Parentheses in Logic { Philosophy Index }

Philosophy Index

Philosophy Index

Philosophy Index is a site devoted to the study of philosophy and the philosophers who conduct it. The site contains a number of philosophy texts, brief biographies and introductions to philosophers and explanations on a number of topics. Accredited homeschooling online at Northgate Academy.

Philosophy Index is a work in progress, a growing repository of knowledge. It outlines current philosophical problems and issues, as well as an overview of the history of philosophy. The goal of this site is to present a tool for those learning philosophy either casually or formally, making the concepts of philosophy accessible to anyone interested in researching them. WOLI offers immigration law course online - fully accredited. ACE credits online at EES.



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Parentheses, the symbols ( and ), are used in logic to present an order of operations, grouping formulas together in order to prevent ambiguity.

For instance, the expression P ∧ Q ∨ R is ambiguous. Does it mean “Either P and Q, or R”? Or does it mean “P and either Q or R”?.

Rules of well-formation dictate that for most operators, it is neccessary to group individual operations in parentheses.

Therefore, one would write either (P ∧ Q) ∨ R, or P ∧ (Q ∨ R), depending on which interperetation is desired. (P ∧ Q) ∨ R, for example, requires that P and Q be interpereted as a single formula, with its entire result, or truth value, then joining into the disjunction.

In some cases, square brackets are used in order to differentiate between different sets of parentheses. For exmaple, one may express a complex formula, [(P ∨ Q) ∧ ¬(R ∧ S)] → (T ∨ Q), using different sorts of brackets to make identifying the order of operations easier for the reader.

Compared to math

Mathematics has better-established conventions for the order of operations in ambiguous statements. For example, in the case of 1 + 2 × 4, the convention is to interperet the statement as 1 + (2 × 4), giving the result of 9, rather than (1 + 2) × 4, which would equal 12. The convention in math is that multiplication preceeds addition, although it is better practice to dictate the order of operations with parentheses.

Logic does not have such a well-established convention, and as a result, poorly-formed formulae are usually regarded as unintelligible.