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Tag Archives: Logic

nature and application of logic

Contraries and contradictories

September 28, 2018 3:03 pm ⋅

Propositions are contrary if they cannot both be true (though they both may be false). Propositions are contradictory when the truth of one implies the falsity of the other, and conversely.

Two properties are contraries if their intersection is null and their union is a whole. A merism is a rhetorical combination of two contrary terms that refers to an entirety. E.g., “young and old” refers to everyone.

The privation of a property is the contradictory opposite property, as disease is the privation of health (Aristotle Metaphysics Z 7.1032b3-5). St. Augustine argued that evil is the privation of good. There can be good without evil, but not evil without good.

A negation presupposes an affirmation but an affirmation doesn’t presuppose a negation. There must be something to negate for negation to have an effect. Contraries presuppose one another. E.g., there is no young without old, no up without down.

In symbolic logic, contradictories are represented by contrary symbols. The closest to representing contradictories is to represent the positive as unmarked and the privation as marked. This is what the Laws of Form does.

See also here and here.

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Form and logic

May 30, 2018 3:37 pm ⋅

I’ve written before about Laws of Form (the calculus of indications); see here and here.

In the beginning is an undifferentiated state, an unmarked space. The first distinction is the first differentiation, the advent of a mark, a cross, a form. The unmarked state is the urgrund of the form, its origin and basis. The marked state is the form of the mark, a cross into markedness. The form encapsulates the progression from undifferentiated unity to differentiated hierarchy.

Spencer-Brown applies the calculus of indications to propositional logic in his book Laws of Form (Julian Press, New York, 1972), Appendix 2. This requires an interpretation of the mark as either true or false. He acknowledges that the choice is arbitrary, and then takes the mark to represent truth. This corresponds to the common logical convention that a proposition is assumed to be false unless it is shown to be completely true.

True propositions can be conceived as arising islands of truth in a sea of falsehood; or they can be conceived as the sea itself, interrupted by islands of falsehood. They are opposite conventions, and there is no logical reason to prefer one or the other.

But they encode different conceptions of truth. Is truth rare or common? Is falsehood rare or common? Do we give a proposition the benefit of the doubt or accept it if it has some truth? Or do we reject all doubtful propositions or partially true propositions?

The form is pre-logical; it assumes no convention about truth values. The form also assumes no convention about classes. A logic of classes can adopt a convention that the mark is the universal class or the null class. It depends on whether classes are conceived as arising from a unmarked space of nullity or void, or from an unmarked space of everything or pleroma.

Laws of form easily fits universal forms of the logic of classes but existential sentences are problematic. Spencer-Brown observes this interpretative theorem:

An existential inference is valid only in as far as its algebraic structure can be seen as a universal inference.

This theorem holds for both conventions about classes, that is, whether existence is considered to stand out from a void or against a pleroma.

These dual conceptions lead to a distinction between sets, which are composed of members that may have certain attributes, and classes, which are composed of attributes that may have certain members. The null set is a set with no members. The universal set has all members within a given context. The universal class has no attributes. The null class has all attributes within a given context.

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Propositional logic calculation

March 6, 2018 9:32 am ⋅

George Boole is known for introducing a logical calculus for propositions in the mid-19th century. Although others before him such as Leibniz worked on logical calculi, Boole developed the first systematic one. Later C. S. Peirce and Gottlob Frege developed calculi that took into account the difference between universal and existential propositions. Since then many logical calculi have been developed, such as the Calculus of Indications previously noted here.

However, these calculi are not necessarily easy to calculate with. For that it is best to use something close to the familiar arithmetic and algebra. Here are two examples:

MIN-MAX LOGIC

The Boolean operations are negation (NOT, ¬, ~), conjunction (AND, ∧), and disjunction (OR, ∨), with the constants 0 (contradiction) and 1 (tautology). These correspond to the set operations complement (c, ´ ), intersection (∩), and union (∪) with constants ∅ (null set) and U or Ω (universal set).

Boolean logic may be represented by the following min and max operations:

¬a = 1 – a

a ∧ b = min(a, b)

a ∨ b = max(a, b)

Other operations may be defined from these such as material implication, a → b = ¬a ∨ b, which corresponds to the subset proposition a ⊆ b, and is represented by max(1 – a, b).

FINITE FIELD LOGIC

Propositional logic may be represented by any functionally complete binary calculus such as the finite (Galois) field of order 2. The constants are 0 and 1 with 1 + 1 = 0. Since ordinary arithmetic is a field, this representation is mostly familiar:

¬a = a + 1

a ∧ b = a · b

a ∨ b = a b + a + b

Then a → b is represented by a b + a + 1.

