Note that Logic as the One law of the universe or as the Immanent and necessary pattern of all being (1) contains logic as the study of principles of inference but (2) does not draw a divide between the logical and the empirical—instead it points to the distinction between the empirically necessary and the empirically contingent

The immanent and perceived versions of Logic

The perceived version of Logic is ‘the entire system of consistent descriptions’ and so on

The immanent version of Logic is the absolutely indeterministic Universe

The immanent and perceived versions of Logic

The perceived version of Logic is ‘the entire system of consistent descriptions’ and so on

The immanent version of Logic is the absolutely indeterministic Universe

On the common origin of necessary empirical truth and Logic… and perhaps of contingent empirical truth and science

See also Being, Metaphysics and Objects… and Cosmology

The system

u_E = u, the empirical universe (which is context dependent)

U = the Universe (all being)

u_P = P = the universe of the possible

u_L = L = the universe of Logic or Logos (this defines Logic)

The fundamental problem is to show that

LIM u = U = P = L

Note that the only question is whether LIM u = U...

Material from Journey in Being-New World-essence.html

Logic and meaning

The idea of Logic arose naturally in Metaphysics where Logic was seen to be capable of a variant, more inclusive, metaphysical meaning. This chapter takes up that thread. The primary objective is to formalize, evaluate and apply this new notion of Logic

Introduction—a variant and ultimate notion of Logic

In Metaphysics a new or variant notion of Logic was suggested; the notion was connected to variant meanings of actuality, possibility and necessity; to a theory of consistent descriptions; and to ‘the one law of the universe.’ The concept was not a finished one and, in some of its expressions, it had a certain awkwardness—what is ‘the entire set of consistent conceptions or descriptions?’ Despite the awkwardness, the variant notion appeared to be related to the traditional notions of logic, to include them and go beyond them. The one law of the universe—that has an ultimate ring to it

Goals of the chapter

1. To formalize this new notion of Logic; and to see its relations to similar reflections on the nature of logic. To further analyze relations and identities among Logic and Metaphysics. Note the commonplace that the elucidation of a concept is not the giving of a definition; definition, application, elaboration, forging or seeing relations to other significant concepts is an ongoing process—and the achievement of closure and perhaps even completion or an ultimate character may be ascertained in the process and not at outset. Therefore, the new conception of Logic will not be altogether complete—as far as this narrative is concerned—until completion of the Cosmology. Even subsequent discussion may add to the conception of Logic

2. To evaluate the relation between the new and traditional conceptions of logic and to show that Logic includes logic

3. To establish relations among Logic, grammar, and meaning. Of course the work of Wittgenstein is an inspiration for these developments. To include emotion and will in the framework of grammar and meaning. The importance of the incorporation of emotion and will in the framework is of such fundamental importance that it cannot be relegated to a ‘special topic’

4. To discuss some special problems of concern—Logic, reference, the problem of the infinite, relations to the semantic and set-theoretic paradoxes and to Zermelo-Fraenkel-Skolem and von Neumann-Bernays-Gödel systems… To apply the considerations of Logic and Metaphysics to—valuation or revaluation of the nature of—science and mathematics

A traditional notion of logic

A traditional notion of logic—the science or art of inference and, more generally, of argument—sees inference as arguing from premises to conclusions. An aspect of this view is that inference is always questionable because, ultimately, there must be unfounded premises; however, it has been seen that experience and existence are facts that require no further foundation. Thus these facts and a host of necessary inferences from them may serve as absolute premises for further inference. In the traditional notion, inference is classed as deductive in which the conclusions necessarily follow from the premises and inductive inference in which the conclusions are likely but not necessary. A model for deductive inference is inference regarding compound propositions. If A is the compound proposition B&C then the truth of A implies the truth of B and C. Based on such examples it is possible to arrive at a set of rules of inference and a calculus of propositions—the propositional calculus—that may be regarded as an abstract system and that can be shown to be consistent and complete. Other logical systems such as the predicate calculus in which the structure of the propositions figure in inference are harder to found and regarding these open questions remain… Inductive inference generally involves generalization or inferring a rule from a finite set of data and, except in the relatively uninteresting case of domains that are a collection of points that is known to be finite in number, cannot be certain. The Aristotelian development—primarily of the syllogism—was, for the most part, regarded as the definitive treatment of deduction for about 2000 years; and, though not altogether devoid of interest, western logic may have been regarded as a dead subject studied only in the schools. In the nineteenth and twentieth centuries, spurred by the revolution in the foundations of mathematics and its needs, logic came to life and a variety of deductive ‘logics’ emerged and the focus in logic concentrated on deduction—‘logic’ became synonymous with deduction. Induction came to refer primarily to the ‘method’ of the sciences

