Sentence, Proposition, Judgment,
Statement, and Fact: Speaking about
the Written English Used in Logic
Dedication: for Professor Newton da Costa,
friend, collaborator, co-discoverer of the
Truth-set Principle, and co-creator of the
Classical Logic of Variable-binding Term
Operators-on his eightieth birthday.
abstract. The ﬁve ambiguous words—sentence, proposition, judg-
ment, statement, and fact—each have meanings that are vague in
the sense of admitting borderline cases. This paper discusses sev-
eral senses of these and related words used in logic. It focuses on a
constellation of recommended primary senses. A judgment is a pri-
vate epistemic act that results in a new belief; a statement is a public
pragmatic event involving an utterance. Each is executed by a unique
person at a unique time and place. Propositions and sentences are
timeless and placeless abstractions. A proposition is an intensional
entity; it is a meaning composed of concepts. A sentence is a linguistic
entity. A written sentence is a string of characters. A sentence can be
used by a person to express meanings, but no sentence is intrinsically
meaningful. Only propositions are properly said to be true or to be
false—in virtue of facts, which are subsystems of the universe. The
fact that two is even is timeless; the fact that Socrates was murdered
is semi-eternal; the most general facts of physics—in virtue of which
propositions of physics are true or false—are eternal. As suggested
by the title, this paper is meant to be read aloud.
The words—sentence, proposition, judgment, statement, and fact—are am-
biguous in that logicians use each of them with multiple normal meanings.
Several of their meanings are vague in the sense of admitting borderline
cases. This paper juxtaposes, distinguishes, and analyzes several senses of
72 John Corcoran
these and related words, focusing on a constellation of recommended senses.
According to the recommendation, a judgment is a private epistemic event
that results in a new belief and a statement is a public pragmatic event,
an act of writing or speaking. Both are made by a unique person at a
unique time and place. Judgments are often not voluntary, although they
cannot be coerced. Statements are usually voluntary, although they can be
coerced. In contrast, propositions and sentences are timeless and placeless
abstractions. A proposition is an intensional entity; in some cases it is a
meaning of a sentence: it is a meaning composed of concepts, a complex
sense composed of simpler senses. A [declarative] sentence is a linguis-
tic entity. A written sentence is a string of characters; it is composed of
character-strings—usually words or “symbols”1that can be used to express
meanings, but which are not in themselves meaningful. A spoken sentence
is composed of [articulate] sounds. Only a proposition is properly said to be
true or to be false—although in certain contexts, or with suitable qualiﬁca-
tions, judgments, statements, or even sentences, may be said to be true or
false in appropriate derivative senses. Propositions are true or false in virtue
of facts, which are either timeless or temporal subsystems of the universe.2
A fact is timeless if it is only about timeless entities such as numbers. The
fact that two is even is timeless. The proposition that two is even can only
be expressed using the timeless-present tense.3It is incoherent to say that
two is presently even or that two is still even. A fact is temporal if it is about
temporal entities such as material objects. Temporal facts are semi-eternal
or eternal. A semi-eternal fact comes into being in an interval of time and
it persists eternally thereafter. The fact that Socrates was murdered can
never be erased. An eternal fact exists without having come into being. The
most general facts of physics—in virtue of which propositions of physics are
true or false-are eternal. In the sense intended here, a fact never changes.
Facts are prior to propositions in the same sense that the past is prior to
1‘Character’, ‘sign’, and ‘symbol’ are often exact synonyms in logic usage. Many
logicians show no awareness of the awkwardness of the fact that, in the senses they prefer,
characters do not characterize, signs do not signify, and symbols do not symbolize—except
under an interpretation, i.e., extrinsically, never intrinsically. See Corcoran, Frank, and
2The proposition that Plato taught Aristotle, which is composed of concepts or senses,
is true in virtue of a fact composed of historical entities that are not concepts. The fact
in question is the fact that Plato taught Aristotle, which has Plato and Aristotle as
constituents. The word ‘fact’ has been used in many other senses. For example, Frege
(1918/1956, 307) takes facts to be true propositions, in a sense of ‘proposition’ very close
to the one recommended in the current article. Later in the same article on page 311, he
uses the word ‘fact’ in yet another sense: he says that thinking, judging, understanding,
and the like are “facts of human life”. Austin (1961, 91) disapproves of taking ‘fact’ as
synonymous with ‘true statement’.
