iSoul In the beginning is reality

Two kinds of negation

This is a follow-up to the introductory post on Laws of Form here.

There are two kinds of negation: contraries and contradictories, and Laws of Form (LoF) represents both types. Furthermore both types apply to terms and propositions.

Contraries are two complete opposites; the negation of one is the other. The poles of a magnet for example are contraries: not South is North and not North is South. This negation is represented by the Law of Crossing.

Contradictories are partial opposites; the negation of one is different from the other but not necessarily the exact opposite. The negation of a proposition is a proposition that is inconsistent with it. This negation is represented by the Law of Calling.

Calling is a kind of negation but calling again doesn’t return to the original proposition; it reiterates the negation and remains in the same place.

If we negate North as a direction, do we get South? Not necessarily; we could get East or West which are different from North. As directions, North and East are contradictories.

It’s unusual to completely negate a proposition but it can be done. “The place is on North Main Street” is contradicted by “No, it’s on East Main Street.” The contrary proposition is “The place is on South Main Street” in the context of a north-south oriented Main Street.

To model both negations in ordinary arithmetic with 0 and 1, use the two operations: standard multiplication for calling and an alternate multiplication for crossing defined as:

x alt y = (x-1) * (y-1).

Then zero represents the marked state and one the unmarked state.

The beauty of LoF is that these two kinds of negation are combined into one symbol — as is the word “not”.


Laws of form

The remarkable book Laws of Form by George Spencer-Brown was published in 1969 and is almost forgotten today. The best expositors have been William Bricken with his boundary mathematics, Louis Kauffman with his knot theory, and Francisco Varela with his work on self-reference. Otherwise it has become something of an underground classic but otherwise forgotten. There are several reasons for the latter, including the exaggerated claims of the author and some enthusiasts. That said, I think it’s worth rehabilitating the Laws of Form (LoF) and rightly discerning its significance.

LoF is a work on diagrammatic reasoning in the tradition of Leibniz and CS Peirce. It is a calculus, complete with arithmetic and algebra, based on the act of making and indicating a distinction. Thus it is a work of mathematical realism, which begins to explain why it is not of interest to anti-realists. Its greatest accomplishment is the unified treatment of injunction and indication, of implication and negation via a single symbol, called a cross.

Here are the arithmetic axioms of the calculus of indications:


That’s it. The inverted “L” is the cross symbol. A cross next to another cross is equal to one cross; this is the Law of Calling. A cross inside another cross is equal to blank, that is, as if no cross had been written. This is the Law of Crossing, hence the name of the symbol, Cross.

This is a two-dimensional calculus, which gives it advantages that one-dimensional notation does not have. It also makes it hard to display typographically. The best alternative is simply to use parentheses or brackets:

( ) ( ) = ( ) and (( )) =  .

These arithmetic axioms can be used to derive two algebraic axioms:

((A) (B)) C = ((A C) (B C)) and ((A) A) =   .

From this a complete calculus can be constructed. It is isomorphic to Boolean algebra and other functionally-complete binary calculi, which is another reason LoF hasn’t stirred a lot of interest.

Things get more interesting as we review where this calculus comes from. Again this exhibits its realism; the standard approach for mathematics and symbolic logic is to begin with algebraic axioms or postulates without reference to any model or reality.

Let’s begin with a blank surface, say a blank page of paper. Now draw a distinction on this surface; that is, draw a closed curve or divide the page into two parts. Notice what has happened: part of the paper is distinguished from the rest of the paper by being to one side of the curve, say the inside. The curve separates the other side from the inside; call it the outside. But the original piece of paper is still there. We can still consider the whole piece of paper.

This process is symbolized by LoF as follows: what is outside the cross (or parentheses) can be seen inside the cross (or parentheses) if we change perspectives to the whole page. This is symbolized in a theorem:

(A) B = (A B) B.

So the distinction that is drawn is not between two contraries but within one space, represented by the whole page. It also shows the distinction can be undermined. This has been exploited to represent self-reference.

Much more could be said about LoF but that’s it for now.

