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.

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.

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:

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:

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

See also here.