3Frege made a similar point (1918/1956, 309–310).
Sentence, Proposition, Judgment, Statement, and Fact 73
historical truths and nature is prior to laws of nature.
People use sentences to express propositions. People express propositions
they state when making statements; they also express propositions they
do not state. For example, the proposition that zero exists is expressed
both in stating that zero exists and in asking whether zero exists.4A
proposition can be entertained in many ways. What we investigate, doubt,
assume, postulate, believe, and know are often propositions. One and the
same sentence is routinely used on diﬀerent occasions to express diﬀerent
propositions. Likewise, one and the same proposition is commonly expressed
by diﬀerent sentences in diﬀerent languages and, quite often, even in the
Diﬀerences between sentences and propositions are involved in describing
diﬀerences between direct and indirect quotation. In some cases, when we
quote directly we intend to give the exact sentence quoted without convey-
ing the proposition expressed. Abe said: “She loves him”. In some cases,
when we quote indirectly we intend to give the proposition expressed with-
out conveying the sentence used. Abe said that Betty loves Carl. We can
quote a person directly and without awkwardness add “But I did not un-
derstand what was meant”. Quoting indirectly presumes understanding the
proposition indirectly quoted; quoting directly does not.6
Although both propositions and sentences are abstractions, some sen-
tences can be seen: when we read an inscription or token of a sentence we
see the sentence—not the proposition, if any, the sentence is used to ex-
press. Sometimes we can look at an inscription for some time before seeing
the sentence inscribed. Most sentences have not been inscribed and are thus
not seen: such sentences will not be seen until inscribed and they will not
be seen when all inscriptions are obliterated.
Ontology is important in this paper as shown in the following consider-
ations. Socrates taught Plato. My judgment that Socrates taught Plato
was made many years ago. My statement of the proposition that Socrates
taught Plato was made seconds ago using the sentence ‘Socrates taught
Plato’, which involves three English words but which contains no meanings
and no persons. The proposition that Socrates taught Plato involves three
meanings but it contains no words and no persons. The fact that Socrates
taught Plato involves two persons but it contains no meanings and no words.
In the senses recommended, facts, propositions, judgments, sentences, and
4To the best of my knowledge, this type of observation is due to Frege (1918/1956,
5The last few points have been standard for years. The result of adding ’in fact’ and
a comma to the front of a sentence expressing a given proposition is a diﬀerent sentence
expressing the same proposition. See Cohen-Nagel 1934/1962/1993, xxii–xxv, 27–35.
6Indirect quotation is a subspecies of indirect discourse (Audi 1999, 424).
74 John Corcoran
statements comprise mutually exclusive ontological categories. However,
questions of “ontological status” play no role in this paper: it is irrelevant
whether sentences, propositions, or facts—or for that matter, characters,
numbers, or truth-values, are “real entities”—are “idealizations”, or are
“theoretical constructs”.7As is suggested by the title, this paper is written
to be read aloud.
2 Sentences, Judgments, Statements
A sentence is made up of words; a statement
is made in words.—Austin 1961, 88.
Statements are made; ... sentences are used.—
Austin 1961, 88.
Since context is important in speech, it might help to begin with examples.
Along the 3000-mile northern border of the United States with Canada, the
weather is changeable, to say the least. The weather is a frequent topic of
conversation. There is keen interest not only in the weather itself, but also
in the nature of communication about weather, in weather discourse.
