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An empirical test is a decision procedure consisting of a modus tollens deduction from a set of one or several universally quantified theory statements expressible in a nontruth-functional hypothetical-conditional schema proposed for testing together with an antecedent particularly quantified description of the initial test conditions. These statements jointly conclude to a consequent particularly quantified description of a produced (predicted) test-outcome event, which is compared with the observed test-outcome description.

In order to express explicitly the dependency of the produced effect upon the realized initial conditions in an empirical test, the universally quantified theory statements can be schematized as a nontruth-functional hypothetical-conditional statement, i.e., as a statement with the logical form “For every A if A, then C.” The hypothetical-conditional statement represents a system of one or several universally quantified related theory statements or equations that describe a dependency of the occurrence of an event described by “C” upon the occurrence of an event described by “A”. The dependency may be expressed as the range of stochastic boundary limits for the values of predicted probabilities. For advocates who believe in the theory, the hypothetical-conditional statement is the theory-language context that contributes meaning parts to the complex semantics of the theory’s constituent descriptive terms including the terms common to the theory and test design. But the theory’s semantical contribution cannot be operative in the test for the test to be independent of the theory.

The antecedent “A” also includes the set of universally quantified statements of test design that describe the initial conditions that must be realized for execution of an empirical test of the theory together with the description of the procedures needed for their realization. These statements are always presumed to be true or the test design is rejected as invalid. They contribute meaning parts to the complex semantics of the terms common to theory and test design, and do so independently of the theory’s semantical contributions. The universal logical quantification indicates that any execution of the experiment is but one of an indefinitely large number of possible test executions, whether or not the test is repeatable at will.

When the test is executed, the logical quantification of “A” is changed to particular quantification to describe the realized initial conditions in the individual test execution. In a mathematically expressed theory the test execution consists in measurement actions and assignment of the resulting measurement values to the variables in “A”. In a mathematically expressed single-equation theory, “A” includes the independent variables in the equation of the theory. In a multi-equation system whether recursively structured or simultaneous, the exogenous variables are assigned values by measurement, and are included in “A”. In longitudinal models with dated variables the lagged-values of endogenous variables that are the initial condition for a test and that initiate the recursion through successive iterations to generate predictions, must also be included in “A”.

The consequent “C” represents the set of universally quantified statements of the theory that describe the predicted outcome of every correct execution of a test design. Its logical quantification is changed to particular quantification to describe the predicted outcome in an individual test execution. In a mathematically expressed single-equation theory, “C” is the dependent variable in the equation of the theory. When no value is assigned to any variable, the equation is universally quantified. When the prediction value of a dependent variable is calculated from the measurement values of the independent variables, it becomes particularly quantified. In a multi-equation theory, whether recursively structured or a simultaneous-equation system, the solution values for the endogenous variables are included in “C”. In longitudinal models with dated variables the current-dated values of endogenous variables that are calculated by solving the model through successive iterations are included in “C”.

The conditional statement of theory does not say “For every A and for every C if A, then C”. It only says “For every A if A, then C”. In other words the conditional statement of theory only expresses a sufficient condition for the production of the phenomenon described by C upon realization of the test conditions given by “A”, and not a necessary condition. Alternative test designs described in “A” are sufficient to produce “C”. This may occur for example, if there are theories proposing alternative causal factors for the same outcome described in “C”. Or if there are equivalent measurement procedures or instruments described in “A” that produce alternative measurements falling within the range of their measurement errors, such that the errors are small relative to the predicted values described by “C”.

Let another particularly quantified statement denoted “O” describe the observed test outcome of an individual test execution. The report of the test outcome “O” shares vocabulary with the prediction statements “C”. But the semantics of the terms in “O” is determined exclusively by the universally quantified test-design statements rather than by the statements of the theory, and thus for the test its semantics is independent of the theory’s semantical contribution. In an individual predictive test execution “O” represents observations and/or measurements made and measurement values assigned after the prediction is made, and it too has particular logical quantification to describe the observed outcome resulting from the individual execution of the test. There are three outcome scenarios:

Scenario I: If “A” is false in an individual test execution, then regardless of the truth of “C” the test execution is simply invalid due to a scientist’s failure to comply with its test design, and the empirical adequacy of the theory remains unaffected and unknown. The empirical test is conclusive only if it is executed in accordance with its test design. Contrary to the logical positivists, the truth table for the truth-functional Russellian logic is therefore not applicable to testing in empirical science, because in science a false antecedent, “A”, does not make the hypothetical-conditional statement true by logic of the test.

