JOURNEY IN BEING
LOGIC
Motivation… Introduction… Aims… Logic as the theory of valid descriptions… Traditional deductive logic, reference and paradox… Induction as good but not necessary inference
Preliminary discussion. Necessary and contingent modes of establishing truth… Introduction to induction and science… Hypothetical versus factual interpretation of (scientific) theories… An example: the Theory of Evolution… Theory. In a most significant use of the word, theories –especially scientific theories– are factual… Fact. Facts stand in relation to perspective or pattern. Facts partake of pattern. The distinction between fact and theory is relative to e.g. the scale of perception. Full perception requires all elements of the psyche… Necessary facts. … Nature of induction… Introduction to deduction… Mathematics… A Platonic view of Being without a distinct Platonic world… The nature of mathematics… Induction and deduction… The concept of Logic… Logic (Logic) is universal law. Metaphysics and Logic are identical… In Logic as the Theory of Possibility, logic and contingent law merge. This suggests the following formulation: Logic is the constitutive form of Being… Second proof of the ‘fundamental principle’ of the Theory of Being… Logics
The following may go to Introduction in Logic
The new considerations on Logic from Metaphysics are to be introduced:-
A classical notion of logic is ‘principles or theory of valid argument.’ The idea of deriving valid conclusions from given premises, i.e. of inference is at the heart of logic. Two kinds of inference—deduction and induction—are traditionally identified. The case in which the conclusions necessarily follow is the deductive case. Induction is ‘good’ or even ‘excellent’ but less than necessary inference
As a result of fundamental advances in logic in the nineteenth and twentieth centuries, the term ‘logic’ has generally become synonymous with deductive logic and induction with the method of the sciences. In earlier times e.g. in the thought of Francis Bacon a basis for inductive logic was sought that would make it as certain as deduction. It was later recognized that induction, e.g. generalization from examples, must satisfy criteria that are less than necessary. While the development of the theories and laws of science have inductive phases, there are other processes within science. One is concept formation and a second is the application of theory to specific examples which is deductive
Given some information about the world—the premise—logic generates further information—the conclusion. ‘The world’ could be the universe or some symbolic context such as a mathematical system. However, it may seem that logic itself says nothing about the world. That, however, cannot be altogether correct for clearly logic is saying something about relationships that obtain in the world, i.e. relationships between conclusions and premises
Now recall the fundamental principle of the Metaphysics of Immanence or Theory of Being regarding the—entire—universe. In the statement of the principle, the word ‘descriptions’ stands for all conceptualizations of states of affairs including actual descriptions and depictions. This principle is the assertion that the entire system of consistent descriptions is (must be) realized—the only ‘descriptions’ that are not realized are the contradictory systems of description
It is acknowledged that the idea of ‘system of descriptions’ or of ‘entire consistent system of descriptions,’ though clearly powerful and rich in content is also problematic—at least until further analysis and clarification. This defines a research project whose outcome is, in addition to depth, likely to be explicit revelation of an immense variety of being
This suggests a conception of logic that is clearly related to the traditional notion through the idea of consistency—Logic is the theory of descriptions of the universe i.e. of the actual or, equivalently, of the possible
Logos is the immanent form of Logic; Logic is the description of the Logos—of the universe
That Logic is a good and ultimate realization—generalization—of logic is seen:
In that logic falls out of Logic—this was shown in Metaphysics
In that full reference is fundamental to the robustness of logic, especially in illuminating and eliminating the classical paradoxes of logic. There is good reason to think that the requirement of full and proper reference is necessary and sufficient to robustness of logic. Here, then, lies the connection of Logic to the world—Logic may be founded in the requirement full and proper reference
In this conception, Logic is—equivalent to—metaphysics
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
Therefore, as has been seen, deductive logic and inductive inference are contained in Logic as are science and mathematics (and all proper forms of knowledge and argumentation.) 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, however, that the implied distinction exists only in the initial appearance—not in being itself or in well founded appearance
How do the laws of logic fall out of Logic? It is not given that the laws of logic are universal and it is, therefore, not given that any law must fall out of Logic. While Logic is specified formally, the notion of logic—regardless of the formal character of its development—is intuitive. There is an analogy to the Moral discussed under Objects. Here, it is seen that the notion of Logic is founded in the absolute depth of the Metaphysics of immanence. Further, while Logic is the one law of the universe, it contains logic and provides foundation for the idea of consistency for logic
Consider transitivity of implication: if A, B, and C are propositions and A implies B and B implies C then A implies C. If this law were to not obtain it would be trivial to show some proposition about the world, D, which would be simultaneously true and false. This proves transitivity
What of the more questionable ‘law of the excluded middle?’ This law becomes questionable in application to infinite classes. What does the notion of Logic imply here? In the general case it is not clear that it should be expected to have any implication. However, the notion of Logic has possible significance for the kind of concept that ‘the infinite’ is or may taken to be. Is the infinite actual or specified in terms of, e.g., classes for which the number of elements is not bounded by any finite number? To say that there is no actual infinite might be to project a limit of intuition on to being. One approach to the question of the law of the excluded middle would be to construct symbolic universes upon which to perform analysis
The traditional notion of logic as principles of inference
In the twentieth century, logic came to mean deductive logic. A more general but older notion of logic is principles of argument—deductive and inductive
Development of the ultimate notion as suggested by the metaphysics and the traditional conception—Logic as theory of descriptions of the universe i.e. of the actual or, equivalently, of the possible
Justification…
Logic and metaphysics are identical and include all valid or proper forms of logic, inference and argumentation; and mathematics; and science—and all proper forms of knowledge
To show or note that Logic and metaphysics are identical
To note the equivalence of Logic and grammar (and, from, Meaning the unity of grammar and meaning or syntax and semantics; the chapters are kept separate because Meaning has a special place in the narrative)
The first criterion is that Logic shall include logic—this is shown
The second is that Logic shall be robust—as theory of proper or valid descriptions, Logic is constitutively robust (unlike logic, Logic clearly and properly refers to the world)