Perhaps interest in weather discourse is increased by the fact that Amer-
icans report weather in one system of units while Canadians report it in an-
other. Americans report temperature in Fahrenheit degrees, where thirty-
two is freezing. Canadians report temperature in Celsius degrees, where
zero is freezing but thirty-two is very hot. People on one side of the bor-
der converse with friends and relatives on the other. It is not unusual for
someone to get a weather report in one system and convey its content to
others in another system. This situation gives rise to sentences that may
seem strange in other contexts.
Here are some examples.8
7I have discussed such questions elsewhere (e.g. 2006s).
8As indicated in the abstract, this paper is primarily meant to be read aloud and
then later to be discussed as a written text. As usual, the reader must “impersonate”
the author. In order to make the text look “normal”, instead of displaying a sentence
as such, e.g. ‘ten is ﬁfty’ (no uppercase ﬁrst letter, no period), I display the so-called
assertoric form of it which begins with an uppercase tee and ends with a period, ‘Ten
is ﬁfty.’(sic) in this case. Where logic matters, this becomes important. The sentence
‘ten is ﬁfty’ is literally a part of its own negation ‘it is not the case that ten is ﬁfty’,
but the assertoric form of the sentence is never a part of the negation of the sentence,
or of the assertoric form of the negation of the sentence. Moreover, the sentence is part
of its quotes-name and the assertoric form of the sentence is part of the quotes-name of
the assertoric form of the sentence. But neither is part of the quotes-name of the other.
Tarski (e.g. 1969/1993, 103) is one of the few writers who usually keep the sentence apart
from its assertoric form. And he never explains what he is doing or why. See Corcoran
2006s for more details.
Sentence, Proposition, Judgment, Statement, and Fact 75
Zero is thirty-two.
Ten is ﬁfty.
Twenty is more than sixty-ﬁve.
Twenty is between sixty-ﬁve and seventy.
Zero is to one hundred as thirty-two is to two-twelve.
Both hundreds are hot but only one is boiling.
Even without conversions between Celsius and Fahrenheit, weather dis-
course sometimes involves sentences that are otherwise unusual.
Zero is freezing.
Ten is cold.
Twenty is comfortable.
Thirty is hot.
Forty is sweltering.
Freezing is zero.
Freezing is thirty-two.
Thirty-two is freezing.
Thirty-two is hot.
One-hundred is boiling.
We can swim in eighteen.
In doing conversions from Celsius to Fahrenheit, it helps to know that ten
Celsius degrees is eighteen Fahrenheit degrees. Zero, ten, twenty, and thirty
degrees Celsius are respectively thirty-two, ﬁfty, sixty-eight, and eighty-
six degrees Fahrenheit. This is calculated by adding eighteen Fahrenheit
degrees whenever ten Celsius degrees have been added.
Notice that ten degrees Celsius is a deﬁnite single point on the Celsius
scale whereas ten Celsius degrees is not a point but an interval-size that has
no location on the scale. It would be incoherent, a so-called category mistake,
to say that today’s temperature is ten Celsius degrees or that the diﬀerence
between yesterday’s high and low was ten degrees Celsius.9A person saying
something incoherent is sometimes making a category mistake.
I have been using sentences made up of English words to make statements
based on judgments. In the words of Austin quoted above: “A sentence is
made up of words; a statement is made in words”. In several places Frege
9Of course, if someone writes the expression ‘the current temperature is ten Celsius
degrees’ we should not jump to the conclusion that there was a category mistake and
not just an inadvertent writing error, a transposition of words. We could ask whether
they intended to write ‘The current temperature is ten degrees Celsius’. An aﬃrmative
answer would suggest a writing mistake and not a category mistake. Notice that the
quotes name of the assertoric form of a sentence does not have a period at the end unless
it occurs at the end of the sentence using it.
76 John Corcoran
said that the answer to a question is a statement based on a judgment (1918,
1919,1952, 117; 1997, 299).
The statements, or assertions10, that I have just made aloud in this spo-
ken delivery were based on judgments that I made earlier, silently of course.
