02x02 - Vocabulary Atomic Formulas And Logical Operators

Welcome to Fast Philosophy. This video is part of our Translating Into Propositional Logic series and explains the vocabulary of propositional logic.
In natural language, we talk about individual propositions. For example, ‘Socrates likes pineapple’ is an individual proposition. In formal logic, individual propositions are called ‘atomic formulas’. An atomic formula is represented using a ‘sentence letter’, which is any uppercase letter from the Roman alphabet. You should use each letter only once in each translation to avoid confusion. If you exhaust all 26 letters, you use subscript numbers to the right of uppercase letters, though this is seldom necessary. It doesn’t matter which letter you choose because you’ll write a key – for instance, P = ‘Socrates likes pineapple’ – however, for ease of use it’s best to pick a letter that’s related to the proposition, if possible.
In natural language, we often negate or connect individual propositions together using sentence connectives. For example, ‘Socrates does not like pineapple’ negates the proposition ‘Socrates does like pineapple’ using ‘not’; and ‘Socrates likes pineapple and quince’ connects ‘Socrates likes pineapple’ with ‘Socrates likes quince’ using ‘and’. In formal logic, sentence connectives are called ‘logical operators’ and are represented using five logical symbols. Arguably, all sentence connectives in natural language are reducible to one of the five logical operators.
‘Not’ is unique among the sentence connectives of natural language because strictly speaking it does not connect anything. Instead, it negates one thing. In formal logic, this unique feature of ‘not’ means that it is a ‘one-place logical operator’. This means that it only affects one thing rather than two. Where ‘P’ stands for any atomic formula, this means ‘not P’ and shows the negation of ‘P’. ‘Not’ is represented using one of the five symbols for the logical operators. This symbol is called the tilde.
The other four logical operators are all two-place logical operators, meaning that, unlike the tilde, they affect two things rather than one. Where ‘P’ and ‘Q’ stand for any atomic formula:
This means ‘P and Q’, and shows the conjunction of ‘P’ with ‘Q’. ‘And’ is represented using the caret symbol;
This means ‘P or Q’, and shows the disjunction of ‘P’ with ‘Q’. ‘Or’ is represented using the vel symbol;
This means ‘if P, then Q’, and shows that P is antecedent to its consequent Q. ‘If… then’ is represented using the horseshoe symbol; and lastly,
This means ‘P if and only if Q’, and shows the truth-functional equivalency of P with Q, which we’ll return to in a later video. ‘If and only if’ is represented using the triple bar symbol when writing formal logic and is also commonly written as ‘iff’ in textbooks.
When we combine atomic formulas such as ‘P’ and ‘Q’ using these logical operators, we call the combination a ‘formula’. All premises and conclusions are formulas; this includes atomic formulas because they are, strictly speaking, a species of formulas. For example, ‘Socrates likes pineapple and quince’ could look like this. This combines two atomic formulas using one logical operator, resulting in a longer formula.
Our logical vocabulary consists in atomic formulas and logical operators. Next, we must understand the grammar and punctuation of our logical language.