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Chapter 1:
On the Nature of Formal Science

1.1 - Quine's Rejection of Analytic and Synthetic Distinction:  A Challenge for Platonists
1.2 - Taking Up the Challenge
    1.2.1 - Who Does Quine Criticize?
    1.2.2 - A Reply to Quine's Criticism
1.3 - Formal Science:  A Neglected Platonist Perspective on Mathesis Universalis

    To be able to provide an adequate doctrine on the relationship between formal science and natural science, we need to explore the nature of formal science itself, which is indispensable to understand its relationship with natural science. I wish to address one of the most important problems that arose in the twentieth century and still continues today, which is the division between analytic and synthetic propositions, and the problem between platonism and antiplatonism in philosophy of mathematics.

1.1 — Quine's Rejection of Analytic and Synthetic Distinction: A Challenge for Platonists

    Willard van Orman Quine's position is the most popular philosophical doctrine that rejects the traditional analytic/synthetic distinction.¹² It must be clarified that Quine's criticism is only directed at Rudolf Carnap's way of making such a distinction in virtue of meanings. In “Two Dogmas of Empiricism”, the two dogmas Carnap wished to attack were the analytic/synthetic distinction and the verifiability criterion of science. We will deal in this chapter only with the former, and we will deal with the latter in the second volume of Underdetermination of Science.

    Quine made a distinction between two kinds of definitions of analyticity in philosophy. There are those who define analyticity as propositions that are logically true, like: “No unmarried man is married”, and there are also those who define it in virtue of meanings, such as “No bachelor is married”. In the latter case, there is a synonymy of terms: we can replace “bachelor” with “unmarried man”. In this way, any analytic statement built on these notions would be a kind of a “second class” of analytic statements. Quine argues that this can only serve to reconstruct logical truth, but not analyticity as such.¹³

    For Quine, to establish the definition on analyticity on the basis of synonymy of meanings is doomed to failure. It can be argued that there are judgments that are analytic in virtue of their definitions, for example, “bachelor” is defined as “unmarried man”. However, the way that the word “bachelor” is defined depends greatly on linguistic usage in people's daily lives. Another apparent criterion of analyticity is the interchangeability of terms salva veritate, as Leibniz suggested. Let us take this case:


    In this sentence, the word “bachelor” cannot be substituted by “unmarried man”. Also, cases like “bachelor of arts” present us counter-instances of interchangeability of terms. In the former case we could say that we mean the word “bachelor,” while in the latter the word “bachelor” means something different from “unmarried man.” We must take into account cognitive synonymy. To claim that “bachelor” and “unmarried man” are synonymous means that the proposition “All and only bachelors are unmarried men” is analytic. This is to say that “Necessarily all and only bachelors are unmarried men.” Let us carry out the substitution and say: “Necessarily all and only bachelors are bachelors.” In both propositions, even though they interchanged terms “bachelor” and “unmarried men,” the cognitive information they offer is very different.¹⁴

    We could try saving the argument by appealing to extensions. For example, two terms are interchangeable salva veritate if they have the same extension. However, extensions that fall under concepts depend greatly on accidental matters-of-fact. For example, the concepts of “creature with heart” and “creature with kidneys” have the same extension (presumably), but they are not interchangeable salva veritate. It seems that the only way to assert the synonymy is by supposing that the identification of the terms “bachelor” and “unmarried man” is analytic. Hence, we see, not a circularity, but a “closed curve in space”, because all of these concepts and theories of meanings are closely related.¹⁵ It can be expressed simply in this way:


    It could be that the failure to establish the interchangeability salva veritate is due to the vagueness of language. We could use instead, as Carnap certainly tried, an artificial language to avoid such vagueness and establish semantic rules to distinguish some propositions from others. Now, we have to explain why there are semantic rules that make a difference between analytic and synthetic propositions, and how they are different from other semantic rules. The answer is its adoption because they can pick up analytic propositions and distinguish them from synthetic ones. Here we find a circular reasoning: What distinguishes analytic propositions from synthetic ones is the supposition of semantic rules, which are themselves adopted because they can distinguish between analytic and synthetic propositions. In seeking the definition of analytic, we are led to presuppose the notion we wish to define in the first place.¹⁷

    From this analysis, Quine presents this analytic/synthetic distinction as a kind of article of faith held by logical empiricists. For him, propositions that are logically true (“No unmarried man is married”), and those of science and experience are all posits, no different epistemologically to the Greek gods of antiquity which were also posited to explain what people perceived at that time. For Quine, mathematics and logic are qualitatively no different from empirical statements, or scientific laws, and can be revised in light of recalcitrant experience. Therefore, the difference between posits of formal laws and posits of science is only of degree of abstraction, which means that there can be no analytic/synthetic distinction between propositions.¹⁸

1.2 — Taking Up the Challenge

1.2.1 — Who Does Quine Criticize?

