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An Introduction to Symbolic Logic

Oftentimes, it seems like math is a smorgasbord of different fields, ranging from algebra to geometry to calculus. These fields seem to have little to no connection with each other other than the fact that they use numbers and letters to represent quantities and how they change. When you consider that, it’s surprising that all of mathematics can be considered a subcategory of another field: symbolic logic. Symbolic logic is the field that explores relationship between ideas denoted by letters, and is important in electrical engineering, programming, and even day-to-day argumentation.

Let’s start with the basics. Symbolic logic deals with propositions, a suggested argument that can be true or false, represented with capital letters. For example, P can stand for “pigs are flying” and Q can stand for “I will buy a Tesla” – both are either true or false. One of the simplest things you can do with a proposition is the negation (~), which just takes the original proposition and says the opposite, so ~P would be “pigs are not flying.”

Negation only deals with a single proposition, but let’s say you want to combine P and Q. A major connective (or proposition that deals with two statements) in logic is implication (→), which means the outcome/truth of the second proposition depends on the truth of the first proposition. If I were to say P → Q, you can write that as “If pigs are flying, then I will buy a Tesla.” That means, if, indeed, pigs are flying then the second proposition – I will buy a Tesla – is guaranteed to be true. However, note that if pigs are not flying, I cannot say anything about whether or not I will buy a Tesla because I cannot conclude anything from just that implication.

Finally, two other major connectives in logic are conjunction (^) and disjunction (ⅴ). Conjunction essentially requires both of the individual propositions to be true for the whole statement to be true. For example, “Pigs are flying and I will buy a Tesla” or P^Q – the whole statement is only true if both P and Q are true. If either one or both are false, then the whole sentence/statement would be false. Disjunction requires at least one proposition to be true for the whole statement to be true. For example, “Pigs are flying or I will buy a Tesla” or PⅴQ – if P is true, Q is true, or P and Q are both true, then the whole statement is true. The whole statement is only false if both propositions are false. You might recognize the resemblance of these symbols to those from set theory representing intersection (∩) and union (U) respectively, because they are the mathematical equivalent – in a Venn Diagram, the values in the middle represent the conjunction or intersection of the two sets because they satisfy both of the individual requirements or propositions.

One application of symbolic logic is logic gates, which are similar to miniature circuits in digital computers. Logic gates represent the outcomes of inputted signals, denoted with a 1 for true or a 0 for false. For example, the and logic gate would need both inputs to be a 1 for the outcome to also be a 1. Logic gate circuits deal with multiple propositions, so to keep track of the truth or falsity of the propositions, we use truth tables as a guideline for creating logic gates.

Figure 1: A logic circuit example (left) with truth tables (right).

In truth tables, to represent all the possible initial input possibilities for A and B, the columns underneath them read 00, 01, 10, and 11. From these initial inputs, we can list out all the possible outcomes of the logic gates. Logic gates are not denoted by the same and/or symbols as in symbolic logic, but have the same relationships at heart. For example, in the logic circuit of Figure 1, logic gates C-X above all use the NAND symbol below, a specific connective that can be made by putting together some of our basic connectives from symbolic logic.

Fig 2: A list of all the major propositions in logic gate form. Note NAND and NOR, which have the circle at the end, essentially negate or reverse the AND/OR conditions, respectively.

While AND gates, or a conjunction in symbolic logic, are true only when both inputs/propositions are true, NAND gates are a negation — they are false/0 only when both inputs are true/1. Looking at all the combinations from A and B to C, the outcome will be 1 unless both A and B are both 1. Going forward to D and E, similarly, the outcome is 1 unless both C and A or B, respectively (in other words, the respective inputs) are 1. Finally, gates F and then X reveal that the logic gate will only yield 1 when the inputs all the way back at A and B were both either 0 or 1 – this logic circuit can be condensed to the XNOR Condition, where both of the initial signals/lack thereof had to be the same for the signal to be transferred at the end. This mini-circuit can be expanded and used in more complex real-world situations with switches and circuit-breakers to find where and what signals are being passed.

Symbolic logic is one of the most omnipresent fields, capable of linking STEM and the humanities, through its applications in deductive reasoning and argumentation. From symbolic logic’s beginnings dating back to the Greco-Roman times, to its development by figures like Boole and Leibniz, symbolic logic continues to hold significance today. This field will continue to hold significance in the future, given its various applications in important fields.

Works Cited:

Coates, Eric. “Digital Logic – Logic Gates.” Learn About Electronics. December 29, 2020. https://learnabout-electronics.org/Digital/dig21.php

Gregersen, Erik. “Logic Design – computer technology.” Encyclopedia Britannica. November 19, 2021.