A Mathematical Theory of Communication

Claude E. Shannon · The Bell System Technical Journal · 1948 · information-theory · 10.1002/j.1538-7305.1948.tb01338.x · record shannon1948-communication
tag layer · index · concepts · authors · עברית
Where this record's text came from. faceted 2026-08 by reading the locally held text (pdftext); every evidence string verified as an exact substring of that text before merging. Claim type='existence' is chosen for the paper's central and most counter-intuitive result, that a definite channel capacity exists and can be approached with arbitrarily small error; a defensible alternative reading is 'method', since the bulk of the paper constructs a measurement framework (entropy, equivocation, rate relative to a fidelity criterion) rather than reporting a phenomenon. The Fano relation is typed 'replicates' in the sense of independent arrival at the same construction, which is the nearest of the four permitted types; the same applies to the paper's note that 'Formulas similar to C for the white noise case have been developed independently by several other writers, although with somewhat different interpretations', which underpins replication status='replicated'. The extraction is heavily damaged in the mathematical passages (equations reflowed into token soup, (cid:xxx) glyph placeholders), so all quotes are taken from prose sentences that survived intact. 2026-08-14 and 2026-08-15 schema-0.3 passes: every quote verified as an exact substring of the staged text and anchored at merge time; concept terms verified present; every edge endpoint an existing node. Update, 15 September 2026: EXCEPTION, measured with the merge gate's own predicate (Paper.locate): 1 quote(s) on this record are not found as one exact string in the current staged text - content_tags.statistics[0].evidence. The sentence above does not hold for them as written. Where a row records why - the quote is grounded in another copy of the paper, joins two passages with an ellipsis, or differs from the staged text by extraction noise - that note stays on the row. Update, 16 September 2026: EXCEPTION TO THE DATED PASS ABOVE, on this record only: this record carries 1 composite quote field - a quote assembled from two or more passages joined by an ellipsis. A COMPOSITE IS NOT AN EXACT CONTIGUOUS SUBSTRING of the staged text and is deliberately not anchored: each of its parts is verified separately at the offsets in evidence composite spans, and a part occurring more than once carries span null rather than a guessed offset. So the 'every quote verified as an exact substring' clause of the dated pass above DOES NOT HOLD for those fields, and never did - the parts were verified, the whole was not. Nothing is deleted, re-grounded or withdrawn: this writes down an exception that was always true of composites and had not been recorded, so a reader was told more than the pass had checked.
The quality of the text these quotes were cut from. symbol-font-not-extractable: mathematical notation is lost to '(cid:NN)' markers - 1136 of them, 5.3% of an internal file by character, but only 13.6% sit isolated in running prose (the rest cluster inside formulae). the summation, log and integral operators render as markers. 24 distinct codes with the top two accounting for 39% - ONE SYMBOL FONT WITH NO ToUnicode MAP, not a general extraction failure; a broken text font needs hundreds of codes. THE ARGUMENT OF THE PAPER IS READABLE AND TAGGABLE; THE EQUATIONS ARE NOT. 53 live quotes, 0 carrying a marker. THE SECOND COPY IS NOT A REPAIR: the web page we fetched shows zero '(cid:NN)' and 2332 '/H#####' operator codes instead - the same glyphs in a different notation, which is why a cid-only check calls it clean. And its the web page we fetched uses a THIRD spelling - '/#28 t/#29' for '( t )', hex escapes that neither the '(cid:' nor the '/H#####' check can see.
How to read this page. The article itself is not reproduced here — this paper is not cleared for redistribution, so what follows is the tag layer alone. The evidence sentences are withheld with the article. The table at the foot lists every tag the record carries; each row names the facet it belongs to, and the whose words column says whether the tag records the paper's own wording or the tagger's sentence about the paper. Every tag links to each other paper in the corpus carrying it. The machine-readable edition of the same thing is the TEI file beside this one, where the tags live in standOff, and the article is at its publisher via the link below. When we have changed our mind. Some entries in the table begin with a word in capitals and the reason for it. EVIDENCE CORRECTED — the finding still stands, but we had quoted the wrong sentence from the paper, and the quotation has been replaced with the right one. RETRACTED — the entry itself was wrong; it stays on the page, marked, with the reason, rather than being deleted. WITHDRAWN AND REPLACED — the entry was wrong, and a corrected version of it appears in the entries below. Each mark carries its own reason, in brackets, because the reasons differ from one entry to the next. An entry we have WITHDRAWN is removed from this page and kept in the record, with its reason. A line reading Audit and a date is a later re-reading of the entry against its whole sentence: DOUBTFUL means it was kept, and the reason says what is loose about it. Quotation marks inside the tagger's sentence. Where a tag is the tagger's own sentence, a phrase in quotation marks inside it is borrowed from the paper — the tagger's wording around it is not. The paper's own sentence is the evidence quote for that row; on this page it is withheld with the article, and the article is at its publisher. Nothing inside quotation marks anywhere on this page is ours: if it is quoted, it is the paper's. Whose words a concept carries. A concept entry records a term the paper itself uses, so the term, the sense given for it, and its evidence are the paper's own words, quoted where the paper fixes them. This is the opposite of a sense entry in the tag table, which is the tagger's sentence about how the paper uses a term. Where a concept entry could not be grounded in the paper's words, the entry says so on its own row.
Other fields define these differently the same word, defined another way in another discipline
uncertainty 6 fields · sample-size 5 fields · information 4 fields · meaning 4 fields · communication 3 fields · standard-deviation 3 fields · and 1 more →