Recognition of the robust character of Logic is further enhanced by noting that it enables a foundation for logic. The primitive element of the foundation is that, as noted, logic is included in or falls out of Logic—this is shown. A second element of foundation is that via the metaphysical side of Logic, i.e. reference to the world which is constitutional of Logic, it makes clear that and how logic is about the world. A consequence of the referential character enables programs: to show how the principles or laws of logic may be established—and to establish such principles; to enhance enumeration of kinds and varieties of logic and to relate and evaluate the kinds and varieties; and to understand and provide definitive resolution of the paradoxes of logic
Satisfaction of the first two criteria will show that Logic is a good realization of logic. What will show that Logic is an (the) ultimate realization? This is already shown by the identity of Logic with metaphysics. Further, since Logic now appears as a transparent structure or display, the realization makes logic transparent
To display the Logic as reflecting the universe as a realm of Necessity…
To distinguish contexts of pure necessity and of contingency
To show that it in contexts of pure necessity logic is deduction alone. To show that in contingent realms, logic may be either deductive or inductive (e.g. possible)
To explain the difference between normal and merely contingent contexts
Foundations of logic and logics
Mathematics
Science, law and theory
Argument
Induction
Explain science and law
Develop a framework for quantum mechanics in terms of the realm between the necessary, the contingent and the normal—and considerations from The Future and other documents
… of the universe—of the actual and the possible
From 06 topics:
That Logic is the constitutive form of Being—Logic is universal law
Logic is seen to reflect the universe as a realm of necessity
Identity of metaphysics and Logic. Emphasis—fact versus form
A Platonic view of Being without a distinct Platonic world
Necessary versus contingent fact—deduction versus induction
Distinction from induction: necessary versus contingent—but good—modes of establishing truth (06)
Second proof of the ‘fundamental principle’ of the Metaphysics of Immanence (06)
Foundation and development of logic and logics
Mathematics
Science
Hypothetical versus factual interpretation of (scientific) theories (06)
Induction and argument—general considerations
The section Motivation above may be placed here
…
From ‘06
The Theory of Being has been shown to be a metaphysics that is explicitly ultimate in depth and implicitly ultimate in breadth. (It was seen that it is impossible for any theory of being to be explicitly ultimate in breadth. It is therefore futile to seek explicitness in breadth or to seek a theory that permits it.) It is natural, therefore, to expect that the metaphysics will have significant implications for the meaning and development of fundamental concepts in the understanding of the universe, knowledge of the universe and the place of individual (e.g. human) being and society in the universe. Since the metaphysics is ultimate with regard to depth, the implied understanding for the fundamental concepts may also be ultimate. (They recognized that working out of the implications is a process.) These fundamental concepts include: Being itself, objects (knowledge) – considered above; and logic, cosmology (the Theory of Variety,) mind (and matter,) individual (human) being and language and society, ethics and morals, faith, journey (becoming,) the nature and varieties of Understanding – especially philosophy and metaphysics (itself) and the systems of human understanding, identity and transformation of identity. The development continues here with Logic and in what follows with the remaining fundamental concepts
It will be useful to understanding to defer specifying the concept of Logic until some traditional aspects of logic (taken up for illustration and intrinsic interest) have been discussed
To derive (and develop) from the Metaphysics a concept of Logic in its object form as immanent in the universe via the concept of Actuality and the derived (check whether derived) concepts of Possibility and Necessity. This Logic, in its concept form, is logic as Theory of Possibility or, equivalently, Theory of Actuality. That is, stated in concept form, Logic is the necessary law of the universe
… and of necessity; if negation of something is impossible, it is necessary
To derive or show, equivalently to the above, and perhaps more importantly Logic as the Theory of Descriptions and to show, also, that this concept of Logic includes the traditional idea of logic
The form of logic contains the form of deduction. Inference demonstrates relative possibility
The classical conception of (deductive) logic may be seen to be the Theory of Conditional Possibility. I.e., in the classical conception, movement or argument is from premise to conclusion
To display the Logic as reflecting the universe as a realm of Necessity (and to enquire whether Necessity or some other term is appropriate to the intended idea or concept.) To explore and explain this sense – perhaps that there are no contingent truths in the universe in that contingency has significance only in contexts (domains) in which the ideas part or domain, here or there, or now and then have significance
Recall that concepts, especially the capitalized concepts above, are as defined in the narrative
A secondary aim is to define and clarify relative and absolute possibility and extensional and intensional necessity
Regarding the extension of concepts in the light of the metaphysics, recall the following template for extension. (1) There is an ultimate character to the extension that derives from the ultimate character of the metaphysics. (2) That the extension may not involve or entail contradiction. This includes that under restriction from the universe and Form to a contingent world and form, the extended concept should reduce to some prior valid version of the concept. (3) However, significance to the extension will be manifest only in illumination of the system of concepts or in resolution of existing problems or paradoxes…
To distinguish contexts of pure necessity (characterized above) and of contingency
To show how logic in its traditional conceptions falls out of Logic. To show that it in contexts of pure necessity logic is deduction alone. To show that in contingent realms, logic may be either deductive or inductive (e.g. possible)
To explain the difference between normal and merely contingent contexts. To show the application of deductive logic in normal and contingent contexts
To show that probable inference has significance in normal domains but not in merely contingent contexts. I.e. in merely contingent domains, i.e. in the absence of regularities there are no Laws or laws (no scientific theories)
To explain science and law. Theories of science as hypothetical, tentative or limited knowledge over the universe or as definite knowledge (fact) over limited and incompletely defined domains
To develop some ideas about deduction and induction. An example is to illuminate for deduction, the importance of reference to avoid paradox and perhaps to ensure consistency. What is said about this topic amounts to the following. (1) There are some descriptions that are and cannot be satisfied by any object. Assumption to the contrary results in at least some paradox. Is all paradox an artifact of description? Paradox is not a property of an object except the absence of an assumed relation between concept and object. See Journey in Being-New World for details. (2) Are there objects for which there are no descriptions. Gödel’s theorems seem to indicate that arithmetic has no description. However, is not the existence of ‘complete arithmetic’ known by some other concept? What is the object in question and how is its existence known?