Science in history

Scientific theories are in principle subject to underdetermination in that multiple theories could account for the data. In the natural sciences this possibility is strongly resisted. When Darwin proposed his theory in 1859, he could not show that a version of special creation would not account for the data. What he and Huxley did instead was to advance a new definition of science that was completely naturalistic so that special creation was no longer science, no matter what evidence it might have in its favor.

The late 19th century also showed the rise of agnosticism and secularism, which are tied to the new definition of science. William Dembski has a chapter on this in his book Intelligent Design, The Bridge Between Science & Theology (Chapter 3 on the Demise of British Natural Theology). Dembski is right to call naturalism idolatry.

Darwin’s strategy won the day so that few people now are even aware of the older definition of science that allowed special creation. One could say that creationists and intelligent design advocates have this in common: both reject the naturalistic definition of science. In that sense both want to return to a former definition of science, though it is not being stated as a return but as a better definition. Perhaps that reflects the anti-historical bias of our day — who cares about the past that has been superseded?

One could argue that the 19th century naturalistic and positivistic turn was a result of Enlightenment thinking in the 18th century. And one could argue that Enlightenment thinking arose as a result of mechanistic thinking in the 17th century. And one could even argue that that was a result of the nominalism that arose in the late medieval and renaissance periods. But even so there has been much continuity throughout this time of what natural science is.

Is it better to promote a post-naturalistic, post-secular science or focus on critiquing the mistakes of the past? Both are worth doing, though the latter has been neglected. Several points could be made by those who accept a form of special creation: (1) they are not scientific newcomers or rebels; (2) they are in continuity with a long past; (3) their opponents have broken from the historic mainstream of science.


Nature and what?

A recent meta-analysis on the old question of nature vs. nurture (Meta-Analysis of Twin Correlations and Heritability) concluded that overall the variation of human traits and disease is 49% due to genetic factors and 51% due to environmental factors. There is a similar question about nature vs. culture in anthropology and education.

Evolutionary theory is fraught with the combination of innate and environmental effects. The emphasis has been on the innate that is reproduced with the environment providing whatever is missing (as in the conditions that ensure the next step of evolution).

Naturalism throws a wrench in all of this because it says there is nothing outside of nature. The universe has no environment so ultimately it’s all nature. Nurture, culture, and the like must be reduced to nature.

I dispute naturalism but what exactly is the alternative? It is that the universe is an open system. We say it is open to God but that does not get to the heart of it. A closed system can be just as open to the transcendent God who creates and sustain it.

An open universe is open to more than God; it is open to spirit, mind, intellect (commonly called intelligence). If we want to be the most precise (and the least accurate), an open universe is open to information in the sense of Shannon (for example see Shannon and Weaver’s book, The Mathematical Theory of Communication).

Not coincidently the intelligent design and various creationist movements have been hammering home the necessity of intelligence and information for the fine tuning of the earth and the universe, the diversity of life-forms, and the basis for natural science. Not surprisingly those caught up in naturalism aren’t listening. They are living in a closed world, deaf to the larger reality.


Historical accuracy and precision

Accuracy refers to closeness of measurements to the true value. Precision refers to variation of measurements to each other. So precision is relative but accuracy is absolute.

Accuracy and precision can apply to statements as well as measurements. For example, if I say, “Today is Tuesday. This is the third day of the week.” but it is Thursday, the two statements are not very accurate, though they would be more accurate than saying it is Monday. Since the two statements are synonymous, they exhibit precision. A statement such as “It is early in the week.” is less precise than specifying a specific day because it contains ambiguity about exactly which day is specified.

There is something of a trade-off between precision and meaningfulness. A very precise statement such as “The dial reads 2.698” doesn’t say much. For some purposes, that’s all that’s desired. But as part of a larger conversation, much more could no doubt be said. What was the investigator hoping to accomplish, scientifically and professionally? Will this instrument reading make much difference? And so forth.

The definition of “information” by Shannon and others shows this trade-off. Statements that are more surprising, more unexpected have more information than other statements. But precise statements contain a lot of redundancy so they don’t have much information in this sense. On the other hand, a literary work will likely contain many surprises and multiple meanings that stir the heart and imagination.

We have come to expect historical accounts to include specifics such as dates that are tied to a solar-year calendar, places with widely-recognized names, people with specifics about place and time of birth, death, and other significant events, and contextual information about culture and political system.