Tags on this paper

Every tag the record carries, each linking to every other paper in the corpus that carries it. A tag appears here whether or not its words occur in the article — the spans below are only the ones a sentence in this paper licenses. Of the 9 concept tags on this paper, all 9 are grounded in a quoted sentence.
entropy channel capacity bit redundancy information meaning uncertainty equivocation communication quantify information as the logarithm of the number of possible messages (entropy); a noisy channel has a definite capacity C that bounds reliable transmission message channel signal noise bits logarithmic measure transmitter discrete communication system information source discrete channel set of possible messages logarithmic base the channel mean mean of sample-size sample size standard-deviation standard deviation mathematical-proof

Legend

relation to other workclaim_relationshow this paper stands to another: agrees, contradicts, extends, qualifies
claim typeclaim_typethe shape of the claim: existence, mechanism, method, critique, prediction, null
concept edgeconcept_edgesan asserted relation between two concepts, with the sentence asserting it
conceptconceptsan idea the paper works with, quoted where the paper uses it
content tagcontent_tagsa finding, definition, quantity, statistic or variable named in the paper
limitationlimitationswhat the paper says its evidence cannot support
mechanismmechanismshow the reported effect is said to come about
methodmethodhow the paper produced its evidence
phenomenonphenomenonthe effect or regularity under study
sensesensesa distinct use of a term, quoted where the paper fixes it
transfertransfersa method or concept carried in from another field

The paper

The full text is not reproduced here. This article is not cleared for redistribution, so the page carries the tag layer only. Its evidence sentences are withheld; the grounding status remains with each tag, and the article itself is at https://doi.org/10.1002/j.1538-7305.1948.tb01338.x. The reading view exists for the papers whose licence permits it.

Every tag on this page

claim_relations 5 · claim_type 1 · concept_edges 2 · concepts 9 · content_tags 16 · limitations 8 · mechanisms 1 · method 1 · phenomenon 1 · senses 7 · transfers 1

Relations on this page carry a quotation for the claim. Where a relation was separately evidenced — and so tested against the vocabulary for its type — the row says so; none of 2 relations here carry that mark. The rest are the tagger’s reading of the sentence quoted beside them.