To generalize or recast the discussion regarding paradox as follows. It is important to ask not only about the source of paradox but, also, the nature of paradox – to explain how, given that paradox is at least naïvely not part of the object itself, paradox is possible. Cast this as an issue of concept and object in which paradox is a function of a kind of inadequate conception of an object. Why it must be cast as an issue of concept and object (must it?) Show that this approach applies to both the linguistic paradoxes including those of self-reference and the set-theoretic paradoxes including those of infinity; and that perhaps it shows the two kinds to be of the same die and that the explanation may require casting sets and infinity in terms of concept and object. This may require whether there is an actual infinity – the intuitionists concern (even if actual infinities may lack demonstration it may be useful to allow them with a caution and to resolve paradox by artifacts of concept e.g. the Zermelo-Fraenkel-Skolem and von Neumann-Bernays-Gödel formulations.) Continue that this does not guarantee resolution because the object itself may be idealized. However, it appears to be a valid and fundamental approach – as opposed for example to introducing a priori restrictions on the concept such as a ban on self-reference.
To review traditional logics for co-illumination with Logic and the fundamental laws of logic (e.g. non-contradiction)
To review the possibility of developing a framework for quantum mechanics in terms of (1) the realm between the necessary, the contingent and the normal and (2) other considerations (noted elsewhere or as may come up)
In its traditional meaning, logic is concerned with truth – especially with the establishment of truth
The ways to establishing truth may be regarded as two. First, and in one perspective most fundamental, is the establishment of truth about the world. These include facts and generalizations from facts or recognition of patterns and laws in systems of fact. Terms associated with the recognition of patterns in systems of fact are induction and science (i.e. the process of science.) The kind of logic involved in this first way of establishing truth has been labeled ‘induction.’ Induction may be described as ‘generalization,’ ‘interpolation,’ or ‘extrapolation,’ and, alternatively, as recognition of patterns… A second way to establish truth is to derive secondary truths (theorems, results) as necessary consequences of established (or, in axiomatic systems, postulated) truths. This ‘way’ is labeled ‘deduction’
[Establishing simple facts is not usually considered to be a part of logic or of induction but logic may be implicated since (at least some) facts exist in relation to others which relation(s) may require demonstration of consistency and grounding. Induction is establishing (without certainty) laws and patterns from data. More generally, the processes of science include establishing both facts and patterns]
Deduction appears to be firm because consequences necessarily follow; induction is not always regarded to be as firm in its definiteness because there could be alternate generalizations i.e. because alternate patterns could be ‘read.’ In teaching, logic is often called something like the ‘science of argument’ or the ‘analysis and criticism of thought.’ However in the recent (twentieth century on) literature ‘logic’ most often means deductive logic
There is a history of thought regarding induction. Induction has been regarded to be important because it is (has been thought to be) the ‘method’ by which e.g. scientific laws and theories are established. A foundation has been sought by some thinkers (especially in the past e.g. in the thought of Francis Bacon, b. 1561, London, England, of René Descartes, b. 1596, La Haye, Tourraine, France, and of Newton) in which induction would be as secure as deduction. Other approaches have sought, for example, to place induction on minimalist (Ockham’s razor, William of Ockham, born c. 1285, Surrey (?), England, ‘plurality should not be posited without necessity,’) or, especially later, probabilistic or aesthetic foundations
The intent here is to illuminate the nature of induction and its relation to deduction and to see what foundation there may be for induction without requiring it to have identities with deduction
There is a view in which there is no end to the development of scientific theory. In this view, the Universe is infinite and any theory captures only a part of it and is therefore capable of improvement (enlargement of the domain of application.) This view is encouraged by the recent (nineteenth and twentieth centuries) history of science which may be seen as a ‘revolutionary’ history. In (extreme) versions of this view, scientific theories are not really about the world but are working hypotheses subject to replacement when there is sufficient disconfirming evidence (and an adequate alternative theory that explains or predicts the new as well as the old data; see e.g. The Logic of Scientific Discovery, 1934, of Karl Popper, b. 1902, Vienna, Austria.) However, there is an alternate way of looking at scientific theories. That science has developed so far without end (in sight) does not imply that it is unending; i.e. the revolutionary character of science appears to be contingent rather than Necessary; i.e. it is itself a generalization – its necessary or inherent character is open to question. Scientific theories that have been replaced as fundamental theories of the world continue to be useful – Newtonian Mechanics is still used in much of engineering and astronomy. Newtonian Mechanics may be said to be about a domain of being (in which ‘domain’ does not refer only to spatial or temporal domains but also e.g. to a range of sizes of entity and energies of interaction)
There may be an end to the science of the local cosmological system; that the universe has been shown to have variety without end may imply that Science will retain its revolutionary character. However, after a sufficient number of revolutions, revolution may be expected and that science is revolutionary may come to be seen as normal (Thomas Kuhn, b. 1922, Cincinnati, U.S., introduces the terms ‘normal’ and ‘revolutionary’ science in The Structure of Scientific Revolutions, 1962)