    We must be aware that most of the criticism made by Quine (1953) is only directed to Rudolf Carnap's way of distinguishing analytic and synthetic judgments. For the late Carnap, we must focus on meaning, especially through cognitive synonymy, which determines which propositions are analytic or synthetic. Practically, from its refutation, Quine deduces that there is no qualitative distinction between judgments.

    First, we have to see if this criticism applies to all of the versions of analytic and synthetic judgments. Before Rudolf Carnap, there were some well-known versions of these notions. The first was the one defined by Kant, who believed that synthetic a priori judgments are at the very heart of science. He gave two definitions of analytic judgments and, respectively, two definitions of synthetic judgments.


    Notice that at most Quine's criticism can be applied to version (II) of the analytic and synthetic distinction, but not definition (I).²⁰ Version (I) basically states that if I have a subject-predicate kind of proposition “S is P” in which S is a concept of the subject, and P, the concept of the predicate, is an essential or necessary property of S, then “S is P” is an analytic judgment. This is a qualitative difference from version (II) which says that a proposition “a = b” is an analytic proposition in virtue of the principle of identity. Not only are both of these definitions incompatible with each other, but they are also unsatisfactory for today's semantics. At least in Kant we have one instance where Quine's criticism does not apply, version (I) of these definitions.

    Another version of analyticity can be found in Frege's philosophical masterpiece The Foundations of Arithmetic. Based on version (II) of Kant's definitions, he says:


    For Frege, logical and arithmetical propositions are all analytic a priori, and this fact points to the reducibility of arithmetic to logic. Geometry, on the other hand, is synthetic a priori. This distinction between analytic and synthetic does not fall under Quine's criticism.

    Finally, there is another less studied criterion to distinguish analytic and synthetic propositions, the husserlian criterion. Husserl did not base it on Kant's but was inspired by Bernard Bolzano, who, at that time, was a relatively unknown philosopher and mathematician. In the second volume of Logical Investigations, the Third Investigation, Edmund Husserl establishes some semantic distinctions to develop his mereological doctrine. He distinguishes concepts and propositions that are completely free from all “matter that contains the thing” (or free from all empirical content), and those which are not free from them. He introduces the notion of logical categories which include concepts like: “something”, “one”, “object”, “property”, “relation”, “plurality”, “number”, “order”, “whole”, “parts”, “magnitude”, and so on. These logical categories are qualitatively different from material concepts such as “tree”, “house”, “color”, “sound”, “space”, “sensation”, and “feeling”, which have empirical content.²²

    For Husserl, analytic propositions are those which express a truth exclusively in virtue of forms, devoid of all material concepts, while synthetic judgments do contain them. Also, Husserl's distinction allows for the existence of synthetic a priori judgments. To understand the difference between analyticity and syntheticity, Husserl makes a distinction between “analytic laws” and “analytic necessity”.²³ He describes these analytic laws this way:


Analytic laws are purely formal, while an analytic necessity is an instantiation of an analytic law. Here he gives an example:


    Husserl distinguishes analytic law and analytic necessity from synthetic a priori law and synthetic a priori necessity. If a proposition is necessary, but not formalizable salva veritate, it is a synthetic a priori proposition. Its laws are founded on material concepts or essences. Synthetic necessities are particularizations of synthetic laws, such as the one expressed in the proposition “this red is different from this green”.²⁶ Propositions like “anything colored must be extended” are necessary, but are synthetic, because we are not establishing direct analytic correlatives such as in the case of “There are no fathers without children”. Here, the concepts of “father” and “child” are analytically founded on each other, and the analytic necessity of that proposition relies in its forms (the way they are formally related). On the other hand, in the case of color we are not appealing to a direct correlative. It is true that a color cannot be exist without a colored extension, but the existence of such an extension is not founded analytically on the concept of color.²⁷

    Husserl includes any geometry based on material categories (such as spatial form, geometrical congruence, among others) as a synthetic a priori discipline.²⁸  Notice that under these husserlian criteria, “No bachelor is married” would not be an analytic judgment nor a synthetic a priori proposition, so Quine's arguments do not apply to this case.²⁹

    As a result of this analysis we find in Quine's argument a non sequitur. The fact that Carnap's criterion for analyticity and syntheticity fails semantically, does not mean that there is no criterion to distinguish analytic and synthetic propositions, and much less that there is no qualitative difference between them.

