mathematics

Propositional logic

Propositional logic is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Propositional logic rather than just read about it. In short: Propositional logic is a branch of classical logic. It is also called statement logic, sentential calculus, propositional calculus, sentential logic, or sometimes zeroth-order logic.

Key takeaways

Propositional logic belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Propositional logic to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Propositional logic from memory before moving on to harder problems.

Reference excerpt

Propositional logic is a branch of classical logic. It is also called statement logic, sentential calculus, propositional calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called first-order propositional logic to contrast it with System F, but it is distinct from first-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of conjunction, disjunction, implication, biconditional, and negation. Some sources include other connectives, as in the table below. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Propositional logic is typically studied with a formal language, in which propositions are represented by letters, which are called propositional variables. These are then used, together with symbols for connectives, to make propositional formulas. Because of this, the propositional variables are called atomic formulas of a formal propositional language. While the atomic propositions are typically represented by letters of the alphabet, there is a variety of notations to represent the logical connectives. For the benefit of readers who may only be used to a different variant notation for the logical connectives, the following table shows the main notational variants for each of the connectives in propositional logic. Other notations have been used historically, such as Polish notation. For the history of each of these symbols, see the respective articles as well as the article "Logical connective".

The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic, in which formulas are interpreted as having precisely one of two possible truth values, the truth value of true or the truth value of false. The principle of bivalence and the law of excluded middle are upheld. By comparison with first-order logic, truth-functional propositional logic is considered to be zeroth-order logic.

History

Although propositional logic had been hinted by earlier philosophers, Chrysippus is often credited with development of a deductive system for propositional logic as his main achievement in the 3rd century BC which was expanded by his successor Stoics. The logic was focused on propositions. This was different from the traditional syllogistic logic, which focused on terms. However, most of the original writings were lost and, at some time between the 3rd and 6th century CE, Stoic logic faded into oblivion, to be resurrected only in the 20th century, in the wake of the (re)-discovery of propositional logic. Symbolic logic, which would come to be important to refine propositional logic, was first developed by the 17th/18th-century mathematician Gottfried Leibniz, whose calculus ratiocinator was, however, unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan, completely independent of Leibniz. Gottlob Frege's predicate logic builds upon propositional logic, and has been described as combining "the distinctive features of syllogistic logic and propositional logic." Consequently, predicate logic ushered in a new era in logic's history; however, advances in propositional logic were still made after Frege, including natural deduction, truth trees and truth tables. Natural deduction was invented by Gerhard Gentzen and Stanisław Jaśkowski. Truth trees were invented by Evert Willem Beth. The invention of truth tables, however, is of uncertain attribution. Within works by Frege and Bertrand Russell, are ideas influential to the invention of truth tables. The actual tabular structure (being formatted as a table), itself, is generally credited to either Ludwig Wittgenstein or Emil Post (or both, independently). Besides Frege and Russell, others credited with having ideas preceding truth tables include Philo, Boole, Charles Sanders Peirce, and Ernst Schröder. Others credited with the tabular structure include Jan Łukasiewicz, Alfred North Whitehead, William Stanley Jevons, John Venn, and Clarence Irving Lewis. Ultimately, some have concluded, like John Shosky, that "It is far from clear that any one person should be given the title of 'inventor' of truth-tables".

Sentences

Propositional logic, as currently studied in universities, is a specification of a standard of logical consequence in which only the meanings of propositional connectives are considered in evaluating the conditions for the truth of a sentence, or whether a sentence logically follows from some other sentence or group of sentences.

Declarative sentences Propositional logic deals with statements, which are defined as declarative sentences having truth value. Examples of statements might include:

Wikipedia is a free online encyclopedia that anyone can edit. London is the capital of England. All Wikipedia editors speak at least three languages. Declarative sentences are contrasted with questions, such as "What is Wikipedia?", and imperative statements, such as "Please add citations to support the claims in this article.". Such non-declarative sentences have no truth value, and are only dealt with in nonclassical logics, called erotetic and imperative logics.

Compounding sentences with connectives

In propositional logic, a statement can contain one or more other statements as parts. Compound sentences are formed from simpler sentences and express relationships among the constituent sentences. This is done by combining them with logical connectives: the main types of compound sentences are negations, conjunctions, disjunctions, implications, and biconditionals, which are formed by using the corresponding connectives to connect propositions. In English, these connectives are expressed by the words "and" (conjunction), "or" (disjunction), "not" (negation), "if" (material conditional), and "if and only if" (biconditional). Examples of such compound sentences might include:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Propositional logic

Start with the simplest possible case. Write down what Propositional logic claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Propositional logic before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Propositional logic ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Propositional logic

In research: Propositional logic appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Propositional logic in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Propositional logic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Analytic philosophy, Boolean algebra, Classical logic, so understanding it makes those chapters shorter.
In everyday life: Look for Propositional logic outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Propositional logic in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Propositional logic means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Propositional logic out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Propositional logic in simple terms?

Propositional logic is a branch of classical logic. It is also called statement logic, sentential calculus, propositional calculus, sentential logic, or sometimes zeroth-order logic.

Why does Propositional logic matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Propositional logic?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Propositional logic.