Some judgments were made as I was planning the paper; some were made
long before that. Some judgments involved little beyond conﬁrmation of
memory; some involved perception; some calculation; some deduction; some
induction. Every objective judgment is made in reference to the fact it con-
cerns (Corcoran 2006e). The statements were all public and were all made
in public today. The judgments were all private and were all made in private
before today. No two persons can make the same judgment.
In judging I form a fresh belief, often a completely new belief in the truth
of a proposition not previously believed by me—perhaps even previously
disbelieved by me. In stating per se new beliefs are not formed: some
statements are not based on belief at all; some are contrary to their speaker’s
judgments—a fact repeatedly overlooked by Frege.
In the senses recommended, ‘judgment’ and ‘statement’ are common
nouns whose extensions are classes of events: epistemic events in the case
of ‘judgment’; pragmatic events in the case of ‘statement’ (cf. Austin 1961,
87). But the ending ‘ment’ brings other senses. The two words can be used
as proper names in various senses. For example, ‘judgment’ can be used to
name the human faculty by which judgments are made or with qualiﬁcation
it can be used to name a person’s faculty as ‘Abe’s judgment improved in
time’. Both can be used as common nouns for the objects or results of
the eponymous acts in various senses: Abe’s judgment (statement) [sc. the
proposition Abe judged (stated)] was the same as Ben’s.
3 People Use Words to Mention Things.
When saying something of an object, one
always uses a name of this object and not the
object itself, even when dealing with linguistic
objects.—Alfred Tarski 1969/1993, 104.
Premises rhymes with nemesis, not with
nemeses.—Frango Nabrasa (per. comm.)
10I use the word ‘assertion’ reluctantly as a synonym for ‘statement’ in the recom-
mended sense. However, the fact that it has drastically diﬀerent senses even in logic calls
for caution. Some logicians use it to referred theorems, statements having the highest
level of warranted assertibility. But it is widely used outside of logic for statements with
the lowest level of warrant. At the end of a famous address before the United Nations,
Colin Powell asserted that his previous statements were not assertions. Other uses occur,
e. g. Quine 1970/1986 and Goldfarb 2003. See Quine’s insightful quip (ibid. p. 2).
Sentence, Proposition, Judgment, Statement, and Fact 77
After this section of the paper, I may seem to be at a slight disadvantage.
Up to this point I have been using some spoken sentences to mention other
spoken sentences. The sentence I am now reading is a spoken sentence.
After this paragraph I will of course continue to use only spoken sen-
tences, but I will be mentioning mainly written sentences. From here on,
most of the sentences that will be mentioned are written sentences.11 The
sentence I am now reading is a written sentence.
I will not be able to use what I will be mentioning. Isn’t that the usual
case? No one uses the freezing point to mention the freezing point.
However, we routinely use expressions that we are mentioning. As I
noted above, I used spoken sentences to mention spoken sentences. Before
the invention of writing there was no other way to mention spoken sentences.
As we saw above, it is common to encounter an ambiguous sentence, one
that could normally be used to express two or more propositions. When
I read an ambiguous sentence to an audience, one listener might have a
thought of one proposition at the precise time that another listener has
a thought of another proposition. When I said ‘The sentence I am now
reading is a spoken sentence’, some readers thought that the proposition
thereby stated was false and some thought that the proposition stated was
true. The expression ‘the sentence I am now reading’ is ambiguous. Some
readers might have thought that I was talking about a sentence written on
the paper I am reading from; others might have thought that I was talking
about the sentence I was enunciating.
Using one ambiguous expression twice in consecutive paragraphs intend-
ing diﬀerent meaning each time is a practice to be avoided, other things
Of course, although no two persons can have the same thought, thankfully
nothing prevents their separate thoughts from being thoughts of the same
proposition. Otherwise, all agreements and disagreements would be merely
“verbal” and communication would be of a very restricted nature, if indeed
it could go on at all.