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Logic as arithmetic

August 17, 2017 1:06 pm ⋅

George Boole wrote on “the laws of thought,” now known as Boolean Algebra, and started the discipline known as Symbolic Logic. A different George, George Spencer Brown, wrote on “the laws of form,” which presented an arithmetic system underlying logic. Below are two symbolic logics equivalent to Boolean algebra that resemble ordinary arithmetic in some respects. To resemble arithmetic in other respects, use the Galois field of order 2, GF(2). Zero is taken as representing false, and one as true.

LOGIC OF SUBTRACTION

Subtraction

A – 0 = 1 – A = 1

A – 1 = A

Definitions

– A is defined as 0 – A (and so 0 is ”  “, ground, false)

A + B is defined as  A – (– B)

Tables

A 0 − A A − B 0 1 A + B 0 1
0 1 0 1 0 0 0 1
1 0 1 1 1 1 1 1

Consequences

– (– A) = A

A − B = A ← B

A + B = A ∨ B

A + B = B + A

– is not distributive

 

DIVISION LOGIC

0 / A = A / 1 = 0

A / 0 = A

Definitions

/ A is defined as 1 / A (and so 1 is ”  “, ground, true)

A • B is defined as  A / (1 / B)

Consequences

1 / (1 / A) = A

A / B = – (A → B)

A • B = A ∧ B

A • B = B • A

/ is not distributive

Tables

A 1 / A A / B 0 1 A • B 0 1
0 1 0 0 0 0 0 0
1 0 1 1 0 1 0 1

 

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Classical Model of Science

April 11, 2017 6:10 pm ⋅

Another paper that should get wider exposure: “The Classical Model of Science: a millennia-old model of scientific rationality” by Willem R. de Jong and Arianna Betti. Synthese (2010) 174:185-203. Excerpts:

Throughout more than two millennia philosophers adhered massively to ideal standards of scientific rationality going back ultimately to Aristotle’s Analytica posteriora. These standards got progressively shaped by and adapted to new scientific needs and tendencies. Nevertheless, a core of conditions capturing the fundamentals of what a proper science should look like remained remarkably constant all along. Call this cluster of conditions the Classical Model of Science. p.185

The Classical Model of Science as an ideal of scientific explanation

In the following we will speak of a science according to the Classical Model of Science as a system S of propositions and concepts (or terms) which satisfies the following conditions:

(1) All propositions and all concepts (or terms) of S concern a specific set of objects or are about a certain domain of being(s).

(2a) There are in S a number of so-called fundamental concepts (or terms).

(2b) All other concepts (or terms) occurring in S are composed of (or are definable from) these fundamental concepts (or terms).

(3a) There are in S a number of so-called fundamental propositions.

(3b) All other propositions of S follow from or are grounded in (or are provable or demonstrable from) these fundamental propositions.

(4) All propositions of S are true.

(5) All propositions of S are universal and necessary in some sense or another.

(6) All propositions of S are known to be true. A non-fundamental proposition is known to be true through its proof in S.

(7) All concepts or terms of S are adequately known. A non-fundamental concept is adequately known through its composition (or definition). p.186

The Classical Model of Science is a recent reconstruction a posteriori of the way in which philosophers have traditionally thought about what a proper science and its methodology should be, and which is largely set up, as it were, by abduction. The cluster (1)-(7) is intended, thus, to sum up in a fairly precise way the ideal of scientific explanation philosophers must have had in mind for a very long time when thinking about science. p.186

A proper science according to this Model has the structure of a more or less strictly axiomatized system with a distinction between fundamental and non-fundamental elements. p.186

The history of the conceptualization Science knows three milestones: first of all, Aristotle’s Analytica posteriora, especially book 1; secondly, the very influential so-called Logic of Port-Royal (1662), especially part IV: ‘De la méthode’, written mainly by Antoine Arnaud and relying in many respects on Pascal and Descartes; and finally Bernard Bolzano’s Wissenschaftslehre (1837). p.187

The formulation coming closest to a systematization of the ideal of science we codify in the Model is perhaps the description of scientific method given in the Logic of Port-Royal, ‘The scientific method reduced to eight main rules’:

Eight rules of science

1. Two rules concerning definitions

1 . Leave no term even slightly obscure or equivocal without defining it.
2. In definitions use only terms that are perfectly known or have already been explained.

2. Two rules for axioms

3. In axioms require everything to be perfectly evident.
4. Accept as evident what needs only a little attention to be recognized as true.