Scenario II: If “A” is true and the consequent “C” is false, as when the theory conclusively makes erroneous predictions, then the theory is falsified, because the hypothetical conditional “For every A if A, then C” is false. Falsification occurs when the statements “C” and “O” are not accepted as describing the same thing within the range of vagueness and/or measurement error, which are manifestations of empirical underdetermination. The falsifying logic of the test is the modus tollens argument form, according to which the conditional-hypothetical statement expressing the theory is falsified, when one affirms the antecedent clause and denies the consequent clause. This is the falsificationist philosophy of scientific criticism advanced by Charles S. Peirce, the founder of pragmatism, and later advocated by Karl Popper. Readers seeking more on Popper are referred to BOOK V below.

The response to a falsification may or may not be attempts to develop a new theory. Scientists will not simply deny a falsifying outcome of a test, if they have accepted its test design and test execution. Characterization of falsifying anomalous cases is informative, because it contributes to articulation of a new problem that a new and more empirically adequate theory must solve. Some scientists may, as Kuhn said, simply believe that the anomalous outcome is an unsolved problem for the tested theory without attempting to develop a new theory. But such a response is either an ipso facto rejection of the tested theory, a de facto rejection of the test design or a disengagement from attempts to solve the problem. And contrary to Kuhn this procrastinating response to anomaly need not imply that the falsified theory has been given institutional status, unless the science itself is institutionally retarded. Readers seeking more on Kuhn are referred to BOOK VI below.

Scenario III: If “A” and “C” are both true, the hypothetical-conditional statement expressing the tested theory is validly accepted as asserting a causal dependency between the phenomena described by the antecedent and consequent clauses. The hypothetical-conditional statement does not merely assert a Humean constant conjunction. Causality is an ontological category describing a real dependency, and the causal claim is asserted on the basis of ontological relativity due to the empirical adequacy demonstrated by the nonfalsifying test outcome. Because the nontruth-functional hypothetical-conditional statement is empirical, causality claims are always subject to future testing, falsification, and then revision. This is also true when the conditional expresses a mathematical function.

Furthermore if the test design is modified such that it changes the characterization of the subject of the theory, then even a nonfalsifying test outcome should be reconsidered and the theory should be retested for the new definition of the subject. If the retesting produces a falsifying outcome, then the new information in the modification of the test design has made the terms common to the two test designs equivocal and has contributed parts to alternative meanings. But if the test outcome is not falsification, the new information is merely new parts added to the univocal meaning of the terms common to the old and new test-design language. Such would be the case if the new information were what the positivists called a new “operational definition”, as for example a new and additional way to measure temperature for extreme values that cannot be measured by the old operation, but which yields the same temperature values within the range of measurement errors, where the alternative operations produce overlapping results.

On the contemporary pragmatist philosophy a theory that has been tested is no longer theory, once the test outcome is known and the test execution is accepted as compliant with the test design. The nonfalsifying test outcome makes the theory empirically warranted, and it is thus deemed a scientific law until it is tested again at some future time and possibly falsified. The law is still hypothetical because it is empirical, but it is less hypothetical than it had been as a theory proposed for testing. The law may thereafter be used either in an explanation or in a test design for testing some other theory. But if the theory has been falsified, it is merely rejected language, although it may still be useful in application for its lesser empirical adequacy, realism and truth.

For example the elaborate engineering documentation for the Large Hadron Collider at CERN, the Conseil Européen pour la Recherche Nucléaire, is based on previously tested science. After installation of the Collider is complete, the science in that engineering is not what is tested when the particle accelerator is operated for the microphysical experiments, but rather is presumed true and contributes to the test-design language for experiments performed with the accelerator.