Occasionally, two of the propositions expressed by one ambiguous sen-
tence are of such a nature that in expressing one we use a word that we
mention when expressing the other. For example, consider the three-word
sentence ‘Santiago is Spanish’. In one sense, this expresses a proposition to
the eﬀect that the city of Santiago de Compostela is in Spain. The eight-
11Thus, every sentence is a string of characters, counting the space or blank as a
character. But, of course, not every string of characters is a sentence. The terminology
varies, even within the writings of one author. Tarski says that a sentence is a series
of printed signs (1956/1983, 156). Church writes of a sentence as being a succession of
letters (1956, 27). Boolos writes of sequences of symbols and of strings of symbols on the
same page (2002, 103).
78 John Corcoran
letter word ‘Santiago’ is used (to mention the city), but the word itself is
not mentioned. In another sense, the same three-word sentence expresses a
proposition to the eﬀect that the one-word expression ‘Santiago’ is part of
the Spanish language; it translates as ‘Saint James’. Here the word ‘San-
tiago’ is mentioned (by using the same word, ‘Santiago’ itself). In a third
sense, it expresses the proposition that Saint James is Spanish, which is
false, of course. Here again the word ‘Santiago’ is used (but to mention
the saint). Before leaving this important topic let us see what Church says
Following the convenient and natural phraseology of Quine, we may distinguish
between use and mention of a word or symbol. In ‘Man is a rational animal’, the
word ‘man’ is used but not mentioned. In ‘The English translation of the French
word homme has three letters’, the word ‘man’ is mentioned but not used. In
‘Man is a monosyllable’, the word ‘man’ is both mentioned and used, though in an
anomalous manner, namely autonymously.12
Notice that the last spoken word quoted is ambiguous; the sense intended
by Church was coined by Rudolf Carnap in 1934 (1934/1937, 17). The
third syllable of its written counterpart is spelled en-wye-em as is the case
in the rhyming word ‘synonymously’. It should not be confused with the
phonetically similar and older ‘autonomously’ whose third syllable is spelled
en-oh-em (Chateaubriand 2005, 150). The two contextually homophonic
third syllables nym and nom are not etymologically related - any more
than the number-word ‘two’ is related to the preposition ‘to’ with which it
is a homophone.13
12Church (1956, 61). The italicization is Church’s, but Church uses double quotation
marks where single quotation marks are used above in keeping with the style of this
article. In the same book, actually on the next page (1956, 62), Church uses single
quotation marks in mentioning words and he even explicitly notices the advantage of
using single quotations in this way.
13As far as I know, no meaning has yet been assigned to ‘synonomy’, ‘synonomous’, or
Sentence, Proposition, Judgment, Statement, and Fact 79
4 Written English is not a phonetic transcription of
Now spoken sounds are symbols of mental
ideas, and written marks symbols of spo-
ken sounds. Just as written marks are not
the same for all humans, neither are spoken
sounds. But what the latter are symbols of -
mental ideas - are the same for all; and what
these ideas are likenesses of - actual things -
are also the same—Aristotle, On Interpreta-
The adjectives ‘spoken’ and ‘written’ are ambiguous: each has a broad and
a narrow sense—like the word animal. In the broad sense, every human is
an animal; in the narrow sense, no human is an animal. There are many
spoken sentences (broad sense) that may never have been spoken or uttered
by anyone and which therefore are not spoken sentences (narrow sense).
Likewise, many written sentences (broad sense) have never been written or
inscribed by anyone and which therefore are not written sentences (narrow
For example, the two-word sentence ‘zero sleeps’ made up of the four-
letter word ‘zero’, or zee-ee-ar-oh, and the six-letter word ‘sleeps’, or es-el-
ee-ee-pee-es, may never have been written or inscribed by anyone before,
but even before it was written it was a written sentence.15 Of course, any
proper name can be written or inscribed one space in front of an inscription
of an intransitive verb in order to make an inscription of a written sentence.
You can picture the following examples of written sentences.