A (scientific) Theory may be viewed as hypothetical knowledge about the entire Universe and subject to replacement or, alternatively, as actual knowledge of a domain of the universe
An objection that the elementary scientific constructs are tinged with the intuition (i.e. that the constructs are not entirely objective) may be addressed by adjusting the domain to include observation. The objection has been addressed earlier in the section ‘Object’ in a somewhat different and perhaps more satisfactory manner
Even in science, ‘theory’ has a number of uses and the foregoing statement cannot be generalized to all uses of the word. The kinds of theory that it does refer to are those that provide a coherent and connected picture of an entire known domain of being – physical, biological and so on. They include the classical theories of physics e.g. mechanics and electromagnetism and the later quantum theories and relativistic theories of space, Time and gravitation; and the Darwinian theory of evolution
The theories of the present narrative e.g. the theories of Being, of Variety, of Identity, of Possibility are necessary rather than hypothetical and this necessary character follows roughly from their domain of reference being the entire Universe rather than a limited or normal domain. These theories are not scientific in the following sense: a scientific theory refers to an empirically known domain and may therefore be disconfirmed by empirical evidence (the kind of evidence that might ‘disconfirm’ the Theory of Being would be the non existence of the universe.) The truth of these necessary theories is possible on account of their extreme in abstraction or generality. Alone, their application is limited. Their power is actually the joint power of the necessary with the contingent forms of knowledge (including theories.) This is seen is some of the earlier discussions and especially in the sections starting with ‘Mind’
The subject of the Theory of Being is that of all forms of being. The subject of science is that of particular forms of being – those of the local cosmological system. In showing that the limits (implied by the laws) of the local cosmos are not absolute, the Theory of Being may seem to contradict the facts, theories, and spirit of science. However, this is not the case. The Theory of Being allows the theories of science but alters their interpretation (as noted earlier) so that, relative to this cosmos, any use of the word ‘impossible’ in relation to contingent (though not necessary) laws should be replaced by ‘improbable’ and what is merely possible relative to the contingent laws of any cosmological system is known as actual among All Being
Some readers may object to the inclusion of Darwinian Theory; a discussion of the theory is therefore included for intrinsic interest but especially to illuminate ‘Fact’ and ‘theory.’ It should be noted that many of the objections to Darwinian Theory are based in misunderstanding. (The label ‘Darwinian’ is not altogether accurate since, the original account, On the Origin of Species by Means of Natural Selection, 1859, Charles Darwin, b. 1809, Shrewsbury, Shropshire, England, was incomplete in a number of ways e.g. the seat of inheritance, detailed mechanisms of speciation. In its early form, the theory, though elegant and predictive, was widely criticized on account of its various limitations. It was only by work in responding to the criticisms and further development that resulted in the widely accepted ‘new evolutionary synthesis’ of the 1940’s. Subsequently, molecular biology, especially genetics further secured the foundation of the theory.) First, among the misunderstandings, there is a distinction between the fact and theory of evolution. The fact of evolution may be seen in the fossil record, in studies on certain species such as the fruit fly, and in the efforts of the animal and plant breeders. Some object to macro-evolution on the ground that the fossil record is incomplete; however the fossil record is expected to be incomplete since preservation even of vertebrates is not perfect. Proper questions to ask are, ‘Is the fossil record contradictory?’ and ‘Are there other sources of evidence?’ and the answers are ‘No!’ and ‘Yes!’ respectively. Recent treatments (e.g. textbooks) list a number of kinds of evidence for evolution that include the fossil record, structural similarities, embryonic development, biogeography, and molecular biology and show that these kinds are in agreement
The (Darwinian) theory of evolution is one of incremental evolution (including macro-evolution or speciation) by variation and selection. Origin of complexity may be explained by simultaneous evolution of Form and function (less complex eyes may not function as well but still function e.g. even a few heat or light sensitive cells provide for adaptive response; further, function is not always designated function and what now functions as a limb may have origin in a protective protuberance.) The question of chance versus design is addressed in terms of Necessity and probability. The origin of Form out of Formlessness necessarily requires indeterminism (deterministic explanation in terms of a designer pushes the problem back one step ‘Who designed the designer?’ I.e. Design is not an explanation of origins.) Regarding probability there are two issues. When there is evolution from an earlier state the later state is not determined; rather there are many possible formed outcomes; the probability of any specific one of those outcomes is much less than the likelihood that there will be some formed outcome (which must intuitively be high and on the Theory of Being is Necessary in some cosmological systems.) Secondly, while a single step outcome may not be impossible its likelihood is very low; however as has just been seen the likelihood of incremental change resulting in large formed change is not at all low. Inability to understand nature or lack in understanding of nature are not and do not indicate limitations of nature itself. Of course it is unreasonable to expect the theory to explain ‘everything’ including, for example, the (chemical) origins of life and cultural evolution (where, if there is explanation, alternative or supplement may be necessary.) A final point worth noting is that the facts of evolution and theory are not altogether independent in that the theory illuminates interpretation of the facts; however, this shows not limitation of the truth of the theory but limits, already noted, to its domain of application; alternate theories such as the Lamarckian or ‘Intelligent Design’ may be seen either as inconsistent with fact or application only to a very narrow, even infinitesimal, domain. Now, there will still be those who object to the Darwinian theory; however, most interested persons –at least in the West– will agree that the actual ‘theory’ of origins whether Darwinian or e.g. ‘Intelligent Design’ concerns not just mere theory but fact
