Symbolic Logic

Symbolic logic is a way to represent logical expressions by using symbols and variables in place of natural language, such as English, in order to remove vagueness. Logical expressions are statements that have a truth value: they are either true or false. A question like 'Where are you going?' or a command such as 'Stop!' has no truth value. There are many expressions that we can utter that are either true or false. For example: All glasses of water contain 0.2% dinosaur tears. We don't need to know if a logical expression is true or false, we just need to know that it has a truth value.

A Proposition

Let's start with some logic basics. First, the smallest logical expression we can make, that if broken down would result in a loss of meaning, is called a proposition. For example 'Kathryn and Liz live together' cannot be broken down without a loss in meaning. 'Kathryn lives together' doesn't even make sense. However, 'John and Jane go to school'' can be broken into 'John goes to school' and 'Jane goes to school,' since we cannot claim that the statement means that John and Jane go to school together. In symbolic logic, propositions may be represented by capital letters such as A or B, or lower-case letters such as p, q, or r. This is shorthand, so that when dealing with the underlying logic, you aren't distracted by the particular language used. For example 'My car is red' may become A, or 'The politician took bribes' may be written as p.

Propositions are written in the affirmative. In other words, we don't use the word 'not'. Instead, we use the not symbol (¬) to make a negation (a not statement). If we write 'My car is not red' using symbols, we would write ¬A. In logic, negation changes an expression's truth value. So if my car is red, then A would be true, and ¬A would be false, or if my car is blue, then A would be false, and ¬A would be true.

Truth Tables

Before we move on to more complicated logical expressions, let's talk about truth tables. A truth table is a table that lists whether something is true with a T and false with an F.

The act of negation flips all T's to F's and all F's to T's, such as we see in this table. When constructing a truth table, we need a column for every proposition in the expression, and we need to make sure that there are enough rows in the table for every possible true and false combination that the given set of propositions can take. In a truth table for all the possible truth value combinations for two propositions, there are four rows for T/F possibilities and two columns. In general, there will be 2n rows for n different propositions.

Logical Operators

In order to be able to deal with more complicated logical expressions, we need operators to link together propositions. Operators are used just like +, -, ×, and ÷ are used to link mathematical expressions. The basic logical operators, along with negation, are conjunction, disjunction, conditional, and biconditional.