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Tag Archives: Logic

nature and application of logic

Set theory and logic and their dual

(1) Set theory and logic, (2) number and algebra, and (3) space and time are three foundational topics that each have duals. Let us begin with the standard approaches to these three topics, and then define duals to each of them. To some extent, the original and the dual may be used together.

(1) Set theory and logic

A set is defined by its elements or members. Its properties may also be known or specified, but what is essential to a set is its members, not its properties. The notation for “x is an element of set S” is “x ∈ S”. A subset is a set whose members are all within another set: “s is a subset of S” is “s ⊆ S”. If subset s does not (or cannot) equal S, then it is a proper subset: “s ⊂ S”.

The null set (∅) is a unique set defined as having no members. That is paradoxical but not contradictory. A universal set (Ω) is defined as having all members within a particular universe. An unrestricted universal set is not defined because it would lead to contradictions.

The complement of a set (c) is the set of all elements within a particular universe that are not in the set. A union (∪) of sets is the set containing all members of the referenced sets. An intersection (∩) of sets is defined as the set whose members are contained in every referenced set.

Set theory has a well-known correspondence with logic: negation (¬) corresponds to complement, disjunction (OR, ∨) corresponds to union, and conjunction (AND, ∧) corresponds to intersection. Material implication (→) corresponds to “is a subset of”. Contradiction corresponds to the null set, and tautology corresponds to the universal set.

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Combining equations

Given two equations with the same variable, how can they be combined? If the equations are consistent, they may be solved as simultaneous equations. But what if the equations are inconsistent? There are two ways to combine them in that case, one is OR, the other is AND.

Consider the equations x = a and x = b, where a ≠ b. If we multiply these equations together, we get

x² = ab,

in which the solution is x = √ab, so that x is the geometric mean of a and b.

If we make the equations homogeneous first, then multiply them together, we get: 0 = x − a and 0 = x − b, so that

0 = (x − a) (x − b) = x² − (a + b) x + ab.

The solution of the combined equation is either x = a or x = b. To combine equations with AND, multiply homogeneous equations together.

Another way to combine equations is to add them together. In this case, we get

x + x = 2x = a + b, or x = (a + b)/2,

so that x is the arithmetic mean of a and b. Homogeneous equations added produce the same result: 0 = x − a + x − b = 2x − (a + b), so that x = (a + b)/2.

Contraries as duals

Contrariety is a property of pairs of propositions, but it also applies to pairs of terms or concepts. “Two general terms are contraries if and only if, by virtue of their meaning alone, they apply to possible cases on opposite ends of a scale. Both terms cannot apply to the same possible case, but neither may apply.” (Aristotelian Logic, Parry and Hacker, p. 216) Opposite ends of a scale are also called extremes, which are contrasted with means between the extremes.

Every pair of contraries forms a duality by inverting the scale of which they are opposites. For example, quantitative contraries such as rich and poor become poor and rich when the scale is inverted. Every measurement scale can be inverted so in this sense a measurement and its inverse are a contrary pair that forms a self-duality. Every ratio or function of two variables, f(x, y), can be interchanged and form a duality, f(y, x). For example, the equation v = Δst can be interchanged to become u = v-1 = Δts.

The scale may be qualitative, too. For example, the qualitative contraries up and down become down and up, respectively, by looking upside-down. The contraries left and right become right and left when looked at facing the other way. Extension and intension are opposites that may be inverted by interchanging them with each other. Compare the duality of top-down and bottom-up perspectives.

“A pair of terms is contradictory if and only if by virtue of their meaning alone each and every entity in the universe must be names by one or the other but not both.” (Aristotelian Logic, Parry and Hacker, p. 216) May the terms X and not-X be made into duals? That depends. If not-X is the contradictory of X and means everything other than X, that includes things that are non-dual. But in some cases, not-X means the opposite of X, so that contraries are indicated.

Balancing contraries

Other posts on contraries include this.

Contrary opposites entail one another. There is no north without south or tall without short, for example. Some things such as sex are contraries in some respects but not in all respects.

Contrary opposites are symmetric. Contraries can be reversed or inverted, and they are still there. Since mirror opposites do not necessarily exist, mirror images are not contraries, though they exhibit a symmetry.

Because contraries entail one another and are symmetric, it is arbitrary to always prefer one to the other. One could just as well prefer the opposite contrary.

Contrary opposites can be unified into a higher perspective that contains them both. Unification is an expanded position that incorporates contraries.

Contrary opposites can be balanced in a duality that resists unification. A static equilibrium or dynamic harmony favors contrary opposites equally.

Ancient science prefers static contraries in balanced duality. Modern science prefers dynamic contraries in progressive unification.

New fallacies

There are several online lists of fallacious arguments: Fallacies, Full alphabetic list of fallacies, Logical Fallacies Handlist, List of fallacies, List of Fallacious Arguments, and especially Master List of Logical Fallacies, A list of Latin names is here.

Aristotle categorized rhetorical strategies under ethos, logos, and pathos. Ethos is an appeal to credibility or authority. Logos is an appeal to reason or evidence. Pathos is an appeal to feeling or emotion. Fallacies are also categorized as formal or informal, with many informal fallacies.

There is perhaps nothing new under the sun when it comes to informal fallacies, but there are at least new variations on old fallacies or fallacies that have not been adequately described. Here are some notes about these new fallacies:

Assailment-by-entailment is the fallacy committed by person B when they attribute to person A a belief that person B thinks is entailed by something person A has said, especially if person A has denied the offending belief. Its object seems to be ostracism or reputation damage. See here.

Controlling the conversation is a no discussion fallacy in which one party rejects reasoned dialogue with a dissenting party. This is usually committed by a stronger party in order not to allow a weaker, dissenting party to be heard. Common in issues that relate to a stronger party’s status. For example, the status of science in modern society is partly based on agreement among scientists, so dissent among scientists is ignored or controlled by the dominant party. This includes scientific publication and conference presentation.

Contraries and contradictories

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.

Form and logic

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 multiplicity.

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 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. But the metaphysical realist prefers to begin with truth, with being and reality.

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.

Propositional logic calculation

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. Below are two examples of Boolean logic calculation methods. Note that every Boolean expression has a dual expression by interchanging the OR and AND operations and the 0 and 1 elements of the expression.

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 Ω (universal set).

Boolean logic may be represented as 0 (contradiction, null set) and 1 (tautology, universal set) with the following minimum and maximum operations:

¬a = ac = 1 – a

ab = ab = min(a, b)

ab = a ∪ b = max(a, b)

Other operations may be defined from these such as material implication, ab = ¬ab, which corresponds to the subset proposition ab, 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

ab = a · b = ab

ab = ab + a + b

Then ab is represented by ab + a + 1.

See also here.

Logic as arithmetic

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

 

Classical Model of Science

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).

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