3 . Two rules for demonstrations

5 . Prove all propositions that are even slightly obscure, using in their proofs only definitions that have preceded, axioms that have been granted, or propositions that have already been demonstrated.
6. Never exploit the equivocation in terms by failing to substitute mentally the definitions that restrict and explain them.

4. Two rules for method

7. Treat things as much as possible in their natural order, beginning with the most general and the simplest, and explaining everything belonging to the nature of the genus before proceeding to particular species.
8. Divide each genus as much as possible into all its species, each whole into all its parts, and each difficulty into all its cases. pp.187-188

… the Model is a fruitful analytical tool. Its influence lasted until recently; having persisted at least to Lesniewski, it in fact extended far beyond what one might expect at first glance. It is certain, however, that at a some point the Model was abandoned without being replaced by anything comparable. p. 196

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Induction with uniformity

December 30, 2016 3:41 pm ⋅

John P. McCaskey has done a lot of research (including a PhD dissertation) on the meaning of induction since ancient times. He keeps some of his material online at http://www.johnmccaskey.com/. A good summary is Induction Without the Uniformity Principle.

McCaskey traced the origin of the principle of the uniformity of nature (PUN) to Richard Whately in the early 19th century. In his 1826 “Elements of Logic” he wrote that induction is “a Syllogism in Barbara with the major Premiss suppressed.” This made induction an inference for the first time.

There are two approaches to inferential induction. The first is enumeration in the minor premise, which was known to the Scholastics:

(major) This magnet, that magnet, and the other magnet attract iron.
(minor) [Every magnet is this magnet, that magnet, and the other magnet.]
(conclusion) Therefore, every magnet attracts iron.

The second is via uniformity in the major premise, which was new:

(major) [A property of the observed magnets is a property of all magnets.]
(minor) The property of attracting iron is a property of the observed magnets.
(conclusion) Therefore, the property of attracting iron is a property of all magnets.
(conclusion) Therefore, all magnets attract iron.

The influential J.S. Mill picked this up and made it central to science. Mill wrote in 1843:

“Every induction is a syllogism with the major premise suppressed; or (as I prefer expressing it) every induction may be thrown into the form of a syllogism, by supplying a major premise. If this be actually done, the principle which we are now considering, that of the uniformity of the course of nature, will appear as the ultimate major premise of all inductions.”

Mill held that there is one “assumption involved in every case of induction . . . . This universal fact, which is our warrant for all inferences from experience, has been described by different philosophers in different forms of language: that the course of nature is uniform; that the universe is governed by general laws; and the like . . . [or] that the future will resemble the past.”

So Mill generalized Whately’s major premise into a principle of the uniformity of nature. McCaskey writes:

“This proposal is the introduction into induction theory of a uniformity principle: What is true of the observed is true of all. Once induction is conceived to be a propositional inference made good by supplying an implicit major premise, some sort of uniformity principle becomes necessary. When induction was not so conceived there was no need for a uniformity principle. There was not one in the induction theories of Aristotle, Cicero, Boethius, Averroës, Aquinas, Buridan, Bacon, Whewell, or anyone else before Copleston and Whately.”

McCaskey goes on: “De Morgan put all this together with developing theories of statistics and probability. He saw that, when induction is understood as Whately and Mill were developing it, an inductive inference amounts to a problem in ‘inverse probability’: Given the observation of effects, what is the chance that a particular uniformity principle is being observed at work? That is, given Whately’s minor premise that observed instances of some kind share some property (membership in the kind being taken for granted), what are the chances that all instances of the kind do? De Morgan’s attempt to answer this failed, but he made the crucial step of connecting probabilistic inference to induction. The connection survives today, and it would have made little sense (as De Morgan himself saw) were induction to be understood in the Baconian rather than Whatelian sense of the term.”

That’s how the problem of induction was born, which is essentially the problem of justifying the principle of the uniformity of nature. But this depends on an inferential understanding of induction instead of the older conceptual understanding.

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Negation and logic

October 30, 2015 4:46 pm ⋅

Two propositions are contrary if they cannot both be simultaneously true but it is possible for both to be simultaneously false. For example, the proposition that “every man is just” is contrary to the proposition that “no man is just,” since both propositions may be false if some men are just.

Two propositions are contradictory if both cannot be simultaneously true and both cannot be simultaneously false. The proposition that “not every man is just” is contradictory to the proposition that “every man is just,” because both cannot be simultaneously true, nor can they be simultaneously false.

Note that contraries are two universal propositions and contradictories must have one universal and one existential proposition. And note that one proposition is the negation of the other — but there are two kinds of negation: contrary and contradictory.