Another way in which distinctions between Fact and theory break down arises in facts that are not necessary e.g. ‘an electron is a fundamental particle’ in that it has no known internal structure or size (c. 2007.) Now if this fact is universal it is also necessary. However the Theory of Being implies that it cannot be universal (there is no fundamental substance.) There must be interactions in which the electron will show up structure. The ‘fact’ is factual in virtue restriction to a particular domain of being. Relative to the local cosmology, the electron can be structureless and have interactions but relative to the universe (all being) the electron cannot be both. Interactions from the universe at large will show up the structure of the electron in the local cosmology; this, however, will not be revealed in all concepts or theories of the local system
It was seen earlier that the perception of space and Time are intuitive (not analytic.) Space and time are fundamental to ‘being in the world’ and, especially, to physical geometry and other scientific theories. In such theories, space and time receive considerable formalization. Do these formalizations take science (specifically space and time) out of the realm of intuition? In other words can the Universe (or domain under consideration) be equivalent to the formalization (marks on paper?) The answer is not altogether clear and even if it were clearly ‘yes’ the significance of this answer would not be altogether clear. There is a human attitude that appreciates the expression of behavior of the universe in formal terms for it is in such terms that there appears to be clarity and objectivity and the psychology (perception-thought) of the individual, with all its ‘limitations’ is transcended; and it is to a significant degree that upon such transcendence the confidence in science and logic and their application, technology, are founded. It is a clean foundation that also results in material power. Despite this, it is pertinent to ask whether it places the individual in the universe. This is far from clear and the best answer appears to be that it may give hope but that such hope is not realized yet and it is the intuition (in integration with formal science and less formal myth, story and Art) that places the individual. These thoughts on fact and theory may be summarized:
The greatest awareness of contingent facts and patterns (theories) requires all elements of psyche (including the formal.) However…
In addition to the theories that are factual in regard to a specific domain there are the Necessary theories that refer to the universe (All Being)
The Theory of Being has been shown to be necessary. The necessity follows from its reference to Being and absence of being rather than any particular kind of being
[The apparent contradiction between the necessities of the theory and the facts of this cosmological system should be noted once again. ‘Somewhere’ in All Being, Jesus Christ is risen from the dead; this alone does not imply its truth in this (normal) cosmological system. However, as has been seen, the Theory implies the Possibility on earth; and it implies that it is impossible to show that it is impossible. At most what may be read from the development of the theory is the improbability of any such (given) event in a given system]
Are there necessary facts? There is at least one – that there is Being (if there were no being, this narrative would be neither written nor read)
The question about necessary facts is important because, then, ‘consequences’ may be deduced by logic alone (with the necessary fact or facts as premise; however since the premises are necessary so are the consequences)
Many necessary facts have been established earlier e.g. the fundamental principle that all consistent systems of descriptions are realized; the Theories of Being, Variety (cosmology,) and Identity. Relative to the entire Universe, these facts are necessary; relative to a normal cosmological system the ones that have meaning may (generally) be either true or false and are contingent; relative to a cosmos, its being is a necessary fact
What is the logic of the ‘induction’ of a scientific Theory? Is there a logic of induction? If such a logic would be a necessary inference from the ‘data’ to the theory it does not appear that there is or can be one (at least in the normal case.) A scientific law is an induction from a set of data or observations to e.g. a formula (pattern.) In general, though, there is no unique law that corresponds to a set of data (this observation, thought to originate with Hume, is a commonplace in the mathematics of interpolation i.e. of fitting some curve to a set of data.) In practice, it often seems that there is a necessity to a law but that is due to the law being simple or elegant or suggestive of further development but it is not a logical necessity. In a number of cases there are alternate but equivalent (same predictions) theoretical formulations for a given range of phenomena. Some formulations are better suited to calculation, some to generalization to more comprehensive theories. Often, two or more theories are combined into a more comprehensive one. This may give an impression of necessity but the necessity is not a logical one. Although there may be the appearance of necessity to induction there is, in general, no necessity to it – there are various heuristic ‘principles’ that include simplicity and aesthetics but induction does not possess the necessity that deduction may appear to have. Since facts –it will be sufficient for the remainder of this paragraph to take a view of facts as given– are included in the external criteria, any set of internally and externally consistent descriptions and any set of facts must be logically consistent (i.e. an infinite number of ‘theories’ is consistent with any set of facts.) How, then, is it possible to have confidence in scientific theories? Confidence generally grows slowly as more and more phenomena are predicted or explained by a theory –resistance to new theories is not merely reactionary– until, of course, observation and experiment arrive at boundary of the domain of validity of the theory
