science

Symbol (formal)

Symbol (formal) is a science 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 Symbol (formal) rather than just read about it. In short: A formal symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term symbol in common use sometimes refers to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in the formal languages studied in mathematics and logic, the term symbol refers to…

Key takeaways

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

Reference excerpt

A formal symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term symbol in common use sometimes refers to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in the formal languages studied in mathematics and logic, the term symbol refers to the idea, and the marks are considered to be a token instance of the symbol, according to the semiotic theory of Charles Sanders Peirce. In logic, symbols build literal utility to illustrate ideas.

Overview Symbols of a formal language need not be symbols of anything. For instance there are logical constants which do not refer to any idea, but rather serve as a form of punctuation in the language (e.g. parentheses). Symbols of a formal language must be capable of being specified without any reference to any interpretation of them. A symbol or string of symbols may comprise a well-formed formula if it is consistent with the formation rules of the language. In a formal system a symbol may be used as a token in formal operations. The set of formal symbols in a formal language is referred to as an alphabet (hence each symbol may be referred to as a "letter") A formal symbol as used in first-order logic may be a variable (member from a universe of discourse), a constant, a function (mapping to another member of universe) or a predicate (mapping to T/F). Formal symbols are usually thought of as purely syntactic structures, composed into larger structures using a formal grammar, though sometimes they may be associated with an interpretation or model (a formal semantics).

Words modeled as formal symbols The move to view units in natural language (e.g. English) as formal symbols was initiated by Noam Chomsky (it was this work that resulted in the Chomsky hierarchy in formal languages). The generative grammar model looked upon syntax as autonomous from semantics. Building on these models, the logician Richard Montague proposed that semantics could also be constructed on top of the formal structure:

There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians; indeed, I consider it possible to comprehend the syntax and semantics of both kinds of language within a single natural and mathematically precise theory. On this point I differ from a number of philosophers, but agree, I believe, with Chomsky and his associates." This is the philosophical premise underlying Montague grammar. However, this attempt to equate linguistic symbols with formal symbols has been challenged widely, particularly in the tradition of cognitive linguistics, by philosophers like Stevan Harnad, and linguists like George Lakoff and Ronald Langacker.

References

See also List of mathematical symbols List of logic symbols Terminal and nonterminal symbols

Illustrations

Worked examples

Example 1 — a first encounter with Symbol (formal)

Start with the simplest possible case. Write down what Symbol (formal) claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Symbol (formal) 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 Symbol (formal) 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 Symbol (formal)

In research: Symbol (formal) appears in science 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 Symbol (formal) 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: Symbol (formal) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Abstraction, Concepts in logic, Formal languages, so understanding it makes those chapters shorter.
In everyday life: Look for Symbol (formal) 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 Symbol (formal) in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Symbol (formal) 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 Symbol (formal) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Symbol (formal) in simple terms?

Why does Symbol (formal) matter?

Because it connects several science 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 Symbol (formal)?

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 Symbol (formal).