14This translation of On Interpretation, I, 16a3–8 is essentially the one used by N.
Kretzmann (1974, 3–4) in his informative and accessible discussion of this passage, which
he calls “the most inﬂuential text in the history of semantics”. Kretzmann credits the
translation largely to Ackrill 1963.
15This is the ﬁrst spoken sentence today illustrating two of the ways of making spoken
proper names of words: the appositional method is to say the two-word expression ‘the
word’ just before pronouncing the word itself, the phonetic-orthographic method is to
“spell-out” the word, i.e. to pronounce the names of the word’s letters in the order in
which they occur in the word. Thus, the word ‘zero’ is one and the same thing as zee-
ee-ar-oh. These two spoken alternatives correspond somewhat to the two written ones in
Tarski’s truth-deﬁnition paper: the quotation-mark name and the structural-descriptive
name (1956, 156).
80 John Corcoran
Have inscriptions of these sentences ever existed before? Who could know
except someone who saw such an inscription?
A written sentence is not necessarily a sentence that has been written
but rather a sentence of the kind that could be written. In other words, a
written sentence is a string of characters.16 In our case, the characters are
all either letters of the Latin alphabet or punctuation marks. Is the space
that occurs between consecutive words or between consecutive sentences a
character? Or is it a non-character? Or is it a borderline case?
It is certainly signiﬁcant; it is used meaningfully even though it is not used
to express a constituent of a proposition, a concept. The word ‘character’
is in many ways typical of the words we will be focusing on: it is ambiguous
and some of its meanings are vague. In the sense recommended in this
article, the space is a character. Ambiguity (or polysemy) and vagueness
(or indeﬁniteness) are two of the most pervasive of linguistic phenomena.
My advice is to recognize them and to become accustomed to them.17 A
writer might use an ambiguous expression without being equivocal. A writer
might use a vague concept without being guilty of imprecision. Aristotle
tells us not to strive to be more precise than the subject-matter permits
(Nicomachean Ethics, I, 3). Describing a sunrise requires vague concepts.
Written sentences are things I will be mentioning but not using in this
spoken presentation. The sentences that I will be using are strings of sounds,
in one sense of this ambiguous word. Using the perspicacious terminology
coined by Charles Sanders Peirce, a written sentence is one that can only be
embodied visually and a spoken sentence is one that can only be embodied
audibly. The linguists who know about such matters conclude that spoken
English and written English are such disparate systems of communication
16The problem of ﬁnding an exact deﬁnition of “Written English sentence” is far from
settled; it is one of the famous unsolved problems in contemporary linguistics. Noam
Chomsky brought this and related problems to the forefront of linguistic research. Like-
wise, even the problem of ﬁnding an exact deﬁnition of “Written English word” is far
from settled (Lyons 1977, Vol. I, Ch.1). The word ‘word’ is of course ambiguous but, in
the “typographical” sense used in this article, a word is nothing more than a string of
characters. Thus, there are no such things as homonyms, or more precisely, homographs.
No two words have the same spelling, or the same succession of characters. A word is its
spelling, so to speak; “two” words spelled the same are one. And no two sentences have
the same wording, or the same succession of words and spaces.
17The “logically perfect” or “formalized” languages studied in mathematical logic ex-
hibit neither ambiguity nor vagueness—this is an advantage for some purposes, and a
disadvantage for other purposes. (Cf. Church 1956, 2, 3, 32, 50.) There are of course
needless ambiguities, sometimes called misnomers, which are exasperating because they
entered the language by way of a mistake involving what had been a misuse of a “less
ambiguous” word. The stock example is the word ‘Indian’ whose misuse became so wide-
spread that it had to be recognized as a use. There are many other examples, several in
philosophy and logic. Church cites ‘sentence’ as a possible example (1956p, 6).