A traditional account such as the Bible is less precise than histories written today because it lacks many of the specifics expected. However, it can be, and is accurate. It contains primarily qualitative rather than quantitative accounts so prized by science. By following Ockham’s razor science minimizes qualities and allows quantities to expand indefinitely. So Konrad Lorenz was right, “Scientists are people who know more and more about less and less, until they know everything about nothing.”


Complementary Catholics and Evangelicals

There is a kind of complementarity between Catholics and Evangelicals today. Very briefly, accepting the authority of the Pope is the key to being a Catholic today. There seems to be little else that unites the bewildering variety of Catholics, from Latin mass hard-liners to Marxist liberation theologians and from the superstitious to the intellectuals.

Accepting the authority of the Bible is the key to being an Evangelical today. There seems to be little else that unites the bewildering variety of Evangelicals, from fundamentalist hard-liners to breezy popularizers and from stuffy traditionalists to laid-back gen-Xers.

Catholics justify the papacy by a hierarchical conception of the church and an emphasis on unity. Evangelicals justify a multiplicity of denominations by a bottom-up conception of the church and an emphasis on doctrinal purity.

Catholics criticize the sola scriptura of Evangelicals by pointing out that a church is needed to determine what is or is not scripture. Evangelicals defend the authority of the Bible by its holy authorship, self-referential integration, and miraculous quality.

Evangelicals criticize the papacy (and the devotion to saints) by pointing out that Christ is our only mediator. Catholics defend the authority of the papacy by its long-running preservation, doctrinal fidelity, and unifying position.

There are two other groups that aren’t part of this complementarity: the Orthodox and the Liberals. The Orthodox are the extreme traditionalists, trying to preserve the church of late antiquity. The Liberals are the extreme accomodationists, trying to be as consistent with larger social and intellectual trends as possible.

The real scientific method

The real scientific method is the inductive method invented by Socrates and elaborated by Aristotle, Bacon, and Whewell. It is different from the hypothetico-deductive method invented by JS Mill in the 19th century which is passed off as the method of modern science.

Consider Francis Bacon. He called immature concepts “notions”. Induction starts with notions from common experience and iteratively improves them using sense experience until the form or essence is identified. This form is the cause in the full sense of the word; the form is what something truly is — and so should be defined as such. Thus the induction is true by definition. Sound circular or trivial? It’s not because getting the concepts right is what inductive science is all about.

William Whewell described two complementary processes, the explication of conceptions and the colligation of facts: To explicate a conception is to clarify it by identifying what it contains, by unfolding it, for example by surveying and examining examples. The end result is a careful definition of the conception. Colligation is the complementary process of binding facts together by means of a precise conception. The result is an induction, which is the narrowing of a generalization until it is exact and universal.

Yes, induction leads to hypotheses and testing but this is for the purpose of finding the consilience of inductions, the confirmation of inductions in different and multiple ways. The key step takes place before hypotheses and testing: the discovery of a conception of the facts that binds them together.

This understanding of induction was lost in late antiquity until Francis Bacon restored it and laid a foundation for science that lasted two centuries. Then in the 19th century Richard Whately and JS Mill replaced it with a different method, one that came to be called the hypothetico-deductive method, which depends on uniformity and naturalism, and is conceptually confused and logically deficient.

John P. McCaskey and others have explained the history of Socratic induction in science. As examples he gives cholera, electrical resistance, and tides (see here).


Philosophical realism

Philosophical realism, or simply realism, is a philosophy that begins where we all begin: with our common sense, our common everyday experience. When Samuel Johnson famously dismissed Berkeley’s idealist philosophy with his “I refute Berkeley thus” and then kicked a rock, he was asserting realism in contrast with idealism.

There are basically two kinds of anti-realist philosophy: idealism (or ideology) and materialism (or naturalism). Idealism begins with an idea that is asserted to be the principle of reality. Materialism begins with physical matter and everything is asserted to be reducible to this matter (or nature).

Realism as a philosophy began with Socrates, Plato, and Aristotle, the three leading lights of classical thought. Plato’s realism of Forms was rather extreme but his student Aristotle taught a moderate realism with a duality of form and matter. Plato’s realism motivated Neo-Platonism in later ancient and early medieval times. When Aristotle was rediscovered, his realism motivated Scholasticism in the Middle Ages.