Experiments with compound propositions suggest that unless a compound proposition has complete reference, i.e. unless every one of its component propositions has reference, paradox or contradiction results—and that complete reference guarantees absence of paradox. Since the world itself is not paradoxical, these conclusions may be expected. (Although the world contains paradox that fact is not paradoxical.) The consideration that paradox arises on account of incomplete or erroneous reference may be used to resolve a number of logical paradoxes. This suggests that proper reference—either to an object or a transparent model—may be the basis of logical systems and their consistency. It further suggests that, contrary to a common view, systems of logic have meaning

Some experiments with compound propositions may be found in earlier versions of Journey in Being

Preliminaries from Metaphysics and from Objects

From Metaphysics, all entities are in the universe, form is immanent in being, and every consistent concept is realized. The actual and the possible are identical; therefore what is possible is also necessary. This suggests a concept of Logic as the analysis—theory—of the actual or, equivalently, of the possible and appears to define a kind of extensional necessity that contrasts with the common concept of necessity that may be labeled intentional and in which necessary propositions are true regardless of reference. It has been seen, however, that the distinction between intensional and extensional necessity is not an essential distinction. From Objects, every—consistent—concept has an object; the distinction between abstract and particular objects is one of (convenience of) approach, i.e. symbolic versus empirical, and not one of kind. Sense is latent or potential reference; without latent reference, there can be no sense

That the notion of Logic as the theory of the actual / possible / necessary should not be circular follows from the empirical discovery of laws of logic and may also be explained though not—without further consideration—proved by appeal to adaptation or transcendental arguments

Conceptions of Logic

Logic as the one law of the universe

From Logic as theory of the possible and the actual, it follows that Logic is the one law of the entire universe

Whatever is allowed by logic is absolutely possible

A concept of Logic as analysis of the actual, the possible or the necessary

The concept of Logic as analysis of the actual / possible appears to define a kind of extensional necessity that contrasts with the common concept of necessity that may be labeled intensional and in which necessary propositions are true regardless of reference. However, since every consistent concept has an object and therefore there is no essential distinction between intentional and extensional necessities. This is further emphasized in Objects where it was seen that both particular and abstract objects exist in the one universe and that the distinction is one of the mode of study than mode of being, i.e. both particular and abstract concepts have reference to real objects. Again, every—consistent—concept has an object; the distinction between abstract and particular objects is one of—convenience of—approach, i.e. symbolic versus empirical, and not one of kind. Sense is latent or potential reference; without latent reference, there can be no sense

Logic as the theory of descriptions

The foregoing suggest an equivalent concept of Logic as the analysis—theory—of descriptions

In the preceding statement, provided that it is understood with sufficient generality, ‘science’ may replace ‘theory’

The notion of Logic as the analysis, theory or science of descriptions has the following virtues

Deduction concerns truth of one proposition relative to the truth of another. Therefore, the standard concept of logic—logics—deduction, falls within—or out of—Logic as defined here

Wittgenstein’s thought that from the truth of one atomic proposition, the truth of another does not follow concerns independent propositions

It coincides with the idea of Logic as the analysis of the possible (and therefore, also, of the necessary)

Logic as an abstract object

It shows that Logic is an—abstract—object. More precisely, Logic is the concept and the universe is the object. It is therefore reasonable to identify Logos as the universe i.e., Logic and Logos as concept and object

Similarly, the Logics are abstract objects, i.e. concepts that must have objects; Logics have reference—a Logic (logic) may be regarded as a premise that, in deductive proof, is necessary

These thoughts suggest development of logics, e.g. modal logic, from the present concept of Logic as a project

It shows the crucial importance of reference to Logic and Logics

Logic, grammar and meaning

It also points to Wittgenstein’s thought that Logic and Grammar are identical…

This idea could be developed here but it is convenient to defer it to the discussion of meaning in the context of Logic below