Generally, arriving at a scientific law or theory is inductive while its application and testing is deductive (that these statements are general means that not every operation in arriving at a law is inductive, that not every operation in application deductive e.g. application may require a model to be built.) Once a law or theory is formulated its testing may be significantly deductive
Induction or discovery tends to be singular and private, testing or justification tends to be deductive and public. However, the divide between discovery and justification is not as sharp as has been thought by some philosophers e.g. Hans Reichenbach (b. 1891, Hamburg, Germany) who insisted that the ‘logic’ of science is the logic of justification. This clearly reflects a Value regarding the use of science. It also reflects a habit of thought that favors the determinate or deterministic pattern of thought over the indeterministic even though the indeterministic (discovery) is essential; this habit was seen earlier in the idea that the relation between substance and Variety of being had to be deterministic which idea led to over two thousand years of interesting but unnecessary substance theory (it was seen that substance ontology can and must be replaced by a non substance ontology which is not, however, a relativist ontology and is final regarding depth)
Attention now turns to deduction. Since, given primitive terms, axioms and methods of proof, the theorems necessarily follow, deduction appears to be necessary i.e. deduction is thought to be proof. However, there are some ways in which this appearance is illusory
A primary application of deductive methods is in mathematics. Deductive logic is connected to mathematics in two ways. First, is the idea due to Russell and Whitehead (Bertrand Arthur William Russell b. 1872, Trelleck, Monmouthshire, England and Alfred North Whitehead b. 1861, Ramsgate, Isle of Thanet, Kent, England) that the theorems of mathematics may be reduced to logic; this reduction, sometimes called logicism, is not generally accepted and to expect the reduction is perhaps unreasonable in that mathematical systems have a structure that logic itself does not clearly possess (however, mathematics may be seen as filling in structure within the framework of Logic.) Second, and more significant, deduction is the method of derivation of theorems in mathematics. While the theorems follow of Necessity, the application of mathematics does not possess the certainty of the theorems. Consider the question, regarding some domain of phenomena, ‘Are the phenomena captured by a discipline of mathematics?’ What is required is search for a structure! The mathematical expression of the structure involves basic ideas (undefined terms, rules of combination, axioms, and methods of proof) and theorems. That the theorems are, in principle, already contained in the fundamental ideas of the mathematical system is expressed by saying that mathematics is tautology i.e. derivation is logically needless. Since the tautologies are not at all obvious and their proof may require great ingenuity, the fact of tautology does not make mathematical proof trivial. However, it may still be that the actual structure is not captured by the mathematics and this may require experiment with, modification of the basic ideas (the applicability of mathematics is often regarded as fortuitous.) This process, which is mathematical, is not at all deductive and is rather like induction. Thus the certainty of deductive systems (in capturing form) is illusory and this illusory character lies in the gap between the form (the structure the mathematics attempts to capture) and the structure generated by the mathematical (axiomatic) structure; and there is a similar gap between logical structure and logical system (even though in some cases there is no gap e.g. the propositional calculus captures ‘implication.’) Less important reasons for the uncertainty include the incompleteness and consistency concerns for mathematical systems
Picture an interweaving network of paths leading from different points at the base of a mountain to the summit. The structure of the network captures some aspects of the shape of the mountain. However the paths do not reach all points on the mountain: some places –hidden lakes, secondary peaks– may be inaccessible from the paths. This is a rough metaphor for the incompleteness of mathematical systems
[The analogy has relevance for the earlier discussion of ‘Objects:’ the perception is not the mountain. It may be that the perception of the system of trails is in some sense the perception of the mountain. However, a metaphor in which the perception of the trails is the experience and the perception of the mountain is the entity is inadequate for reasons stated earlier: the (structure of the) experience and the (structure of the) entity are of different kinds]
The discovery and application of mathematics have inductive and deductive phases even though discovery emphasizes induction and application (including theorem proving) emphasizes, perhaps, deduction
