. Among other things, it shows that if the assessment outcome is

inconclusive any one of the three scenarios is a possibility. Inconclusive

Acceptance

Unacceptance

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the case when a theory does not satisfy the method of the time and yet it becomes accepted. There are also several other forbidden scenarios illustrated by the following diagram:

It is not hard to imagine a scenario forbidden by the second law. Consider a hypothetical community with employed Method 1. Suppose the community assesses Theory 1, and Theory 1 satisfies the requirements of Method 1. Suppose also that Theory 1 remains unaccepted by the community. In this case, the new second law would be clearly violated. Such a scenario seems logically conceivable, as it does not seem self-contradictory. That is, it seems quite conceivable that a theory can satisfy the requirements of the method and yet remain unaccepted.

However, this scenario is problematic given the current definition of employed method. According to the current definition, “a method is said to be employed at time t if, at time t, theories become accepted only when their acceptance is permitted by the method” (Barseghyan, 2015, p. 54):

In our hypothetical case, Method 1 could be said to be employed by our hypothetical community if and only if Theory 1 were assessed by it and consequently became accepted. If Theory 1 were to remain unaccepted (as our case suggests), it would indicate that Method 1 was not employed in the first place. So our hypothetical case is actually impossible given the current definition of employed method.

The major flaw of the current definition is that it defines method employment through its indicators. In other words, the definition confuses method employment with our scientonomic ways of detecting method employment, despite the fact that employed methods can, in principle, be detected in many ways. That is, we can, in principle, detect the fact of a method’s employment by analyzing the record of transitions from one accepted theory to the next in a particular community at a particular time or, alternatively, we can try to infer the method of the time from our knowledge of the body of accepted theories using the third law (recall, according to the third law, all employed methods are deductive consequences of theories accepted at the time). However, the current definition of employed method equates method employment with one particular means of identifying method employment. It is as though a type of subatomic particle were to be defined through the trace it leaves on a photographic plate,

Satisfied

Not Satisfied

The new second law forbids the following four logical possibilities:

• Satisfied → Unacceptance • Satisfied → Mosaic Split • Not Satisfied → Acceptance • Not Satisfied → Mosaic Split Inconclusive

Acceptance

Unacceptance

Mosaic Split

A method is said to be employed at time t if, at time t, theories become accepted only when their

acceptance is permitted by the method.

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despite the fact that it can also be detected in other ways, such as with a bubble chamber or cloud chamber. Evidently, the fundamental attributes of a particle are one thing (i.e. mass, spin, electric charge, etc.), while the means that we currently use to detect the particle and its attributes is another. The two clearly do not coincide because a particle can be detected using many different techniques (e.g. by photographic plate, bubble chamber, cloud chamber, etc.). Likewise, in our case, method employment is distinct from the means of its identification. Therefore, the current definition of employed method is deficient; a new definition is needed.

It is safe to say that another tacit definition of employed method permeates the scientonomic discourse. On many occasions, when talking about a method being employed, what we are really referring to is the fact that the community of the time had certain expectations towards new theories. For instance, when we say that the Aristotelian-Medieval community of natural philosophers employed the method of intuition schooled by experience, what we are really referring to is the fact that this community expected new theories to be intuitively true (Barseghyan, 2015, pp. 143-145). Similarly, when we say the double-blind trial method is employed, we mean that the community expects new drugs to be tested in double-blind trials (Barseghyan, 2015, pp. 134-142).

This tacit definition of employed method has been repeatedly conflated with the official definition of the term given on page 54 of The Laws of Scientific Change. For example, in an attempt to explicate the requirements of the Aristotelian-Medieval method concerning axioms, Barseghyan notes that “[t]he axioms were expected [emphasis added] to be intuitive” (Barseghyan, 2015, p. 144). In the same way, when trying to explicate the method of contemporary empirical science, Barseghyan asks, “[i]ndeed, what are the implicit expectations [emphasis added] of our contemporary scientific community?” (Barseghyan, 2015, p. 145). Evidently, Barseghyan seeks a community’s expectations when explicating the method of the time. But a community’s expectations are never even mentioned in the current definition of employed method.

Our suggestion is to make this usage of the term explicit by redefining employed method. The new definition is:

As we can see, this definition makes no mention of any possible indicators of method employment, for it takes into account that these indicators might be both multiple and changeable. Importantly, with this new definition, it becomes logically conceivable that a theory can satisfy the method of the time, yet remain unaccepted – just as our hypothetical scenario suggested. Since Method 1 constitutes the actual expectations of the community, there is no logical contradiction in saying that Theory 1 met these expectations without becoming accepted. Albeit clearly bizarre, this scenario is nevertheless logically conceivable.

Thus, we should appreciate that the new second law is not a tautology, as it clearly forbids certain logically conceivable scenarios.