This record holds no controls. A layer is empty when the tagger recorded nothing in it — which is not the same as the paper having nothing to record.

facettagfieldconf.whose words
claim_typeexistence
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claim_typethe paper's words
phenomenoninformation transmission over noisy channels
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phenomenonthe paper's words
methodmathematical-proof
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methodthe paper's words
mechanismsmechanisms
Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.
mechanisms[0]the tagger's words
limitationsthe theory deliberately excludes the semantic/meaning aspects of communication
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what this limits: everything the theory is ever used to say about content: THESE SEMANTIC Quotation withheld - meaning is declared out of scope in the founding paper
limitations[0]the tagger's words
limitationsMeaning is explicitly excluded from the theory's scope.
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duplicate of: limitations[0] · what this limits: the same scope as limitations[0] - meaning is excluded by construction, not by oversight
limitations[1]the tagger's words
limitationsThe continuous-case results are deliberately not developed with full generality or measure-theoretic rigour.
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what this limits: the continuous-case results: WE WILL NOT ATTEMPT, IN THE CONTINUOUS CASE, TO OBTAIN OUR RESULTS WITH THE GREATEST GENERALITY, OR WITH THE EXTREME RIGOR OF PURE MATHEMATICS
limitations[2]the tagger's words
limitationsThe existence proof of the noisy-channel coding theorem does not yield a usable construction.
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what this limits: any practical reading of the noisy-channel coding theorem: AN ATTEMPT TO OBTAIN A GOOD APPROXIMATION TO IDEAL CODING BY FOLLOWING THE METHOD OF THE PROOF IS GENERALLY IMPRACTICAL - existence, not construction
limitations[3]the tagger's words
limitationsFor non-white noise the capacity cannot be computed explicitly, only bounded.
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what this limits: capacity results for arbitrary noise: IT DOES NOT APPEAR THAT THE MAXIMIZING PROBLEM INVOLVED IN DETERMINING THE CHANNEL CAPACITY C CAN BE S[olved] explicitly
limitations[4]the tagger's words
limitationsSource rates relative to a fidelity criterion have only been computed in a handful of simple cases.
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what this limits: the rate-versus-fidelity theory: THE ACTUAL CALCULATION OF RATES HAS BEEN CARRIED OUT IN ONLY A FEW VERY SIMPLE CASES
limitations[5]the tagger's words
limitationsShannon states that the uniqueness theorem for H, and the assumptions behind it, are in no way necessary to the theory; the justification is deferred to the implications of the definitions.
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what this limits: the axiomatic derivation of H as the theory's foundation: the paper says the theorem is not necessary and that the definitions are justified by their implications instead
limitations[6]1.0the tagger's words
limitationsThe 50% redundancy figure for English is bounded to statistical structure within about eight letters.
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what this limits: the 50% redundancy of English as an unqualified figure: it excludes statistical structure beyond about eight letters
limitations[7]1.0the tagger's words
transfersentropy itself - the quantity H, its form, and its identification with the H of Boltzmann's theorem, carried out of statistical mechanics and made the measure of information in a communication channel
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transfers[0]0.9the tagger's words
conceptsentropy
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also in: entropy
concepts[0]the paper's words
conceptschannel capacity
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concepts[1]the paper's words
conceptsbit
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also in: bit
concepts[2]the paper's words
conceptsredundancy
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also in: redundancy
concepts[3]the paper's words
conceptsinformation
Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.
also in: information
concepts[4]the paper's words
conceptsmeaning
Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.
also in: meaning
concepts[5]the paper's words
conceptsuncertainty
Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.
also in: uncertainty
concepts[6]the paper's words
conceptsequivocation
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also in: equivocation
concepts[7]the paper's words
conceptscommunication
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also in: communication
concepts[8]the paper's words
sensesinformation - the engineering sense: which message was selected from a known set, with meaning explicitly set aside
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senses[0]1.0the tagger's words
sensesinformation - the significant aspect for design: one message drawn from a set of possible messages, any of which the system must handle
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senses[1]1.0the tagger's words
sensesinformation - a rate at which a source produces, rather than a quantity a message contains
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senses[2]1.0the tagger's words