Fregean logic handles these two kinds of negation by segregating them: contradictory negation goes before the quantifier and contrary negation goes after it. So these expressions are equal:

All x aren’t y as -∀x: x ⊂ y = ∃x: x ⊂ -y

and

Some x aren’t y as -∃x: x ⊂ y = ∀x: x ⊂ -y.

The other purpose of quantifiers is to bind a variable as universal or existential.

George Spencer Brown’s Laws of Form does something similar in two dimensions with his “cross” symbol ( ⏋). Contradiction is represented in the horizontal dimension via the Law of Calling. Contraries are represented in the vertical dimension via the Law of Crossing.

The intersection of horizontal and vertical crosses is a single cross, which in the interpretation for logic represents negation. With a variable under or ‘inside’ it, the cross represents “non” or “no” as in “non-A” or “no A”.

 

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Distinctions of Genesis 1

October 19, 2015 8:03 pm ⋅

In the beginning God created the heavens and the earth. The earth was formless, and indistinct; and darkness was on the face of the deep. And the Spirit of God was hovering over the face of the waters.

Then God said, Let there be light; and there was light. And God saw the light, that it was good; and God divided the light from the darkness. God called the light Day, and the darkness he called Night. The evening and the morning were the first day. So the first distinction was between Day and Night.

Then God said, Let there be a space in the midst of the waters, and let it divide the waters from the waters. Thus God made the space, and divided the waters which were under the space from the waters which were above the space; and it was so. And God called the space Heaven. The evening and the morning were the second day. So the second distinction was between waters below and above Heaven.

Then God said, Let the waters under Heaven be gathered together into one place, and let the dry land appear; and it was so. And God called the dry land Earth, and the gathering together of the waters he called Seas. And God saw that it was good.

Then God said, Let the earth bring forth grass, the herb that yields seed, and the fruit tree that yields fruit according to its kind, whose seed is in itself, on the Earth; and it was so. And the Earth brought forth grass, the herb that yields seed according to its kind, and the tree that yields fruit, whose seed is in itself according to its kind. And God saw that it was good. The evening and the morning were the third day. So the third distinction was between the Earth and the Seas.

Then God said, Let there be lights in the space of Heaven to distinguish the Day from the Night; and let them be for signs and seasons, and for days and years; and let them be for lights in the space of Heaven to give light on the Earth; and it was so. Then God made two great lights: the greater light to rule the Day, and the lesser light to rule the Night–and also the stars. God set them in the space of Heaven to give light on the Earth, and to rule over the Day and over the Night, and to divide the light from the darkness. And God saw that it was good. The evening and the morning were the fourth day. So the Day was marked with the greater light and Night was marked with the lesser light.

Then God said, Let the Seas abound with an abundance of living creatures, and let birds fly above the earth across the face of the space of the Heavens. So God created great sea creatures and every living thing that moves, with which the waters abounded, according to their kind, and every winged bird according to its kind. And God saw that it was good. And God blessed them, saying, Be fruitful and multiply, and fill the waters in the Seas, and let birds multiply on the Earth. The evening and the morning were the fifth day. So the Seas were marked with fish and Heaven was marked with birds.

Then God said, Let the Earth bring forth the living creature according to its kind: cattle and creeping thing and beast of the earth, each according to its kind; and it was so. And God made the beast of the Earth according to its kind, cattle according to its kind, and everything that creeps on the Earth according to its kind. And God saw that it was good.

Then God said, Let us make man in our image, according to our likeness; let them have dominion over the fish of the Seas, over the birds of the Heaven, and over all the Earth and over every creeping thing that creeps on the Earth. So God created man in His own image; in the image of God he created him; male and female he created them. Then God blessed them, and God said to them, Be fruitful and multiply; fill the Earth and subdue it; have dominion over the fish of the Seas, over the birds of Heaven, and over every living thing that moves on the Earth.

And God said, See, I have given you every herb that yields seed which is on the face of all the Earth, and every tree whose fruit yields seed; to you it shall be for food. Also, to every beast of the Earth, to every bird of Heaven, and to everything that creeps on the Earth, in which there is life, I have given every green herb for food; and it was so. Then God saw everything that He had made, and indeed it was very good. The evening and the morning were the sixth day. So the Earth was marked with man.

Thus the Heaven and the Earth, and all the host of them, were finished. And on the seventh day God ended his work which he had done, and he rested on the seventh day from all His work which he had done. Then God blessed the seventh day and sanctified it, because in it he rested from all his work which God had created and made. So the seventh day was marked with the Sabbath.
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Two kinds of negation

August 12, 2015 4:58 pm ⋅

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”.

 

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Laws of form

August 11, 2015 5:56 pm ⋅

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:

lof2

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.

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