1.2.2 — A Reply to Quine's Criticism

    Even if the analytic and synthetic distinction appears arbitrary, from a pragmatic point of view it is logically legitimate to make a qualitative difference between kinds of propositions. Let us assume, for the sake of the argument that analytic propositions are posits like scientific theories. However, Quine is one of those philosophers who refuse to believe that logical laws and mathematical propositions are abstracted from experience. Posits about the world are not abstracted from experience either, they only serve to explain the “sense-data” we receive from our bodily senses. Nothing prevents anyone from making a difference among posits, just as we can make a difference between imaginary objects we cannot represent mentally (the round square or Hegel's universal spirit) and objects we can represent mentally (a table, a unicorn, Pegasus). Some of these objects do actually exist in the physical world, while others do not (or at least no one has good reason to believe they exist, especially when they have no scientific use). This is not merely artificial, this distinction can be made very clearly and legitimately.

    We can say the same concerning posits that have one characteristic and posits that have other ones. For example, we can follow Husserl and identify posits like these as true:


These have nothing to do with material objects or concepts. They only express formal truths, whose variables can represent any propositions in the first case, or, any number in the second. The way they are shown to be true according to logical rules and mathematical axioms does not depend on empirical objects or temporal events.

    These logical and mathematical statements differ from propositions like


which is a newtonian formula to find the gravitational force between two masses. The variables F, G, m, d represent material concepts that can only be applied to the phenomenal or temporal world. These are not pure formal relations like our previous logical and mathematical formulas. Why can't this be taken into account when establishing legitimate criteria to distinguish between analytic and synthetic propositions? It can be argued that this statement is circular in a way, like in Carnap's case, because we make the difference between these kinds of propositions in order to make a difference between analytic and synthetic. However, the undisputed fact is that a difference can be made legitimately, and, furthermore, it is scientifically acceptable to make such a difference.

    In the following sections and chapters I will discuss in depth the important differences between formal truths as analytic and natural-scientific propositions as synthetic.

1.3 — Formal Science: A Neglected Platonist Perspective on Mathesis Universalis

    One author who has been ignored by philosophy of mathematics is Edmund Husserl. Only recently we have rediscovered his influence on analytic philosophy, especially regarding his philosophy of language, of logic, of mathematics, and his influence over eminent figures in logical empiricism such as Rudolf Carnap.³⁰ Some authors have reexamined his philosophy of logic and mathematics in order to solve contemporary problems in these fields.³¹ In this section, I am not going to say anything new about Husserl's philosophy of mathematics, but I think we should look at this relatively unknown aspect of Husserl's philosophy.

    In 1890, when Husserl began his anti-psychologist turn, he was heavily influenced by G. W. von Leibniz, Bernard Bolzano, Hermann Lotze and David Hume. He realized that meanings (logical contents) and mathematical objects were clearly different from entities and events that are subject to changes in time. It is difficult to base the necessary and universal validity of logical and mathematical truths on temporal psychological acts or material abstractions from experience. One thing is the temporal act of recognizing a truth, and another the objective validity of truth itself; the act of counting and the number itself; the act of collecting, and collection (set) itself. Hence, we recognize, on the one hand, the temporal psychological acts and events in the world, and on the other the atemporal meanings and abstract mathematical entities. He says that temporality is the sphere of the real (reell) and the other the realm of the ideal (ideel).³² Husserl retains this platonist doctrine throughout his life, even to the point of extending it to phenomenology.

    Some have tried to argue against the view that Husserl held some kind of platonist doctrine. He pointed out that he rejects a certain form of platonism: that which considers abstract entities as existing as real (reell) outside of thought. He regards this as a mistake he called “metaphysical hypostatization”.³³ He clarifies:


In other words, ideality, an atemporal realm, does exist for Husserl. The fact that he was a platonist cannot be questioned.