The Scholastics were limited in their understanding of Aristotle and ended up giving him a bad name so that the early moderns opposed all things associated with Aristotle. Ironically, people such as Francis Bacon who assailed Aristotle also incorporated key elements of Aristotle in their own philosophy.

Modern realists include the American philosophers C. S. Peirce, W. V. O. Quine, M. J. Adler, and Thomas Nagel. While they mostly lack a religious faith, they are not inimical to true religion. Their realism puts them at odds with many of their contemporaries and closer to true religion than might appear at first.

Realism is an open philosophy, contrary to the anti-realist philosophies which have decided what reality is from the start. Realism is open to reality, however that may turn out to be. Realism is consistent with the common sense that people in general have and so is a way of engaging people in a common pursuit of the true, the good, and the beautiful.

The legal fiction of same-sex marriage

Now that the U.S. Supreme Court has invented a right to marriage that extends (so far) to any two non-married adults, many articles have been written criticizing the Court. Some have called same-sex marriage a legal fiction, apparently using that as a simple pejorative rather than a legal term. I’m not a lawyer but I think legal fiction is an accurate term to use.

It is well-known that corporations can be treated as persons by the law; this is a legal fiction that ensures that groups of people are treated similarly to individuals. It is fiction because a corporation is not literally a person and so should be treated as a person only in limited ways. That applies to same-sex marriage, too.

Same-sex marriage is not literally marriage. No law or interpretation of law can change the reality of marriage, which is only for opposite sexes. But laws or interpretations of law can create legal fictions in which something that is not literally true is treated as if it were true in limited ways. That is what the Court has done.

The question now is what are the ways that same-sex marriage is like real marriage and what are the ways it is not. Unfortunately the Court did not specify this or even acknowledge the issue. They left that for the future.

This ensures a turbulent time for the U.S. while those who want to maximize the treatment of same-sex marriage as real marriage will battle with those who want to minimize it. The dissenting justices were right to point out how this could have been avoided by letting the democratic process deal with the whole matter.

But the Court short-circuited the democratic process and inserted their judgment for the judgment of the people. That is a defeat for democracy. And it undermines respect for the judicial branch of government.

Morality and same-sex civil marriage

Same-sex civil marriage has become legal in various jurisdictions and states. What should those do who do not recognize the moral validity of these marriages? This has become an issue for people personally and in business.

Consider the latter first. There have been cases of a photographer and a baker: what should they do if customers ask for a cake or photographic services for a same-sex wedding? This is not a case about the customers themselves but about the use or customization of a product or service.

In general what a customer does with a product or service is their business, as long as a criminal enterprise isn’t involved. Other objectionable uses are possible, too. Custom T-shirt providers should be free to reject objectionable messages. But a baker should not object to making a standard wedding cake for a customer who uses it as they desire.

Wedding cakes often carry ornaments on top depicting a bride and groom. What if the customer requests an ornament with two grooms or two brides? If the bakery does not have such ornaments, they simply say so. It’s not a product they sell.

Services such as photography are customized to some extent so how they are advertised makes a difference. Until now, advertising “wedding photography” was pretty well understood. The term covers marriage ceremonies in general. If there are particular aspects of an ethnic or religious wedding that the photographer is unfamiliar with, then either the customer hires a different photographer or negotiates their specific requirements.

Photographers who don’t want to photograph same-sex weddings should imply that in their advertising. “Traditional wedding photography” would tell customers what services the photographer offers. Customers should not expect other services. It’s possible the courts won’t accept that, but it remains to be seen how else they will set the boundaries of what can and can’t be done.

An example of a personal issue would be an invitation to a same-sex wedding ceremony from a relative. There may not be one answer for everyone. On the one hand, God “makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust.” [Mt 5:45] On the other hand: “Bad company ruins good morals.” [1 Cor 15:33] “Take no part in the unfruitful works of darkness, but instead expose them.” [Eph 5:11]

The best policy is to have a policy and stick with it. Then people know where you’re coming from and aren’t arbitrarily picking on them. And be able to defend your policy.