Sentence, Proposition, Judgment, Statement, and Fact 81
that the two should not be regarded as diﬀerent forms of the same language
but rather as two diﬀerent languages. My language, [American] English, is
actually two languages.18
In keeping with the usual convention of capitalizing proper names of
languages, I will be using Spoken English (written as a proper name) to
discuss Written English (also with capitals); henceforth, unless explicitly
indicated otherwise, the single word ‘sentence’ is elliptical for the two-word
expression ‘Written sentence’—in the broad sense of a writable sentence,
not one that has necessarily been written.19
There is no such thing as an English sentence that has two “forms”—a
“Spoken form” and a “Written form”. Every English sentence is a sentence
of Spoken English or of a Written English, and not both, of course.
Using the terminology of modern logic, we can say that in the discussion
to follow, Spoken English is the metalanguage and Written English is the
object-language. Actually I should say ‘an object-language’ or ‘one of the
object-languages’, not ‘the ob ject-language’, because I will be mentioning
Spoken English sentences also, as well as sentences of a few foreign writ-
ten languages. In our case, Spoken sentences of the metalanguage will be
used by a speaker, namely me, to mention mainly Written sentences of one
object-language, Spoken sentences of another object-language, and written
sentences of other object-languages. More importantly, I will be using sen-
tences to make statements about object-language sentences to the members
of the audience, namely you, who are responsible for critical evaluation. The
members of the audience routinely make judgments20 about the speaker’s
18It is obvious that Written English is not a phonetic representation of Spoken English.
There are no Spoken sight rhymes and no Written sound rhymes. Moreover, Written
English can represent logical groupings with parentheses and other typographical devices
where Spoken English is without equally adequate devices. On the other hand, there are
propositions expressible in a small number of words of Spoken English that can not be
expressed in Written English at all. In these cases, every Written sentence expressing a
proposition implying one intended by such a Spoken sentence also implies propositions
not implied by the intended one. The gulf between the two languages becomes even
more evident if we try to ﬁnd a pair of sentences one Written and one Spoken which are
uniformly translations of each other in the sense that they express exactly the same range
of messages. To the question of whether there is a peculiarly English system of messages
that can only be exactly conveyed in one or both of the two media Spoken English and
Written English, I would answer yes, tentatively and with qualiﬁcations.
19Accordingly, the two two-word expressions ‘Spoken sentence’ and ‘Written sentence’
are used as ellipses respectively for the two three-word expressions ‘Spoken English sen-
tence’ and ‘Written English sentence’.
20Gottlob Frege, the meticulous logician regarded as the founder of Analytic Philos-
ophy, uses the word ‘judgment’ in a sense close to that used here. He says (1980, 20):
“According to my way of speaking, we think by grasping a thought, we judge by recog-
nizing a thought as true, and we assert by making a judgment known [to others]”. Also
see Frege 1918/1956, 294. However, not everyone is this careful. For example, David
82 John Corcoran
statements and about the subject matter of the speaker’s statements. My
next statement is a good example to reﬂect on.
It is obvious that the Written English sentence ‘zero is cold’ can be used to express
a true proposition if and only if zero—in some sense—is cold—in some sense.
By using this Spoken sentence, I mention but do not use the Written sen-
tence ‘zero is cold’; and I use but do not mention the Spoken sentence zero
There is no way to confuse a statement with a sentence or with a propo-
sition, since it is only the statement that inherently has a speaker, who is
responsible for its accuracy, and it is only the statement that per se has an
audience, which is responsible for its critical evaluation. Every statement is
an event. Like every other event, every statement has eﬀects, ramiﬁcations
which are also events, not propositions; and like any other action, it changes
the world in which it is made. No proposition is an event. Every propo-
sition has consequences, implications which are propositions, not events;
the proposition per se is causally inert. No statement is a proposition. No
proposition is a statement. Here I am using the standard, traditional ter-
minology that Church so adequately describes in his classic 1956 article
“Propositions and Sentences”, perhaps the clearest thing ever written on
the topic, certainly the best thing I know of (1956p, 3–11). To be explicit,
the word ‘statement’ is used in the transactional sense, the word ‘propo-
sition’ in the intensional sense, and the word ‘sentence’ in the syntactic
5 Sentences Express Propositions and People Make
When a sentence expressing a proposition
is asserted we shall say that the proposition
itself is thereby asserted.—Church, 1956i, 27.