Logic, reference and the problem of the infinite

An immensely important concern regarding Logic as theory of description and the requirement of reference is the infinite case—for what is an infinite object… what is the object whose concept refers to an infinite extension or an infinite collection? There are preliminary thoughts on the object side of ‘infinity’ in Objects and in Logic

Is the requirement of proper reference necessary to validity in Logic and Grammar? Since various semantic paradoxes (Russell…) and set-theoretic paradoxes (Zermelo-Fraenkel-Skolem and von Neumann-Bernays-Gödel) have been resolved by non-referential artifacts, the requirement of proper reference may be unnecessary

These thoughts define a research project

That the paradoxes have been resolved by non-referential artifacts is not clear. The valid aspects of the various analyses—of Russell and others—should be studied to see if reference is the root justification—Kripke employs the term grounding

Secondly, a general study of the nature of ‘logical objects’ and infinite objects may be undertaken to analyze necessary and sufficient conditions of validity including the important case of the necessity and sufficiency of proper reference. The abstract and Logical objects may be studied directly—where—possible or semi-directly in terms of models

In any case, however, it appears reasonable that requiring proper reference may be rich in consequences

Logic and metaphysics

In this conception, Logic is—equivalent to—metaphysics; Logic is the constitutive form of being. It may be noted that although Logic and metaphysics are identical, metaphysics initially emphasizes facts or states of being and Logic initially emphasizes structure or relationships among facts, and that any apparent distinction lies only in the initial appearance. Whereas metaphysics emphasizes the study of being from the object side while Logic may be regarded as study that emphasizes the concept side

There is a project to develop this concept of Logic and its consequences

Logos

In this conception, Logic is the one law of the universe. The immanent form of Logic may be called Logos. More accurately, perhaps, Logos contains the immanent form of any actual Logic. Then, Logos is simply the universe. This also follows from the idea of Logic as theory of descriptions

Thus while Logic is the analysis of the actual, the possible and the necessary; the one law of the universe; and theory of—consistent—descriptions; its immanent form Logos is simply the Universe. As long as nothing is said or nothing is desired to be said about Logos-Universe, nothing needs to be said in order to exclude that that cannot be. When something is said, then the various analytical specifications of Logic may be necessary

In the narrative, Logic and Logos are occasionally conflated

The immanent character of Logic—and of Language and Grammar—makes it clear that reference is crucial in Logic and Grammar

It may be shown by examples, that improper reference may result in paradox and that a number of the classical paradoxes may be resolved by paying proper attention to reference

It certainly appears that proper reference is sufficient to valid Logic or Grammar

Mathematics, science, and Logic

Therefore, it also follows that the results of inductive inference—science—are contained in—or fall out of—Logic as is mathematics—and all proper forms of knowledge and argument whether inductive or deductive

Below, reflect on Mathematics as the study of form in terms of symbolic representation—modeling

…and Scientific theory—theory in general—as fact versus best hypothesis

What is mathematics?

Mathematics as the study of form in terms of symbolic representation—modeling

What is science?

It is not a primary objective of the narrative to answer this question and it is not assumed that there is a simple answer. The term ‘science’ has different families of meaning according to whether science refers to a body or bodies of information or theories, to an activity, to an institution, to an approach or method or to some combination of these possibilities. It is not the intent, either, to evaluate the logic or the value of science. However, since the metaphysics of immanence and science so clearly intersect within the domain of science, since there may appear to be conflict between the metaphysics and science, since it is argued that the metaphysics illuminates science and that science illustrates and elaborates the metaphysics, it will be useful to make some comments on science

There is no conflict between metaphysics and science. Science, as it is usually understood, is conceptual but remains close to its empirical ground—in this cosmos. Within that contingent realm, science reigns—at least with regard to material reality—and provides elaboration, illustration and grindstone for the metaphysics. Outside this cosmological system, where science is suggestive, the metaphysics and science may be mutually illuminating. In phases or domains of the universe that are extremely remote from this cosmos, science may have occasional application but has no necessary general application and the metaphysics reigns

