Symbols That Express Our Thoughts: The Evolution of Symbolic and Boolean Logic

A set of vocabulary flashcards reviewing key historical figures, terminology, and foundational principles in symbolic logic and Boolean algebra.

Lingua characteristica universalis

A symbolic or universally characteristic language proposed by Gottfried Wilhelm Leibniz to simplify complicated logical arguments so disputes could be settled through exact calculation.

Alphabet of human thought

A universal method of expression envisioned by Gottfried Wilhelm Leibniz that anyone of intelligence could read or speak regardless of their native language, reducing all thought to exact reasoning and computation.

Symbolic number

A determinate number assigned to each term in a proposition in Leibniz's logical system, allowing truth to be determined through divisibility and complex concepts to be represented by numerical products.

Composite concept

In Leibniz's logic, a concept formed by combining more basic concepts and represented mathematically as the product of their symbolic numbers (for example, rational animal = man, or $2 \times 3 = 6$).

Composite number

Any number that can be expressed as the product of whole numbers greater than $1$, as opposed to a prime number.

Binary arithmetic

A mathematical system developed by Leibniz in 1679, influenced by the dualities of yin and yang, that represents all numbers and calculations using only $0\text{s}$ and $1\text{s}$.

Bit

The smallest unit of data in a digital computer, standing for binary unit, which holds a single binary value of either $0$ or $1$.

Stepped drum calculator

A mechanical calculating machine invented by Leibniz in 1673 (also called a stepped wheel calculator) based on a pedestrian step counter, capable of addition, subtraction, multiplication, and division.

Universe of discourse

The complete realm or class of everything under discussion, introduced to modern logic by Augustus De Morgan and represented in George Boole's algebra by the value $1$.

Negative terms

In Augustus De Morgan's logic, complementary terms (referred to by De Morgan as contraries) symbolized by lowercase letters (e.g., $x$ for not-man) that together with positive terms exhaust the entire universe of discourse.

Law of the excluded middle

The logical principle asserting that everything under discussion must either be a term or its negation (such as 'man' or 'not-man'), expressed in Boole's algebra by the equation $X(1 - X) = 0$.

Boolean algebra

An algebraic system of logic created by George Boole in which variables represent classes or propositions and operations can be evaluated numerically using the values $1$ (true) and $0$ (false).

Logical product

In Boolean logic, the conjunction of two classes or propositions ($XY$), representing the intersection or overlap consisting of elements that belong simultaneously to both class $X$ and class $Y$.

Logical sum

In Boolean logic, the disjunction of two classes or propositions ($X + Y$), representing the union of elements that belong to class $X$, class $Y$, or both.

Ockham's Razor

A philosophical maxim formulated by William of Ockham stating that assumptions introduced to explain a phenomenon should not be multiplied beyond necessity—meaning that, all else being equal, the theory with the fewest assumptions is best.

De Morgan's Rules

A pair of logical transformation rules stating that the negation of a disjunction is the conjunction of the negations, and the negation of a conjunction is the disjunction of the negations.

Sheffer stroke

A single logical operation symbolized by $|$ (interpreted as 'neither/nor' or 'not both') introduced by Henry M. Sheffer, through which all standard logical connectives can be uniquely formulated.

NAND gate

An electronic circuit component representing the 'Not AND' operation based on the Sheffer stroke, serving as a fundamental functional building block in digital electronics.