Logical Connectives

Logical connectives, also called logical operators, are symbols which are used to form a more complex proposition from two or more simple ones. To define a logical connective, one must define how it joins together the simple propositions, and also how the truth of the resulting complex proposition depends upon the truth value of its constituent propositions. Some common logical connectives are defined below, in terms of some arbitrary simple propositions X and Y.

  • Not: NOT X is true whenever X is false, and false whenever X is true. It operates on a single proposition, inverting its truth value.
  • And: X AND Y is true only when both X and Y are true, and false otherwise.
  • Or: X OR Y is true when either X is true or Y is true or both are true. In this sense it differs from the everyday conversational usage of “or”, which typically means ‘one or the other but not both’. For example, if I were to say “for lunch I always eat an apple or a banana”, in everyday language we would usually interpret this to mean that I do not eat both an apple and a banana for lunch. Logically speaking, eating both fruits would be perfectly consistent with my claim.
  • Conditional: The conditional operator has the form IF X THEN Y, and is true if X and Y are both true, or if X is false. The only time IF X THEN Y can be false is when X is true but Y is false. The proposition that occupies the X position is known as the antecedent, while the proposition filling the Y position is called the consequent. The behaviour of the conditional operator is quite different in logic to how the phrase ‘if…then’ works in ordinary language. For example, consider the conditional proposition “if the sky is green, then grass is also green”. Intuitively it seems this is false, since the sky isn’t green, and furthermore the colour of the sky has nothing to do with the colour of grass. Logically speaking, however, this proposition is actually true, since the antecedent “the sky is green” is false, and whenever the antecedent of a conditional is false, the conditional as a whole is said to be true.
  • Biconditional: the biconditional is like a stricter version of the conditional operator. It has the form X IF AND ONLY IF Y, and is true either when both X and Y are true, or when they are both false. The phrase ‘if and only if’ is sometimes abbreviated in philosophy as ‘iff’, spelled with two fs. Failure to distinguish between the conditional and biconditional operators is a common cause of confusion in many arguments. For example, suppose I were to say “if my favourite candidate wins the election, the economy will improve”. This is a conditional statement, and logically speaking, if my candidate loses the election (making the antecedent false) this proposition will always be true, regardless of what the economy does. It is not equivalent to the statement “the economy improve only if my candidate is elected”, which is a biconditional statement, and claims that the economy will improve if my candidate is elected, but won’t improve if they are not elected.

Further Reading

Logical connectives: a brief introduction to negation and disjunction

Indicative conditions: a discussion of the philosophical issues related to the material conditional from the Stanford Encyclopedia of Philosophy

Paradoxes of material implication: a discussion of some of the counter-intuitive properties of the ‘if’ conditional used in philosophy

Logical connective: a detailed introduction to logical connectives in philosophy and computer science from New World Encyclopedia

Logical symbols: an introduction to some of the symbols used in denoting logical arguments in philosophical publications (more advanced)