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      For example there are transformation rules for colloquial discourse that change a sentence from declarative to interrogative mood. The object language of science is typically expository, and philosophy of science therefore principally considers the declarative mood for the descriptive discourse as in theories and laws. The imperative mood is also of interest for describing procedural instructions in test designs for executing the tests.

      Transformation rules are used in logical and mathematical deductions. But logic and mathematical rules are intended not only to produce new grammatical sentences but also to guarantee truth transferability from one set of sentences or equations to another to generate theorems, usually by the transformation rule of substitution that makes logic extensional.

      In 1956 Herbert Simon developed an artificial-intelligence computer system named LOGIC THEORIST, which operated with his “heuristic-search” system design. This system developed deductive proofs of the theorems in Alfred N. Whitehead and Bertrand Russell’s Principia Mathematica. The symbolic-logic formulas are object language for this system. But Simon correctly denies that the Russellian symbolic logic is an effective metalanguage for the design of discovery systems.

      Transformation rules are of greater interest to linguists, logicians and mathematicians than to contemporary philosophers of science, who recently have been more interested in formation rules for generative-grammar discovery systems.

      3.06 Mathematical Language

      The syntactical dimension of mathematical language includes mathematical symbols and the formation and transformation rules of the various branches of mathematics. Mathematics applied in science is object language for which the syntax is supplied by the mathematical formalism. Whenever possible the object language of science is mathematical rather than colloquial, because measurement values for variables enable the scientist to quantify the error in his theory, after estimates are made for the range of measurement error, usually by repeated execution of the measurement procedure.

      3.07 Logical Quantification in Mathematics

      Mathematical expressions in science are universally quantified when descriptive variables have no associated numerical values, and are particularly quantified when numeric values are associated with any of the expression’s descriptive variables either by measurement or by calculation.

      Like categorical statements, mathematical equations are explicitly quantified logically as either universal or particular, even though the explicit indication is not by means of the syncategorematic logical quantifiers “every”, “some” or “no”. An equation in science is universally quantified logically when none of its descriptive variables are assigned numeric values. Universally quantified equations may contain mathematical constants as in some theories or laws. An equation is particularly quantified logically by associating measurement values with any of its descriptive variables. A variable may then be said to describe an individual measurement instance.

      When a numeric value is associated with a descriptive variable by computation with measurement values associated with other descriptive variables in the same mathematical expression, the variable’s calculated value may be said to describe an individual empirical instance. In this case the referenced instance has not been measured but depends on measurements associated with other variables in the same equation.

      Individual empirical instances are calculated when an equation is used to make a numerical prediction. The individual empirical instance is the predicted value, which makes an empirical claim. In a test it is compared with an individual measurement instance, which is the test-outcome value made for the same variable. The individual empirical instance made by the predicting equation is not said to be empirical because the predicting equation is correct or accurate, but rather because the predicting equation makes an empirical claim, which may be falsified by the empirical test.

      B. SEMANTICS

      3.08 Semantical Dimension

      Semantics refers to the meanings associated with syntactical symbols.

      Semantics is the second of the four dimensions, and it includes the syntactical dimension. Language viewed in the semantical metalinguistic perspective is said to be “semantically interpreted syntax”, which is merely to say that the syntactical symbols have meanings associated with them.

      3.09 Nominalist vs. Conceptualist Semantics

      Both nominalism and conceptualism are represented in contemporary pragmatism. There are several variations of nominalism, but all contemporary nominalist philosophers advocate a two-level semantics, which in written language consists only of syntactical structures and the ontologies that are referenced by the structures, or as Quine says “word and object”. The two-level semantics is also called a referential theory of semantics, because it excludes any mid-level mental representations variously called ideas, meanings, significations, concepts or propositions. Therefore on the nominalist view language purporting to reference nonexistent fictional entities is semantically nonsignificant, which is to say literally meaningless.

      On the alternative three-level view terms symbolize universal meanings, which in turn signify such aspects of extramental reality as attributes, and reference ontologies that include individual entities. When we are exposed to the extramental realities, they are distinguishable by the senses as perceived stimuli, which in turn are synthesized by the brain and registered in memory. The sense stimuli deliver information revealing similarities and differences in reality. The signified attributes are similarities found by perception, and the referenced entities manifesting the attributes are recognized by invariant continuities found in perceived change. The signification is always more or less vague, and the reference is therefore always more or less indeterminate or what Quine calls “inscrutable”. The three-level view is also called a conceptualist thesis of semantics.

      The philosophy of nominalism was common among many positivists, although some like the logical positivist Carnap maintained a three-level semantics. In Carnap’s three-level semantics descriptive terms symbolize what he called “intensions”, which are concepts or meanings effectively viewed in simple supposition. The intensions in turn signify attributes and thereby reference in personal supposition what he called “extensions”, which are the individual entities identified by the signified attributes.

      While the contemporary pragmatism emerged as a critique of neopositivism, some philosophers carried the positivists’ nominalism into contemporary pragmatism. Pragmatist philosophers such as Quine adopted nominalism. He rejected concepts, ideas, meanings, propositions and all other mentalistic views of knowledge due to the notational conventions of the Russellian predicate calculus, a logic that Quine liked to call “canonical”. However, in his book Word and Object (1960) Quine also uses a phrase “stimulus meaning”, which he defines as a disposition by a native speaker of a language to assent or dissent from a sentence in response to present stimuli. And he added that the stimulus is not just a singular event, but rather is a “universal”, which he called a “repeatable event form”.

      Nominalism is by no means essential to or characteristic of contemporary pragmatism, and most contemporary pragmatists such as Hanson, Feyerabend and Kuhn, and most linguists except the behaviorists have opted for the three-level semantics, which is assumed herein. Also, computational philosophers of science such as Simon, Langley and Thagard, who advocate the cognitive-psychology interpretation of discovery systems instead of the linguistic-analysis interpretation, reject both nominalism and behaviorism. Behaviorism is positivism in the behavioral sciences.

      Computational philosophers of science recognize the three-level semantics, and furthermore believe that they can model the mental level with computer systems. Thus in his book Mind: Introduction to Cognitive Science Thagard states that the central hypothesis of cognitive science is that the human mind has mental representations analogous to data structures and cognitive processes analogous to algorithms. Cognitive psychologists claim that their computer systems using data structures and algorithms applied to the data structures, can model both the mind’s

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