As a result of its empirical ground that is centered in the immediate world and moves outward with discovery, the ideas and theories of science are often thought to have a hypothetical character. This thought emerged in the twentieth century—the result of a number of scientific revolutions from about 1850 to 1960. Each revolution had such an impact on the view of the world, that doubt emerged that any such world view could be regarded as final. Therefore, the corresponding theories—of evolution; of the molecular basis of chemistry; of space, time, and fields; of the quantum; and of the molecular basis of life—were, at least initially, regarded as hypothetical. Those theories whose domain is relatively restricted—the outlines of a chemistry within the solar system, of a theory of life on this earth—are no longer generally regarded as hypothetical—at least within the scientific community. The theories of space, time and fields and of the quantum, however, are regarded as tentative

In each case of scientific theory there is a concept, the theory, and an object to which the theory refers. In saying that a theory is a concept it should be noted that such a concept is compound and its ‘constituents’ are the individual concepts, laws, and explanatory systems of the theory. If the object of the theory is regarded as the entire universe then the corresponding concept either has no application—little is known about life in the universe at large—or is wrong for there is little doubt that theoretical physics is still incomplete; and, further, the metaphysics of immanence shows that, relative to the entire universe, the theoretical physics of this cosmos must be incomplete. However, the object of the theory may be regarded as limited by its domain of validity. Given the excellent explanatory and predictive power of the theories, each concept or theory may be said to have excellent reference to a limited object—the object defined by the theory and limited by its domain of validity

In summary, science is characterized by concepts that remain in close and ideally precise empirical contact with their objects. The domain of empirical science starts with this—material and mental but not spiritual—world and, in extending the known—macroscopic and microscopic—reaches of the cosmos, it emanates both outward and inward

The status of scientific theories

There is a popular idea that equates theory and hypothesis and that is one use of the word theory. In science, however, theory has a range of connotations. Some theories may begin, perhaps in the mind of the scientist, as hypotheses or systems of hypotheses, but after publication and extensive criticism they may come to be accepted as factual. In the case of physical theories, as our empirical knowledge of the universe expands, those theories may be replaced. However, even the replaced theories may be factual—and extremely useful—over their domain of validity. In the case of life on earth, that life evolved along certain lines is factual beyond reasonable doubt; even the theories of evolution whose function includes explanation of the processes of evolution have a factual though not complete character

In this sense, induction may be seen to have a kind of certainty that is rather different in nature than logical certainty

Do the theories of biology project to the universe? I.e., does life on every world proceed on lines laid out in modern evolutionary theory? It has been seen in Metaphysics that, in contrast to thinking based on the evidence from this cosmological system, there must be infinitely many cosmological systems with life. Further reflection showed that the normal case may well be one in which evolution conforms to evolutionary theory and that it is reasonable to think that the vast preponderance of evolutionary scenarios are normal. However, it must also be true that there are infinitely many non-normal cases

Fact and pattern

The metaphysics of immanence implies that facts may be infinitely divisible. Here it has been seen that patterns—theories—have factual interpretation. The distinction between fact and pattern or theory breaks down; both fact and theory designate objects

Is mathematics a science? Can mathematics have an empirical side?

It has sometimes been regarded as one but, since its methods are—it appears—deductive and not inductive or experimental, it is also clearly distinct from natural and social sciences. The similarities and differences between mathematics and the sciences may be clarified in terms of the theory of objects. Mathematics may be seen as a study of actual objects from the concept side. It is known from the theory of being that a mathematical system must, if it is consistent, have an actual object but it is not always known what that object is and therefore study of concept side may be the only way of study—in addition, as is in the character of mathematical objects, to being a most productive way of study; further, because the actual object side may remain implicit, there is the possibility that a mathematical or logical system may lack reference—and may therefore be inconsistent and it may be required to patch up such inconsistency from the concept side as in the discussion of Logic and the problem of the infinite, above, or to live with the possibility of inconsistency

However, as was seen in Objects, mathematical systems often begin from a study of an object side (e.g. geometry as the measurement of the earth and number in counting) and only later move into a focus on the concept side but as in some kinds of computational proof may have return to the object side. The example of geometry also shows that by changing or relaxing certain assumptions or axioms a broader class of systems (e.g. geometries) is obtained—the domain of reference expands. It is in fact the focus on the concept side without reference, which should have difficulty for non-finite systems, to an object side that makes mathematical and logical systems susceptible to paradox at which, in the absence of known reference, attempts at resolution are made from the concept side (or reference to a transparent model.) The science of physics is studied from the concept side—in theoretical development—and the object side—in experimental study—and seeks consistency between the object and concept side as well as internal consistency in the concept side; and in so doing, the domain of reference expands. Perhaps the decision to distinguish mathematics and science is a function of attitude rather than object—i.e., when mathematics and science are themselves seen as objects. At minimum, it may be admitted that, while mathematics and science may have sharp distinction in some phases of their activity, no sharp distinction can be maintained eternally or even over historical time