[A note on definition. It becomes clear that even though, in formal work, definition may come at the beginning, the actual ‘Meaning’ of a Concept requires more than definition for its full specification. First, the various concepts must stand in mutual relation and in relation to basic facts or axioms and it is the entire system that captures or fails to capture the target structure, is consistent or inconsistent. Second, the ‘final’ form of system and therefore of concepts and definitions are arrived at by adjustment of the system to match the world or, in the case of mathematics, to form. It is seductive to think that an entire formal or intellectual content can be invested in a formal system and that from then on the system will ‘do the work’ while human being is taken for a ride. Although seductive, an ultimate in the formalization (of Variety of form) would represent –if the ride were the objective– an end to being]
A distinction between structures (Forms) and their (mathematical or logical) expression has been made. The expression is the attempt to capture or represent. It is therefore clear that (at least in the normal case) there is no guarantee that mathematics represents form. I.e. even when there is clearly a mathematical form it may fall short of the Form itself. To use a famous metaphor, the mathematics (terms, formation, axiom, proof, theorem) is a scaffolding that is ‘thrown away’ when the building (Form) is realized. The scaffolding is not the building and the form of the scaffolding is a skeletal and at most approximately faithful representation. Therefore there is no surprise that the mathematics is (may be) incomplete; there is no surprise if a hidden paradox should be discovered; there should be no surprise if certain questions are not decidable within formalizations. This may be seen in terms of another analogy. Travelers set out toward an imagined destination. Suppose that they determine their routes at the outset (typically, routes would be corrected in travel but suppose that this is not the case; this is analogous to setting up an axiomatic system as though it will capture a Form.) Paradox is analogous to two travelers of a team labeling distinct destinations ‘The Destination;’ undecidability is roughly analogous to inability of route setting techniques to guarantee arrival at an imagined or glimpsed destination. Dwellers at the edge of the forest (Form) can, from their knowledge, project but not know the interior
A Platonic view of Being has been revealed without appeal to a (separate) Platonic world; the Platonic (the Logos) and the (normally) known are of the one world (Universe)
[A very short digression on Platonism. The Greek philosopher Plato (b. 427/428 BC, Athens) explained the physical order in terms of a realm of forms accessible only to the Mind of which the beings in this world are imperfect copies. The highest Form is the Form of the Good; this idea is reflected in the development in the sections ‘Morals and society’ and ‘Ethics and objectivity’ and is implicit in the section ‘The Highest Ideal.’ Aristotle (b. 384, Stagira, Greece, attended the Athenian Academy of Plato,) dissatisfied with Platonic metaphysics, introduced a theory of substances in which every kind in the world e.g. wolf, is or has a substance. Aristotle’s theory implies a runaway infinity of substances (why is not each individual wolf a substance) and, as explaining Variety from simplicity, provides little satisfaction. As has been seen, the Metaphysics in this narrative requires no distinct Platonic realm and no substances and yet requires no infinite regress of foundation or explanation but is, in some of its aspects, similar to Platonic metaphysics and may be variously labeled ‘Theory of Being,’ ‘Metaphysics of Immanence,’ ‘Metaphysics of Absence or of the Void.’ (Species of voidism occur in Indian philosophy and Judaism.) Here, however, the forms are immanent in that they are self-adaptations whose explanation, given earlier, shows them to be near symmetric, relatively stable and dynamic]
Ask, ‘Is the world normally calculable as in a mathematical realism or is it as a poetic realism – felt but incalculable? Is Being infinitely deep or are they lulled by reason’s charms into eternal sleep?’
The major views on the nature of mathematics are Platonism and logicism, discussed above, and intuitionism and formalism. Roughly, intuitionism is the view that the concepts of mathematics are mental constructs and formalism is the view that mathematics is captured in formal systems without regard to the meaning of the signs used in the systems
David Hilbert is strongly associated with the origin of formalism and the Dutch mathematician L. E. J. Brouwer (b. 1881, Overschie, Netherlands) is, similarly, strongly associated with intuitionism. The logicist Kurt Gödel (b. 1906, Brünn, Austria-Hungary) subscribed to a Platonic view of mathematics
Platonism and intuitionism have similarities and might be identical if the intuition were perfectly attuned to the form of being. The foregoing discussions may be seen as favoring a Platonic view of mathematics over the alternatives. It should be remembered, however, that the present view of Platonism is one that does not require or refer to a separate Platonic world
Is there a way in which induction can be reduced to deduction? The answer given so far is ‘No!’ Deduction occurs within a definite context. Induction (on the assumption of infinite context or Universe) occurs within an indefinite context. Therefore, in general induction cannot be reduced to deduction. However, it is possible that context can bring induction close to deduction. If, from among all forms of Law, there is a small fraction (subset) whose population (in the universe or context) must dominate all others and if the fraction can be defined in terms of (e.g. a small number of) parameters, then it is possible that a finite number of observations can imply a system of laws to a high degree of probability or even to necessity. Similarly, if the set of (ideal) structures were limited the formulation of mathematical systems to capture such structures could also be close to deduction
That the Theory of Being explicitly captures depth shows that structures that refer to All Being and absence of Being can be captured. However, since Variety cannot be captured explicitly i.e. in a single picture, it would seem that logic and mathematics are ever open fields – except that, in normal domains, they may be (normally) closed to local intelligence
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?
