Beautiful Equations Explained... Logical Conjunction
When I started my undergraduate degree in maths, one of the first things I learned is that maths is a language. Well, several languages in fact. Each subject area has its own set of axioms and notation, and gaining understanding and knowledge in any given area requires patience to slowly learn that language. Unfortunately, the time needed to break down the different symbols and definitions and really get to grips with a new concept is not something that many people have outside of the university. In order to continue sharing our beautiful equations with a wider audience, here’s a guide to one of our favourites, the formula for logical conjunction!
In a first-order logic language, it is possible to define the logical conjunction (‘and’) and disjunction (‘or’) using the symbols for logical negation and implication.
First-order logic provides a language from which all mathematical theories can be formulated. The alphabet of this language is made of 8 categories of symbols including the ‘negation’ and implication symbol used above but crucially, to save time when completing logical proofs, not the ‘and’ and ‘or’. This neat definition allows them to still be used freely, again saving mathematicians a lot of time!
The first study of modern logic is credited to George Boole, an English mathematician well known for his contributions to differential equations and algebraic logic such as Boolean Algebra, which he introduced in his first book The Mathematical Analysis of Logic in 1847. The conventions and language used has transitioned over the years as mathematicians searched for greater consistency across different areas. It was not until the end of the twentieth century that first-order logic became standard for research in the foundations of mathematics.