Is the inclusion of Mathematics in Logic the Frege-Russell logicist thesis that mathematics is a chapter of logic? The validity of that thesis shall depend on where logic is thought to stop and where mathematics thought to begin. It is not the case that what is traditionally taken to be logic (as in the Frege-Russell logicism) is shown here to found or contain mathematics

Science and logic

As describing a limited phase of being—this cosmos, life, human mind—the laws and theories of science falls under Logic: although the laws and theories may be seen as having a hypothetical character, they may also be seen as factual over some domain. Since Logical and Mathematical objects have concept and object sides, there is, similarly, the possibility of incompleteness of Logical and Mathematical theories. The ‘method’ of science is induction or generalization from particular cases. This method does not fall under—deductive—Logic. There appears to be no analog in scientific discovery to deduction in logic and mathematics, i.e., while the results of inference in science fall under Logic, scientific inference itself does not—there is a literature on discovery in science that need not be repeated here that refers to simplicity, beauty, following intuition and guesses. However, logic and mathematics are not so different. While the method of deduction in mathematics or logic is characterized by certainty, the approach to arriving at a mathematical or logical system is characterized by trial and error guided by simplicity, beauty, following intuition and guesses. Briefly, then, scientific and mathematical theories have similarities in content, argument within the bounds of the theory, approaches to developing theory, and incompleteness of reference

Logic, grammar and meaning

As noted above, Logic as theory of descriptions points to Wittgenstein’s thought that Logic and Grammar are identical…

Specifically, that conception of Logic shows Logic and Grammar are identical and, further, that Syntax and Meaning are inseparable e.g. Syntax is not devoid of meaning

Meaning, appropriately interpreted, is identical to Logic and Grammar but may be regarded as focus on the experiential side—sense which is latent reference—and experimental side of Logic and Grammar

It shows that full reference is fundamental to the robustness of logic or logics, especially in illuminating and eliminating the classical paradoxes of logic and in formulating logics or Logics as the study of certain kinds of object (it may be practical, however, to study logics and attempt to build consistency in from the abstract or conceptual side—starting with intuition and models.) There is good reason to think that the requirement of full and proper reference is necessary and sufficient to robustness of Logics. Here, again, lies the connection of Logic to the world—Logic may be founded in the requirement full and proper reference

It may be allowed that conception generally, including sensing, depicting, imaging are forms of ‘description’ and therefore there is a grammar or logic of conception—of symbolic expression, of sensing, depicting and or imaging

It is not the possibility of a connection of logic and being with grammar that is surprising—that there may be a connection is obvious once it is pointed out

The clarity and necessity of the identity of Logic and grammar

What is surprising is the clarity and necessity of the connection, that the connection is one of identity rather than mere relatedness. It may also surprising that the connection should have emerged when it was not sought

Rethinking Wittgenstein’s Tractacus Logico-Philosophicus

In reviewing the developments of this narrative, especially those regarding the fact of being as implicit in the fact and content of its meaning, the metaphysics, the discussion of Form, and the present discussion of Logic, it seems that the ideas veer in the direction of Wittgenstein’s Tractacus—whose thought followed a similar patter and has influence and significance here—and go beyond it in some aspects. The ideas that the universe is—in the global mode of description—all its states and that all its states are all states is close to Wittgenstein’s thought that the universe is the sum of its atomic facts. A distinction between the present thinking and that of Wittgenstein is that, here, the kind and enumerability and denotability—reference—of all states is not given at outset or assumed to be possible—even in principle. Additionally, there are parts of the Tractacus e.g. the discussion of Ethics that suffer from an implicit substance thinking regarding the nature of the object

The backward foundation, elimination of substance thought, and elaboration of the ideas of the Tractacus is a project that awaits keen analysis

Meaning

Introduction

The foregoing considerations on Metaphysics, Objects, and Logic allow a clarification of meaning and grammar and their relation to Logic

Note that the substantive aspects of meaning that are important to the present development and that flesh out the following comments have been treated in Being. The concern here is with the formal side of meaning brought out by the considerations on Logic

