Logic

Anil Mitra, © May 2010, © Latest Revision December 26, 2011

Contents

Introduction to logos and logic I 1

Introduction to logos and logic II 2

Concept of Logic and relation to the fundamental principle 2

The concepts of logic and Logic 2

The fundamental principle of metaphysics expressed in terms of Logic 2

Logic, grammar and meaning 3

From Investigation in the modes and means of transformation > Logic 3

Logic 4

Method 5

On the New Conception of Logic 6

Proof of its Validity 6

Character of Logic 6

Introduction to logos and logic I

The title of the section is Introduction to logos and logic

[From Intuition > A preview of the Universal metaphysics]

If we define the logos as the universe of logically possible states then the logos includes the Universe

It does not appear that satisfaction of the principles of logic implies existence. With an indefinite notion of the universe some state whose concept satisfied logic might or might not exist. However, the Universe is all being. What is the significance of a state whose concept is logical but does not exist? That state lies out all actuality; in a not unreasonable sense it could not exist. But this appears to contradict the fact that the state satisfies the principles of logic. Perhaps, then, satisfaction of the principles of logic does imply existence. If that is true every state in the logos is in the Universe and therefore the logos and the Universe in all its details are identical. In considering the Void, this will be seen to be true but will require introduction of an alternative conception of logic

Introduction to logos and logic II

The title of the section is Introduction to logic and logos

[From Metaphysics > The Universal metaphysics > Universe]

Now it is reasonable to assert that if a state exists it must satisfy the principles of logic. What this really means is that the concept or description of the state must satisfy the principles of logic (it is implicit of course that the principles are all ‘the’ relevant correct principles.) Therefore:

If we define the logos as the universe of logically possible states then the logos includes the Universe

Concept of Logic and relation to the fundamental principle

[From Metaphysics > The Universal metaphysics > The Void]

The concepts of logic and Logic

It was noted earlier that every axiom or principle of logic has been subject to reasonable doubt regarding its universality. The thinking is therefore inverted and Logic (capitalized) is defined as the principle of being. The concept of Logic is that it is the set of sufficient and necessary conditions that the conception of a state of affairs must satisfy for that state to exist. A state of affairs exists if and only if it is ‘Logical:’ if it is Logical it must exist, if it exists it must be Logical

The fundamental principle of metaphysics expressed in terms of Logic

The second form of the fundamental principle of metaphysics or principle of reference now follows:

1. Subject to Logic every concept has reference

The principle of reference is the basis of the unified theory of Objects developed in Objects

An obvious system of objections arises. What is the reference of a system of ‘laws of physics’ that are consistent but quite different from the laws of our cosmological system? It will be seen later that corresponding to every consistent set of laws there must be a cosmological system. Next consider that the principle suggests that the laws of physics of our cosmological system do not have necessary or eternal purchase: this should not be an objection for it is consistent with what we know for our laws to be contingent and of finite duration. However, since our laws are ‘a concept’ it is necessary from the system that they should have some purchase as for example in our cosmological system so far and so far as is known. Thus although the principle hints at contradiction, it is not contradictory and although it hints at chaos over law it actually supports law. Thus the Normal behavior of our cosmological system and others is supported by the principle. The concept of the Normal as used here will be developed below; it may be emphasized that it will not receive meaning such as a statistical or functional or normative meaning

That every concept should have reference appears to not make grammatical sense, e.g. what is the Object of the concept of ‘redness.’ The resolution of this concern is addressed in Objects. In anticipation of the later development:

2. Subject to Logic every concept has an Object

Even though Logic is introduced as a definition the conclusion is far from empty because the known logical principles are at least approximations to Logic

Logic, grammar and meaning

[From Objects]

The purpose of this section is to see (from the principle of reference and the developed theory of Objects) grammar as concept on par with Logic and so to further clarify the nature of linguistic meaning

Logic defines the form of concepts or descriptions that is necessary for them to be capable of valid reference

Given that the Universe is all actual being and that there is no distinction between the actual and possible:

Logic defines the form of concepts or descriptions that have reference

In talking of a limited context as though it were the Universe, it would be necessary reintroduce the first italicized form above

Thus the thought due to Wittgenstein that logic is grammar

Meaning involves concept and Object or sense and reference; this was perhaps first pointed out by Frege and taken up by Wittgenstein. Therefore, Logic or Grammar are aspects of meaning; and, further, a full Meaning determines Metaphysics… and a full and final Metaphysics determines Meaning

Since Logic has reference, Logic has meaning

From Investigation in the modes and means of transformation > Logic

Logic

Logic and logics. Logic as the theory of concepts that have andor are capable of reference; as the theory of the actual; equivalently, of theory of the possible and the necessary. Logic as the law on the concept side; as the absence of Law on the Object side; Logic as immanent, therefore as Logos. Reference as crucial to logic. Development of logics

Experiments with ‘toy’ logical systems suggest that a requirement of reference of every ‘atom’ of all logical statements should have reference in order to avoid paradox. 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. It remains true that the requirement of reference may have deep consequences; and these consequences may reveal the artifactual approach to consistency to have an ad-hoc character

Logic

[From Method > Implications for the tradition]

Argument is the means of establishment of knowledge (fact)

Corresponding to the distinction between observation and thought, argument is broken down into establishment of facts and inference of facts (conclusions) from other facts (premises)

In inductive inference the premises support the conclusions; the conclusions do not necessarily follow from the premises and may therefore contain novel elements not already present in premises; i.e. induction produces truly new but not entirely certain knowledge; it is questionable whether the result should be called knowledge but induction is a part of the development of science and therefore essential to that endeavor. Since observation and experiment are not infallible (questions of completeness—the entire range of pertinent facts may be inaccessible—and accuracy) it is perhaps not entirely unsatisfactory that induction is not certain (early in the history of science there had been hopes for a necessary logic of induction but since logic is or has come to connote necessity there can be no logic of induction)

Deductive inference asserts that the conclusions are necessary consequences of the premises; if the premises are true, the conclusions must be true (therefore if the conclusions are false, the premises must be true;) deduction produces nothing new—it amounts to tautology; of course immense creativity may be required to provide the steps that are the deduction… that, in a sense, constitute the tautology. Deductive arguments are valid or invalid, and sound or not sound. An argument is valid if and only if the truth of the conclusion is necessary consequence of the premises. A sound argument is a valid argument with true premises. Given true premises, conclusions by deduction are also true. If all men are mortal and Socrates is a man, then Socrates is mortal; here it is obvious that the conclusion is contained in the premise (tautology;) however, since the premises are not certainly true, the argument is not entirely sound. The point may seem as though to pick a nit but what is trivial in this case might may be an immense gulf when absolute certainty is necessary. Sound argument is an essential element of sound reasoning—especially of sound philosophy

It might seem that a deduction applied to imprecise premises is a waste of a good deduction; however, there is a paradigm case in science in which given some data and a theory by induction, consequences may be derived by deduction and the value of this standard process of science lies in the fact that if the conclusions are at odds with experimental test then it is known that the error does not lie in the deduction but in the data and or theory. Mathematics is a paradigm case of deductive argument. The premises of mathematics are typically axioms expressed symbolically in terms of abstract terms. Although an arbitrary system of axioms cannot be assumed to be true, some axiom systems (e.g., Euclid’s first four, Euclid’s first four and negation of five, Euclid’s first five) can be shown to be consistent and therefore to have true interpretations. The structure of a mathematical system thus founded on consistent axiom systems and theorems by deduction has true interpretations.