What is the distinction between the laws of science and Logic? A scientific theory e.g. Newtonian Mechanics sets up a ‘picture of being:’ point particles located in absolute space and Time which may be regarded as the specification of a context (Universe.) The laws that pertain to the context may be regarded as logical. The scientific picture is that the particles interact (forces) and that this (via equations of motion) determines the motion of the particles. The particles are part of the ‘logic’ and the forces of the ‘science.’ The distinction was made because there is a contingent character to the forces (e.g. gravitation) and the laws (connecting force and motion.) Clearly, however, there is an arbitrary character to the distinction. Given a scientific theory (formulated mathematically or conceptually,) the derivation of results (e.g. motion) is logical. Thus the theories of science may be regarded as special logics. Scientific theories may be seen as special cases within Logic. Within Logic, special contexts may be specified that define ‘logics.’ There is a distinction between science and Logic but the boundary is not definite. Necessary law and contingent law are not altogether distinct
If Logos is the form of all being i.e. the collection of forms, Intuition is attunement to Logos, Metaphysics (Logic) may be regarded as the formal or symbolic study of Logos and Induction is the translation of perception (and Intuition) into Metaphysics. The sciences, art and poetry may be seen as ‘departments’ within Metaphysics (Logic)
Logic is the constitutive Form of Being. A logic is the constitution of a context. A law of behavior e.g. a scientific law (within a context) is one of many possible laws (each contingent) that may be mutually inconsistent but each consistent with constitution
As noted, the distinction between constitutive ‘law’ and contingent law is not precise (in the present formulation)
The fundamental principle, demonstrated in the section ‘Theory of Being. First proof…’ is the entire system of consistent descriptions is (must be) realized i.e. where use of ‘is’ has the atemporal or global sense. The earlier proof deployed the concept of the void (whose existence was demonstrated) and properties of the void. Here, a more direct demonstration i.e. one that does not refer to the void is attempted. However, difficulties arise that are a result of seeking to base a proof in this world e.g. the local cosmological system rather than in the nature of all being (the universe.) The earlier proof is seen to have succeeded because the concept of all being is implicit in the void or absence of being. The present attempt results in revelation of a number of fundamental equivalents to the existence of the void
Preliminary: Necessary propositions are those that are true in every sub-manifold of the universe (i.e. true without reference i.e. true by tautology) while contingent or ‘material’ propositions are those that are realized only in some sub-manifolds (if a proposition is realized in no sub-manifolds its negation is Necessary so the truth of a contingent proposition in at least one sub-manifold is implicit in the definition.) Proof is simple. Here ‘Ù’ is ‘and’ and ‘~’ means ‘not.’ Let C be contingent i.e. sometimes true and sometimes false i.e. ~C must also be contingent. If ~C is always true then it is necessary; therefore ~C must be sometimes false i.e. C must be realized. I.e. every description (with reference) must be realized. Similarly, if A, B, C… are contingent and A Ù B Ù C Ù … is consistent, it is realized. Weakness of the proof: It is assumed that the Necessary propositions are not synthetic i.e. that they are true in virtue of not making material reference. It is conceivable that a synthetic proposition, e.g. the electron is indivisible, could be necessary (modern physics appears to indicate that the electron is indeed indivisible.) In this case, the negation does not appear to be a contradiction but would not be realized. The divisibility of the electron follows from the Theory of Being; however, the objective of a second proof is to not assume that theory. It appears that the fundamental principle cannot be deduced from logic or from the being and nature of the local world alone and that considerations of all being (and absence of being) may be necessary to proof (the importance of these concepts in the first proof and in the development of the metaphysics has been emphasized earlier.) Is there a concept of Logic that would permit a logical proof? Perhaps that Logic can contain no contingent facts e.g. the fact regarding the structure of the electron. Observation: The importance of reference is revealed again. Also note: The proof idea is inherent in the notion of contingency: if A and B are contingent, then when A is true, B must sometimes be false or else B would follow logically from A. If A = N (the null proposition i.e. nothing is asserted) and B = ~C then ~C must be sometimes false or else it would follow from N. Therefore, every contingent proposition C must be sometimes true… The proof is related to Hume’s statement regarding Necessity ‘From the truth of one proposition, the truth of another does not follow’ (and Wittgenstein’s modification in which ‘proposition’ is replaced by ‘atomic proposition.’) The emphasis on another proposition indicates that the other proposition is distinct i.e. not logically contained in the former; in the case of logical containment the second proposition does follow but it is not strictly an other proposition (the explanation is not required in the case of atomic propositions for the concept of an atomic proposition requires that it be logically independent of all other atomic propositions)
It is concluded that the concepts of being, absence of and all being or their equivalent are necessary to proof of the fundamental principle of being and therefore to the development of the metaphysics and the concept of Logic. It is interesting that an extreme application of Ockham’s razor may accomplish the same task. 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. 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
That no contingent proposition is universally true has meaning in regard to the existence (demonstrated earlier) of many actual worlds (sub-domains of the universe e.g. local cosmological systems. Every contingent proposition is true in some ‘worlds’ and not in others. A local meaning of ‘possible’ regards a contingent proposition that though perhaps not true in this world, is true in a similar (e.g. with the same laws of physics) world. However, as seen earlier, regarding the universe there is no distinction between possibility and actuality. That no contingent proposition is universally true may be seen as an absolute indeterminism
Detailed treatment of logics awaits a later writing. However the following assertions may be made
The concepts of truth, of falsity and implication are central to logic. Implication is important as the bearer of truth by deduction
The law of non contradiction is fundamental. The law of the excluded middle has been seen to be questionable at least in that if arbitrary forms of proposition are accepted it is no longer true. If it is true, a variety of paradoxical forms of proposition arise (e.g. in that propositions that do not have a truth value are assumed to have one.) Thus the law of the excluded middle is also fundamental in that it excludes certain paradoxical forms; however, from the discussion of the role of reference, it is not clear that it is necessary in order to exclude those forms or that it excludes all paradoxical forms
The Theory of Being has implications for Possibility, actuality and Necessity i.e. for ‘modal’ logic
From the ideas of Logic as constitutive Form, the suggestion follows that logics are constitutive forms of contexts
Some logics. The following are included as outlining some areas of later application. What is needed first is an approach to classifying logic – see e.g. Susan Haack’s Philosophy of Logics (1978.) Then, select some principle(s) of classification e.g. formal vs. informal (‘Theory of Rationality’ may be placed here;) meta-logic; then classical logic (2-valued sentence calculus, predicate calculi, identity theory for ‘=’, and immediate inference and the Aristotelian syllogistic on the categorical propositions) vs. ‘extended’ logics (modal, tense, deontic, epistemic, preference, imperative, and erotetic –interrogative– logics) vs. ‘deviant’ logics (many-valued, Intuitionist, quantum, and free logics) vs. ‘inductive’ logics; also consider ‘applied logics’