Preliminaries from Metaphysics and from Objects

It is convenient to repeat the preliminaries to the discussion of Logic:

Formal discussion of meaning

In its concern with the experiential and experimental side of Logic—the study of the world—meaning comes, with Wittgenstein, to emphasize context (use) over the lexicon, to emphasize that sense or latent reference is always in process. The ideas of the fixed lexicon and fixed syntax are an arrest. Further, from the connection with Logic, complete meaning can reside only in systems of concepts which may be—the experimental side—in process or evolution; meaning of individual terms is fragmented and may be distributed in more than one way (it is crucial to pay attention to the meanings of terms in this narrative.) From the practical side, an identical situation obtains regarding axiomatic systems; the meaning of the system resides in it taken as a whole; the evolution of such systems occurs in the first place in their genesis and, then, in a sequence of such systems in which each step is a modification in response to the needs—explicit or not—of reference. It is thus that the study of abstract objects and axiomatic systems is both symbolic or abstract and experimental

Sense and reference

It has been seen that the two sides of knowing an entity are concept and object. That distinction corresponds to the sides of meaning as suggested by Frege: sense and reference (here, it has been seen that sense must be latent reference.) The following distinctions are similar, connotation versus denotation and intension versus extension

Grammatical forms; emotion and will

Is there a fixed set of grammatical forms? Wittgenstein argued, for example, that the ideas covered by ‘noun’ are so varied that conflation (mind and brick are nouns) leads to misleading con-fusion. Whitehead questioned the universality of the standard subject-predicate form—taught in schools—for expression. Does emotion have an object? One view is that emotion is simply expression and has no object; however, it is conceivable that emotion has a diffuse and variable, perhaps even latent, object located somewhere in the organism-environment. What is the significance of emotion and will and motivation in relation to the distinctions expression vs. assertion vs. declaration vs. commission vs. direction? These questions have relationship because ‘sentences’ that express emotion or feeling of the affective type are one mode of—apparently—non subject-predicate form as in a cry that expresses delight or determination. Reflect on the metaphysics of immanence—all objects are in the universe, every consistent concept is realized and the theory of objects developed above—the distinction between abstract and particular objects is not one of kind as is commonly thought but is one of which mode of study (conceptual or symbolic and so on versus empirical,) sense is latent reference… These observations reinforce the idea that emotion and may and typically does have an object, that grammar which is cognitive in form and emotion expressed in language may have unification and that there may be a universal mode of expression even if it is not the subject-predicate form. These thoughts, of course, suggest a program of research whose outcome may be glimpsed but is not known (experience suggests that even glimpses may be well off mark with some outcomes being negative and others quite beyond expectation in extent and quality)

Logical proofs of the fundamental principle of metaphysics

The fundamental principle states that every consistent concept has a realization—recall that a consistent concept is one that neither contains nor entails contradiction. The earlier proof depended on the existence and properties of the void. Here follows a proof that does not depend on the void

A concept is necessary—absolute—if it must be realized. A concept may be regarded as contingent—possible—if it could be realized. What, however, does ‘could be realized’ mean? Consider the entire universe. A concept A is not realized but it is said that it could be realized. What does that mean? There are two classes of concept—those that are realized and those that are not. There is nothing outside the universe; therefore if a concept is realized, it must be realized—it is necessary; if it is not it could not be realized. The implicit meaning of ‘is not but could be realized’ is that there is another world in which things are sufficiently different that the concept is realized. However there is no other world

Therefore, every contingent—possible—concept is realized, is actual; the obvious and only exceptions the contradictory propositions. Obviously, every actual concept is realized. Therefore, the possible and the actual are identical

The concept of the void is implicit in the proof; it is the other world

Relation to atomic propositions

Consider Wittgenstein’s ‘from the truth of one—atomic—proposition, the truth of another does not follow.’ Here, atomic means logically independent. Since the contentless proposition is atomic, the thought becomes ‘atomic propositions are true in some worlds and untrue in others.’ Here, world means sub-domain of the Universe. Therefore, every atomic proposition is realized