    Inspired by Leibniz, Husserl held an idea of mathesis universalis, where pure logic and mathematics come together to form the most universal mathematics of all. To understand this, we should examine the way Husserl conceived both formal disciplines, especially in his official position which appears in Chapter 11 of the first volume of his Logical Investigations.³⁶

    For Husserl, there is formal knowledge and material knowledge. This is made clear when we take into account the differences made by Leibniz between “truths-of-reason” and “truths-of-fact”, or by Hume between “relations-of-ideas” and “matters-of-fact”.³⁷ However, as we have seen, his view on analyticity and syntheticity is more complex. From his criticism to psychologism, it is evident that in its theoretical standpoint, logic and mathematics are qualitatively different from any discipline that deals with matters-of-fact. They express truths that are necessary, and, because of that, they prescribe all forms-of-truth and all forms-of-being respectively.³⁸

    This is also true for knowledge. Science, in the widest sense (Wissenschaft), is an activity that has an anthropological origin, that is, it originates in the acts and dispositions of our thinking. This anthropological unity of sciences has as its correlate an ideal and objective unity of propositions and of objects. We must realize that all sciences refer to objects in a specific manner. So we see, in the first place, formal interconnections of objects (objectualities) or, as Husserl called them “states-of-affairs” (Sachverhalte) to which scientific truths refer to. On the other hand, science itself consists in the interconnections among truths which refer to those states-of-affairs. Truths are propositions that are fulfilled in states-of-affairs.³⁹ In the realm of truth, the interconnection of objects and the interconnection of propositions that refer to them are given a priori together and are inseparable. Because true propositions tell us that these states-of-affairs are indeed the case, these objectualities are necessarily correlated to truths about them. In other words, truth-in-itself is a necessary correlate of being-in-itself.⁴⁰

    If we abstract all subtracts (low level substrates of sensible objects) of a given state-of-affairs, we will end up with two different sorts of formal unities: the objectual unity and the unity of truth. In the former, we discover the ideal laws of all kinds of objectual relations, and in the latter we discover the ideal laws of truth. These legalities are supposed a priori for every science in order to provide knowledge. It is up to mathematics to explore the a priori laws of the ideal unity of objects and objectualities, while it is up to logic to explore the a priori unity of truths. In other words, logic is a formal theory of judgment or formal apophantics, and mathematics is a formal theory of object or formal ontology. For Husserl, mathematics is logic's ontological correlate, or in his words, “logic's adult sister”.⁴¹

    He develops an epistemology of mathematics which we discuss in detail in Chapter 2. However, we must point out that formal interconnections of objects are given in intuition along with the objects themselves. In other words, our consciousness constitutes at once the objects and their formal relationship in a state-of-affairs, and not through a process of psychological reflection as many phenomenalists and psychologists thought at the time.⁴²


    Since formal components are intuitively given in a state-of-affairs, and once abstracted from all empirical content these reveal a priori legality, we must explore three groups of problems regarding the laws of pure logic and their relationship with mathematics. These will reveal three logical strata and correlatively three mathematical strata:

• First Problem: We should look for primitive concepts that make such legal interconnections possible in an objective and theoretical sense. On one hand, all the ideal interconnections of truths in science form a complete deductive unity. If we substitute all propositions with indeterminates (variables), we are able to discover their elementary forms of deductive combinations. Husserl calls these elementary forms “meaning categories”, which include: the copulative, disjunctive and hypothetical combinations of propositions in new propositions. So, what we today would call “connectives”, would form part of these meaning categories: conjunction (∧), disjunction (∨), implication (), equivalence (↔), among others. However, meaning categories also include forms of combination between concepts themselves so that meaningful propositions can be expressed. This means that these categories include subject-predicate forms, forms of copulative or disjunctive combinations, forms of plural, and so on.⁴³ In other words, we are talking about a morphology of meanings, or a pure universal grammar, and Husserl calls the ideal laws that govern this sphere “laws to prevent non-sense [Unsinn]”. Thanks to Carnap, today we call them “formation rules”.⁴⁴ In this logical stratum, we can construct an infinity of possible forms of meaningful propositions. For example, in the case of subject-predicate forms, if I say “S is p”, we can also use this proposition to form another proposition “S(p) is q”, and at the same time use this judgment to form a new one “S(p,q) is r”, and we could continue indefinitely.⁴⁵
  
  On the other hand, correlated with the morphology of meanings, we find a morphology of intuitions, which includes all the laws of adequate meaning fulfillment in objectualities.⁴⁶ According to Husserl, once we substitute the sensible or material objects with indeterminates in any state-of-affairs, we find in this stratum the elementary formal components of every objectuality which he called “formal-objectual categories” or, more properly, “formal-ontological categories”. These include concepts of object, state-of-affairs, unity, plurality, cardinal number, ordinal number, sets, part and whole, among others. This morphology of intuitions is nothing more than a morphology of formal-ontological categories or of formal-objectual categories.⁴⁷
  