Although logic has sometimes been called the theory of sound (deductive) inference, it is commonly used to refer to valid (deductive) argument (to repeat, logic connotes necessity and therefore there is no logic of induction)

Today, logic refers almost exclusively to deduction. There really is no method for induction in the way that we think logic is ‘methodical.’ Perhaps the prime example of induction is hypothesis formation (e.g. in science;) there are other ‘kinds’ of induction but there is invariably a leap that makes the ‘induction’ not entirely certain regardless of whether it is called hypothesis or generalization

As deduction, there are two ways to view logic. In the first, logic is the sequence of inferential steps in logical deduction. In the second, perhaps equivalent way, logic is the system of necessary relations among truths

The truths of the Universe must be a subset of all the truths compatible with logic. In the metaphysics it was seen that there is an identity of the two sets but since the actual logics of the tradition might be (and almost certainly are) inadequate with the resulting conception, we named it Logic

Method

But what is the method of logic? We may think of step-by-step deduction as a method of deriving conclusions that are true if the premises are true. In this sense, logic is used in precise argument from premises and especially as in mathematics. The step-by-step is (logical) justification. However, coming up with the sequence of steps is not generally ‘logical’ and may require imagination (creativity.) This aspect to method, therefore, has both discovery and justification (similarly the formulation of axiomatic systems in mathematics is creative as is hypothesis formation in science)

The application of logic just derived is analogous to prediction in science which is, simply, the derivation of conclusions about some configuration of material elements in the world on the basis of the laws (for application and test.) That is part of scientific method; the other part is hypothesis formation

What is the corresponding function of method in logic? Since the conclusions of logical process are regarded as necessary, we do not generally think to test them. However, there is an implicit test: we do not know that logic is perfect and it is possible that even while following given logical principles faithfully we might still come up with a contradiction (whose source might lie in premises / axioms or in the principles of ‘logic’ itself.) Additionally and importantly since logic is expressed symbolically there is the possibility of symbolic approaches to showing consistency

So, in analogy to theory formation in science, how do we come up with the principles of logic? In part, since the principles of logic are time honored we think of them as a priori. However, there is an empirical component which is not the empiricism of science (empirical with regard to the e.g. material world) but an empiricism over symbolic systems—as, e.g., in the predicate calculus (simpler systems are almost true by examination.) As we have seen, logic itself is subject to hypothesis formation and testing (in the simpler parts, e.g. sentence calculus and the syllogism of Aristotle but not in axiomatization of predicate calculus or set theory, the simplicity of the models make examination of consistency transparent)

Thus in relation to logic there are two aspects of method. As inference, logic is a method. We have seen that it is not certain but perhaps the highest of our certainties. This leads into the second aspect of method for logic—what is the derivation of the principles of logic? Clearly there probably can be no general logical derivation of logic. The process is one of drawing into intuition, laying out Logic and developing the logics within that framework and that development includes hypothesis and test

We see, then, that the apparent a priori character of logic is significantly due to its remoteness and its necessity in trivial cases; it is also due to the fact that logic has become concerned with symbol systems. In fact logic is empirical in a special sense and hypothetical (that it is empirical implies that it is testable and tested even if implicitly; initial empiricism lies in the coding of experience into the principles)

We also see method and content overlapping in logic

And we see also Logic as the only universal law (limit on concepts.) Logic is a concept that is perfectly faithful to an Object—Logos. The notions are implicitly known. Explicit realization of Logic as logics may be approximate but reveals its power-as-such. However, the present conception, Logic, frames the logics: maximizes and limits the range (how is a topic for study and if there are no atomic facts the field must be quite open)

On the New Conception of Logic

Proof of its Validity

The Universe has no limits—this is the Principle of Being

The meaning of the statement is brought out in the following. The Universe could not have greater variety than it does. Subject to logical constraint, every referential concept is realized

The second statement has the criticism that the logics are not perfect. Therefore define Logic as the requirement that referential conceptual systems have reference

Character of Logic

It is a conception of Logic

It is not an ad hoc conception because it arises from the Universal Metaphysics

It is not empty because the logics are at least approximations to it

That Logic defines Logos, the Universe in all its detail suggests that Logic is infinitely richer than previous conceptions entail and infinitely richer than the logics

It is interesting in that it defines the Universe

Whereas the view of logic as—means of—valid argument is restrictive (it rules out invalid argument), the new view is permissive in allowing much more than otherwise imagined