‘Proof’ from Ockham’s razor

It is interesting that there is an extreme and rather unconventional—some might prefer to say perverse—use of Ockham’s razor to supply a proof idea of the fundamental principle. In science, Ockham’s principle amounts to making no unnecessary hypotheses; Ockham’s is a minimalist principle—only those ‘hypotheses’ are made that reflect the structure or form of the domain; no additional hypotheses are to be added to the minimalist regimen just to make a theory work. Here, the extreme use is to make no hypotheses whatsoever and shall take the form no contingent proposition is universally true. I.e. every contingent proposition must be true in some sub-domains of the universe and untrue in others and it is this that is the foundation of the second approach to proof of the fundamental principle

Material from previous editions

2006 Edition

On the role of reference in Logic

Following is a section from the 2006 edition

The concept of Logic

Recall that the Theory of Being developed the concept of Possibility. The fundamental assertions were first, whatever may be described (conceived, pictured) without contradiction is realized and, second, the Possible and the actual are identical. This suggests the following conceptions

Logic (Logic) is the Theory of Possibility (of the depth and variety of being) or, equally, from the identity of the possible and the actual, Logic is the Theory of the Actual; Logic is what may be said –what holds– in the absence of hypotheses (the derivation and significance of this form of the concept of Logic is clarified in the subsequent section Second proof of…;) Logic is the form of the facts of all being as they stand in mutual relation: logic is the system of necessary relations among facts or, if consistency is required, among hypotheses; logic is the theory of consistent systems of description; all else, including logic and science, that may pass under the names ‘Logic’ or ‘logic’ concern restriction to a context or domain of being

The principle of non contradiction is that an assertion cannot be both true and false must be essential to logic; without it ‘consistency’ would have no meaning or significance. Paradox is endemic to systems that violate allow contradiction

Logicians and mathematicians are willing to forego this law of logic in search of the possibilities of form – provided that the ‘virus’ of inconsistency may be isolated; however the purpose to the containment is to maintain the practical distinction of truth and falsity

A second principle of classical logic is the principle of the excluded middle that an assertion must be either true or false (i.e. the possibilities for truth values are ‘true’ and ‘false’ and no other such as ‘null’ or ‘in between.’) The following interesting application arises. Consider a solar system with nine planets; one planet ‘earth’ is blue. Consider A = ‘Planet ten is yellow.’ Since there is no planet ten, A is not false; therefore it is true. Similarly, A' = ‘Planet ten is red,’ is also true; however, A' contradicts A (a.) Now consider B = ‘Planet ten is yellow and earth is blue.’ Since there is no planet ten, B is not false; therefore it is true. Similarly, B' = ‘Planet ten is red and earth is blue,’ is also true; however, B' contradicts B (b.) Now, (a) suggests that Reference is necessary to avoid paradox while (b) suggests the improvement that complete reference is necessary. Now consider C = ‘This statement is true.’ Is C either true or false? It is true if true and false if false. Granting the principle of the excluded middle C is equivalent to ‘The truth value of this statement is ‘true’’ or ‘This statement has a truth value and that value is ‘true.’’ The referential character of C does not permit evaluation of a truth value: it makes reference to its truth value as though it is determinate but it is not so; there is no truth value. I.e. there is a paradox associated with C in a logic that accepts the principles of non contradiction and the excluded middle; the source of the paradox is the assumption that it has reference to something definite. In fact the reference is to its own truth value and C may be thought of as an equation on the truth value – for which every truth value (true and false) is a solution. The classic liar paradoxes ‘This statement is false,’ ‘I am lying’ can also be seen in this light. The assumption that the liar statement has a truth value already associates it with paradox. However there is the additional paradox that results from the equivalence of the liar statement to ‘This statement has a truth value and that value is ‘false.’’ (This ‘equation’ has no solution.) Think, also, of another famous paradox that arises in asking, ‘Who shaves the barber?’ of ‘There is a village whose barber shaves all except those who shave themselves.’ If such a village does not exist there is nothing further to say; but it is asserted to exist. If it does exist it then follows that the barber shaves himself if and only if he does not shave himself. The source of the paradox is the assumption that (arbitrary sentence constructions have Meaning and reference and, specifically, that) the village exists (can exist) i.e. that the statement can have reference; the paradox indicates that it can not. In formal logic some paradoxes due to ill founded reference may be avoided by introducing rules of formation of sentences

If paradox is of words but not the world, how can a precise or true description of the world contain paradox? What does this say about paradox? That it is an artifact of description e.g. of Language? What does that say of language? That it is not completely bound to its objects? That it is experimental?