Suppose an observational scientonomist tries to make sense of a certain historical transition – a certain instance of scientific change. On the one hand, the observational scientonomist needs to be able to establish what theory was accepted before and after the transition. In other words, a historian requires indicators of theory acceptance. On the other hand, the observational scientonomist also needs to establish the outcomes of theory assessments during that transition. In other words, the historian needs certain indicators that would tell her what the outcomes of theory assessment were.

In the current state of the field, we have some knowledge about the indicators of theory acceptance. It has been suggested that we can identify which course of events actually occurred from historical evidence. Such evidence could include discussions of the theory in publications that are intended to present the accepted views

A method is said to be employed if its requirements constitute the actual

expectations of the community.

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of the scientific community to a broad audience, such as encyclopedias, or materials used in scientific education and training, such as textbooks and curricula (Barseghyan, 2015, pp. 113-117). While we know a thing or two about what sorts of historical artifacts might be indicative of theory acceptance, we currently lack an understanding as to what sorts of direct indicators there might be of the outcomes of theory assessment. For instance, it might be that an observational scientonomist can infer the outcome of theory assessment from post-debate/conference/etc. transcripts. Whether or not this is possible remains an open question. That said, there is an indirect way to infer the outcome of theory assessment from the actual course of events by means of the new formulation of the second law.

Initially, Barseghyan proposed that the second law can be used to infer implicit methods from the fact of a theory’s acceptance (Barseghyan, 2015, pp. 117-120, 129-130). He writes that “we can only learn about our actual expectations by analyzing transitions in our mosaics, i.e. transitions from one accepted theory to the next. So, when a theory becomes accepted, the question that we should ask is not whether the theory violated our actual requirements (for it obviously couldn’t), but rather what the actual requirements that allowed the theory to become accepted were” (Barseghyan, 2015, p. 130). Barseghyan’s idea was that the method of the time can be somehow inferred from actual cases of theory acceptance/unacceptance. Barseghyan illustrated this intended use of the second law for observational scientonomists by means of a well-known example. It followed from Einstein’s theory of general relativity that a strong gravitational field, such as that of the sun, could detectably bend light. Einstein’s theory became accepted around 1920 after this prediction was confirmed by the astronomer Arthur Eddington in 1919. “From that episode”, Barseghyan writes, “we can reconstruct what the actual requirements of the time were” by means of the second law (Barseghyan, 2015, p. 130). To reconstruct those requirements, i.e. the method of the time, we can first supposedly infer from the second law that an accepted theory received a favorable assessment according to the method of the time. Knowing the acceptance of Einstein’s theory of general relativity around 1920, and knowing that the acceptance took place after Eddington’s observations confirmed a novel prediction of Einstein’s theory, we can supposedly further infer that the method of the time likely had to do with confirmed novel predictions. In other words, the second law was supposed to make it possible to infer the actual method employed for theory assessment (i.e. that the method of the time actually requires confirmed novel predictions) by studying which theories garnered acceptance during a particular era.

While it might eventually be possible to infer the requirements of the method of the time from the record of accepted theories, at this stage we can only reasonably argue that the second law allows us to make some inferences concerning the outcome of theory assessment.

We will first apply the proposed reformulation of the second law to the simple case of assessing a single contender theory, T1, for inclusion in the mosaic. The possible inferences are detailed below:

Note that both the acceptance and the non-acceptance of T1 can be produced by more than one assessment

outcome. For the case of acceptance, we can infer either that T1 has satisfied the employed method or that its

assessment was inconclusive. For the case of unacceptance, we can likewise infer either that T1 failed to satisfy

the employed method, or that its assessment was inconclusive. The only case where a unique conclusion can be drawn is that of a particular kind of mosaic split where one part of the community accepts the contender theory while the other maintains the current state of the mosaic – in which case we can conclude that assessment was inconclusive. The ambiguity of inference in the first two cases is the consequence of inconclusiveness, which can

Theory Acceptance/ Unacceptance

T1 Accepted

T1 Unaccepted

Mosaic Split Occurred

Inferred Assessment Outcomes

Satisfied Not Satisfied

Possible Impossible

Impossible Possible

Impossible Impossible

Inconclusive

Possible

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result either in a theory being accepted, remaining unaccepted, or causing a mosaic split. So long as one supposes that theory acceptance is underdetermined by the employed method, inference of the assessment outcomes from the observed cases of theory acceptance/unacceptance is marred by ambiguity.

We will now consider the more complex case in which two or more new theories are being assessed and an accepted theory is being challenged. The full set of possibilities that can arise when an accepted theory T0 is

challenged by two mutually-incompatible rivals T1 and T2 is summarized in the table below (cf. Barseghyan, 2015,

pp. 206-207):

For the six possible courses of events considered in the table, there are none that can be uniquely produced only by a single assessment outcome. This is, in all cases, due to the possibility of an inconclusive assessment outcome.