sensesH - one quantity under three names: the amount of choice involved in a selection, and the uncertainty of the outcome
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senses[3]1.0the tagger's words
sensesrate of transmission - not the transmitted bits minus the errors; the naive subtraction is shown to credit a channel that carries nothing
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senses[4]1.0the tagger's words
sensesH - a single quantity serving as the measure of three things at once: information, choice and uncertainty
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senses[5]1.0the tagger's words
sensesredundancy - the fraction of a message fixed by the structure of the language rather than freely chosen
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senses[6]1.0the tagger's words
claim_relationsbuilds-on → Nyquist, "Certain Factors Affecting Telegraph Speed" / "Certain Topics in Telegraph Transmission Theory" and Hartley, "Transmission of Information"
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claim_relations[0]the tagger's words
claim_relationsextends → the communication theory of Nyquist and Hartley
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claim_relations[1]the tagger's words
claim_relationsbuilds-on → Hartley's logarithmic measure of information
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claim_relations[2]the tagger's words
claim_relationsbuilds-on → N. Wiener, Cybernetics and The Interpolation, Extrapolation and Smoothing of Stationary Time Series
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claim_relations[3]the tagger's words
claim_relationsreplicates → R. M. Fano's coding method (Technical Report No. 65, Research Laboratory of Electronics, M.I.T.)
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claim_relations[4]the tagger's words
concept_edgesentropy —same-as→ uncertainty
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strength of the relation: 1 Audit 2026-09-28: DOUBTFUL - read against its span and kept, but the span is a question about measuring choice and uncertainty; it does not itself equate entropy with uncertainty.
also in: entropy · uncertainty
concept_edges[1]the tagger's words
concept_edgesEVIDENCE CORRECTED (RE-GROUNDED) · redundancy —operationalised-as→ entropy
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EVIDENCE CORRECTED: RE-GROUNDED. The old span named only one endpoint. This sentence names both and IS the definition - redundancy is one minus relative entropy, which is exactly what the edge records. — strength of the relation: 0.8
also in: redundancy · entropy
concept_edges[2]the tagger's words
content_tags5 — components of a communication system
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content_tags.quantities[1]the tagger's words
content_tags2^N — N binary devices store N bits
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content_tags.quantities[2]the tagger's words
content_tags3 — discrete, continuous, and mixed systems
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content_tags.quantities[3]the tagger's words
content_tagssemantic meaning is irrelevant to the engineering problem
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content_tags.findings[0]the tagger's words
content_tagsthe fundamental problem is reproducing a message at another point
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content_tags.findings[1]the tagger's words
content_tagsthe logarithmic function is the natural measure of information
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content_tags.findings[2]the tagger's words
content_tagsbase-2 units of information are called bits
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content_tags.findings[3]the tagger's words
content_tagsset of possible messages
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content_tags.variables[0]the tagger's words
content_tagslogarithmic base
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content_tags.variables[2]the tagger's words
content_tagsthe channel
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also in: the channel
content_tags.variables[3]the tagger's words
content_tagsbit
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also in: bit
content_tags.definitions[0]the tagger's words
content_tagscommunication system
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content_tags.definitions[1]the tagger's words
content_tagsinformation source
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content_tags.definitions[2]the tagger's words
content_tagsdiscrete channel
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content_tags.definitions[3]the tagger's words
content_tagssample-size
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also in: sample-size
content_tags.statistics[1]the tagger's words
content_tagsEVIDENCE CORRECTED (RE-GROUNDED) · standard-deviation
Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.
EVIDENCE CORRECTED: RE-GROUNDED. The old span quoted the ensemble expression itself, from the web page we fetched, and reached the page as 'f /#28 ai /; t/#29/= /+ ∞ ∑ n/= ...'. The grounding copy states the same fact in prose with no debris at all, and that clause - not the formula - is what attests 'standard deviation'. A TEXT EXTRACTION CANNOT GROUND A CLAIM WHOSE EVIDENCE IS AN EQUATION, so the claim is grounded on the sentence that says it in words.
content_tags.statistics[2]the tagger's words

Interpretive choices

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