  
• Second Problem: We have to deal with the ideal laws of the objectual validity of formal-ontological categories and the truth and falsity of propositions or judgments on the basis of meaning categories. On the side of logic, we find a stratum Husserl called “logic of consequence”, which is concerned with theories of forms of deductions or theories of inference which preserve truth. He would call the laws of this sphere “laws to prevent counter-sense [Windersinn]”, which include syllogistics among other forms of deductions. Today, thanks to Carnap, we call them “transformation rules”.⁴⁸ If we have a logical proposition like “(α→β)↔¬(¬α∨β)”, according to the first logical stratum it would be meaningful, but according to this second stratum, it would be contradictory.⁴⁹
  
  Distinct from the logic of consequence we find the “logic of truth”, where the concepts of “truth”, “falsity” and other related concepts are being considered. Any deductive relationship between judgments can be turned into a deductive interconnection of truths if these judgments are determined to be true on the basis of states-of-affairs.⁵⁰ If the judgments themselves are formally contradictory, this a priori excludes them from being true.
  
  Logic's second stratum has its mathematical correlate. It deals with the being and non-being of “objects in general” and of “states-of-affairs in general”. Each theory in this level of formal ontology (mathematics) is founded on formal-ontological categories. We consider many sorts of theories of plurality founded on the category of plurality, arithmetic founded on the category of number, set theory founded on the category of sets, and so on.⁵¹  Husserl holds the ideal completeness of these categorial theories on ideal grounds for any true and valid judgment as well as any valid or true combination or arrangement of objects in any way. It makes possible a science of the conditions of possibility of any theory in general.⁵²
  
  
• Third Problem: The discovery of a morphology of meanings and forms of inference point to a still higher logical level, which deals a priori with all essential forms or sorts of theories and their corresponding laws. In other words, it points to a theory of all possible forms of theories or a theory of deductive systems. This logical stratum
  

Both, the theories of all possible forms of theories (pure logic) and the theory of manifolds (mathematics in its highest expression) together form a mathesis universalis, the most universal mathematics of all. He gives an example of the partial realizations of this mathesis universalis or theory of manifolds:


Rosado (in Hill & Rosado, 2000) also points out that Husserl's view of logic and mathematics is amazingly very contemporary. General topology, universal algebra, category theory and other mathematical disciplines can be seen as “important partial realizations of Husserl's view of mathematics as ultimately the theory of all forms of possible multiplicities or, to use a more frequent term nowadays, forms of possible structures”.⁵⁸


Footnotes:

¹²Quine, 1953, pp. 20-46.  [Return to Text]

¹³Quine, 1953, pp. 23-24.  [Return to Text]

¹⁴Quine, 1953, pp. 27-31.  [Return to Text]

¹⁵Quine, 1953, p. 30; Katz, 2004, p. 19.  [Return to Text]

¹⁶Quine, 1953, pp. 24-32.  [Return to Text]

¹⁷Quine, 1953, pp. 32-39.  [Return to Text]

¹⁸Quine, 1953, pp. 17-19, 42-46.  [Return to Text]

¹⁹Kant, 1998, A7-10/B11-14.  [Return to Text]

²⁰Notice that Quine (1953) believes his criticism reaches Kant’s original proposal (p. 21).  [Return to Text]

²¹FA. p. 4.  [Return to Text]

²²LI. Vol. II. Inv. III. §11.  [Return to Text]

²³LI. Vol. II. Inv. III. §§11-12. Rosado (2008a, 2008b) states that Quine (1970) and Chateaubriand (2005) arrived separately to almost the same notion of Husserl's “analytic laws” with their respective notions of “logical truth”. (see Chateaubriand, 2005, pp. 254-255, 262)  [Return to Text]

²⁴LI. Vol. II. Inv. III. §12.  [Return to Text]

²⁵LI. Vol. II. Inv. III. §12.  [Return to Text]

²⁶LI. Vol. II. Inv. III. §12.  [Return to Text]

²⁷LI. Vol. II. Inv. III. §11.  [Return to Text]

²⁸LI. Vol. II. Inv. III. §12; Rosado, 2008b, p. 109.  [Return to Text]