The new formulation is a distinct improvement over its predecessor. The old formulation did not allow us to infer that an accepted theory satisfied the employed method of the time, nor that a theory that was not accepted failed to do so. The new formulation can allow us to do so, but not from the mere fact of a theory’s acceptance, or unacceptance as it remains possible that the satisfaction of the method was inconclusive. So, to truly infer theory assessment outcomes from cases of theory acceptance/unacceptance, we require additional information as to whether or not the theory’s assessment is likely to have been inconclusive. Given this information along with facts about the theory’s acceptance, we can draw probable conclusions regarding the outcome of the theory’s assessment.

Establishing the assessment outcomes is an important step in our quest towards explicating the actual requirements of the method of the time. There remains an open question as to how an observational scientonomist can infer the requirements of the method of the time from knowledge of assessment outcomes, the record of accepted theories, and additional relevant historical data. This is a topic for future research.

We have argued that the current formulation of the second law is flawed because it does not specify the actual causal relations between the outcomes of theory assessment and the actual acceptance or non-acceptance of a theory. Instead, the current formulation merely tells us that a theory was assessed by the method of the time. We have proposed a new formulation of the second law that posits a causal connection between theory assessment outcomes and theory acceptance/unacceptance. The law states that if a theory satisfies the method actually employed at the time, then it will be accepted into the mosaic. If it does not, it will not. If it is inconclusive whether the theory satisfies the method, the theory can be accepted. We have shown that this new formulation is not a tautology, for it clearly forbids certain scenarios, such as a theory satisfying the method of the time and yet remaining unaccepted. Finally, we have outlined what inferences an observational scientonomist can make regarding theory assessment outcomes from the record of accepted theories. As for inferring the requirements of the method of the time, more work needs to be done.

Both Unaccepted

Only T₁ Accepted

Only T₂ Accepted

T₀ - T₁ Split Occurred

T₀ - T₂ Split Occurred

T₁ - T₂ Split Occurred

T₁ Satisfied

T₂ Satisfied

Impossible Impossible

T₁ Not Satisfied Possible Impossible Possible Impossible Possible T₂ Not Satisfied Possible Possible Impossible Possible Impossible T₁ Inconcl. Possible Possible Possible Necessary Possible T₂ Inconcl. Possible Possible Possible Possible Necessary

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Thus, we suggest the following modifications:

Accept the following reformulation of the second law:

• The Second Law: if a theory satisfies the acceptance criteria of the method employed at the time, it becomes accepted into the mosaic; if it does not, it remains unaccepted; if assessment is inconclusive, the theory can be accepted or not accepted.

Accept the following definitions of theoryassessment outcomes:

• Outcome: Satisfied ≡ the theory is deemed to conclusively meet the requirements of the method employed at the time.

• Outcome: Not Satisfied ≡ the theory is deemed to conclusively not meet the requirements of the method employed at the time.

• Outcome: Inconclusive ≡ it is unclear whether or not the requirements of the method employed at the time are met.

Accept the following ontology of theoryassessment outcomes:

• The three possible outcomes of theory assessment are “satisfied”, “not satisfied”, and “inconclusive”.

Accept the following redefinition of employed method:

• Employed Method ≡ a method is said to be employed if its requirements constitute the actual expectations of the community.

Reject:

• The previous formulation of the second law.

• The previous definitions of theoryassessment outcomes. • The previous ontology of theoryassessment outcomes. • The previous definition of employed method.

2nd_{Law: Theory Acceptance}

If a theory satisfies the acceptance criteria of the method employed at the time, it becomes accepted into the mosaic;

if it does not, it remains unaccepted; if assessment is inconclusive, the theory can be accepted or not accepted.

Outcome: Inconclusive ≡

It is unclear whether or not the requirements of the method employed at the time

are met.

Outcome: Not Satisfied ≡

The theory is deemed to conclusively not meet the requirements of the method

Outcome: Satisfied ≡

The theory is deemed to conclusively meet the requirements of the method

A method is said to be employed if its requirements constitute the actual

expectations of the community.

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Contingent upon the acceptance of the preceding modification [Sciento-2017-0004], accept that the new second law is not a tautology.

Contingent upon the acceptance of modification [Sciento-2017-0004], accept the following set of inferences of theory assessment outcomes from the acceptance or unacceptance of a single contender:

Also accept the following set of inferences of theory assessment outcomes from the acceptance or unacceptance of two contender theories:

Barseghyan, H. (2015). The Laws of Scientific Change. Springer.

T1 Accepted

T1 Unaccepted

Mosaic Split Occurred

Satisfied Not Satisfied

Impossible Impossible

Inconclusive

Possible

Necessary

Both Unaccepted

Only T₁ Accepted

Only T₂ Accepted

T₀ - T₁ Split Occurred

T₀ - T₂ Split Occurred

T₁ - T₂ Split Occurred

T₁ Satisfied

T₂ Satisfied

T₁ Not Satisfied

Possible

Impossible

Possible

Impossible

Possible

T₂ Not Satisfied

Possible

Impossible

Possible

Impossible

T₁ Inconcl.

Possible

Necessary

Possible

T₂ Inconcl.

Possible

Necessary