²⁹It should be mentioned that, although Husserl's criteria seem to satisfy our intuition of what should be analytic or synthetic, it is not perfect. Rosado (2008a) noticed that, under husserlian criteria, some mathematical statements that should be considered analytic become synthetic a priori. This is the case of the mathematical truth “1³+2³+3³+4³=100” which formalized will give us “x³+y³+z³+w³=100”. The problem with both of these mathematical truths is that the latter can only be instantiated with the former (Rosado, 2008a, p. 134). The same could be said regarding the analyticity of certain metalogical statements such as the Löwenheim-Skolem Theorems, or Gödel's Completeness Theorem for first-order logic, or Gödel's Incompleteness Theorems, and so on. (Rosado, 2008a, pp. 134-135). Rosado proposed a model-theoretic definition: A statement S is analytic if it satisfies the following two conditions: (i) it is true in at least one structure, and (ii) if it is true in a structure M, then it is true in at least any structure M* isomorphic to M. (Rosado, 2008a, p. 137; Rosado, 2008b, p. 110) Under such a definition, all mathematical (excluding usual geometric statements that include what Husserl would consider “material concepts”) and metalogical truths are considered analytic. This definition could also allow the existence of some sort of synthetic a priori definition: A statement is synthetic a priori if: (i) it is true in at least one physical world, and (ii) if true in a physical world W, it is true in any possible physical world. (Rosado, 2008a, p. 139; Rosado, 2008b, p. 110) This definition would distinguish between empirical statements on space-time which can be applicable to this world, while statements such as “anything coloured must have extension” would apply to any world and include what Husserl would consider “material concepts” (Rosado, 2008a, p. 139). These definitions have not been challenged yet, and Quine's criticisms do not apply to them.  [Return to Text]

³⁰See Beaney, 2004; Dummett, 1993; Friedman, 1999, 2000; Hill, 1991; Hill & Rosado, 2000; Hintikka, 1995; Mayer, 1991, 1992; Mohanty, 1974, 1982a, 1982b; and Rosado, 2008b.  [Return to Text]

³¹Many of these problems are addressed in Hill, 1991; Hill & Rosado, 2000; and Hintikka, 1995.  [Return to Text]

³²The term “real” in this context should not be identified with the German words “Realität”, or “Wirklichkeit”. “Reell” in this context only refers to the sphere of the temporal. As we shall see, the realm of the atemporal is also real in the sense of the word “Wirklchkeit”, that which forms part of everything that indeed exists.  [Return to Text]

³³LI. Vol. II. Inv. II. §7.  [Return to Text]

³⁴LI. Vol. II. Inv. II. §8.  [Return to Text]

³⁵LI. Vol. II. Inv. II. §8, my italics.  [Return to Text]

³⁶Rosado (2006) makes a more detailed and thorough exposition of Chapter 11 of the first volume of Logical Investigations.  [Return to Text]

³⁷LI. Vol. I. §51.  [Return to Text]

³⁸LI. Vol. I. §§41, 46, 51.  [Return to Text]

³⁹See a more detailed account of this doctrine of sense and referent in Appendix A, Section A.2.  [Return to Text]

⁴⁰LI. Vol. I. §62.  [Return to Text]

⁴¹LI. Vol. I. §§46, 63, 66.  [Return to Text]

⁴²LI. Vol. II. Inv. IV. §§43, 45.  [Return to Text]

⁴³LI. Vol. I. §67.  [Return to Text]

⁴⁴LI. Vol. II. Inv. IV. §§10-14; Hill & Rosado, 2000, p. 203.  [Return to Text]

⁴⁵LI. Vol. II. Inv. IV. §§10-14; FTL. §§12-13.  [Return to Text]

⁴⁶LI. Vol. II. Inv. VI. §59. See Section A-2, in Appendix A for more details.  [Return to Text]

⁴⁷LI. Vol. I. §67; Bernet, Kern & Marbach, 1999, p. 48.  [Return to Text]

⁴⁸Hill & Rosado, 2000, p. 203.  [Return to Text]

⁴⁹LI. Vol. I. §68; LI. Vol. II. Inv. IV. §12; FTL. §§14-22.  [Return to Text]

⁵⁰FTL. §§15, 19. See Section A-2, Appendix A for more details on truth fulfillment.  [Return to Text]

⁵¹LI. Vol. I. §68.  [Return to Text]

⁵²LI. Vol. I. §§65, 68.  [Return to Text]

⁵³LI. Vol. I. §69.  [Return to Text]

⁵⁴LI. Vol. I. §69; FTL. §28.  [Return to Text]

⁵⁵LI. Vol. I. §70.  [Return to Text]

⁵⁶LI. Vol. I. §70.  [Return to Text]

⁵⁷LI. Vol. I. §70.  [Return to Text]

⁵⁸p. 205.  [Return to Text]