It seems to me that [by imagination vs.
judgment] you have in view the diﬀerence
between grasping a thought [sc. proposition]
and recognizing a thought as true. The latter
is what I call judging.—Frege to Russell in
1904. 1980, 163.
Any one sentence—made of words and ultimately of letters and characters—
can be used to express two or more diﬀerent propositions or, more generally,
Hilbert uses the word ‘judgment’ to mean “proposition”, in a sense close to that of this
paper (Luce 1950, 165–6, Hilbert-Ackermann 1928/38/50, 171).
Sentence, Proposition, Judgment, Statement, and Fact 83
diﬀerent messages—made up of meanings or concepts. The number of words
in a sentence is not usually the same as the number of concepts in a message
it expresses; it could be more but it is often less. Aristotle had already said
that although sentences are not the same for all humans, what they express
is the same for all (On Interpretation, Ch. 1, 16a3-18).
The three-word sentence ‘thirty-two is freezing’ is typical. It can be used
to express the true proposition that thirty-two degrees Fahrenheit is the
freezing point of water. Thirty-two is freezing. I just made a statement
that thirty-two degrees Fahrenheit is the freezing point of water by using
the three-word sentence. The very same sentence can also be used to express
the false proposition that thirty-two degrees Celsius is the freezing point of
water. I never made a statement to that eﬀect. It can be used to express a
proposition about one of the athletes known as thirty-two.
And it can be used to express an incoherent message to the eﬀect that the
number thirty-two is in the process of becoming frozen, or perhaps that the
number thirty-two feels cold. I am using the word ‘message’ for the genus
of what is expressed by a complete sentence. Thus a message is either a
proposition, whether true or false, or an incoherency, which is neither. An
incoherent message cannot be coherently said to be a mistake any more than
a false proposition can coherently be said to be a lie. Some incoherencies
can be poetic.21
Uncomfortable Numbers: A Poem.
Thirty-two is freezing.
Thirty-two is hot.
Thirty-two is sweating.
Thirty-two is not.
The above paragraph including the poem illustrates several facts. First,
sentences are ambiguous, or polysemous: rarely if ever is a sentence unam-
biguous, or monosemous. Second, propositions are about something and it
21There are several fallacies that arise in discussions of incoherencies, i.e. incoherent
messages. Perhaps the most common is to think that there are sentences that somehow
by nature can only be used to express incoherencies. The fact is that people can make
conventions instituting new coherent uses for any given sentence. Power corrupts. Glory
addicts. Fame fades. Logic confuses. Logic clariﬁes. Another common fallacy is to think
that a sentence expressing an incoherency is meaningless. The expression ‘meaningless
sentence expressing’ itself understood in the intended way is an oxymoron: every sentence
expressing something means that which it expresses. Another common fallacy is to
confuse contradictory messages with incoherent messages. The message that some odd
number is not odd is coherent but contradictory and thus false. The message that some
odd number trisects a right angle is incoherent, thus not false, and thus not contradictory.
84 John Corcoran
is in reference to their subject-matter that they are true, or false, as the case
may be.22 Third, not every meaningful use of a sentence involves using it
to express a proposition; a sentence can be used to express an incoherency,
a message that is neither true nor false.23 Fourth, not every pair of sen-
tences that could be used to express a contradiction are so used in a given
discourse—there are “verbal contradictions” that are not contradictory. We
have already seen that it is not necessarily self-contradictory or even false
to say that not all written sentences are written sentences, nor is it neces-
sarily tautological or even true to say that all written sentences are written
sentences. Logicians and grammarians might legislate that no one expres-
sion may be used in two or more senses in one text. Even though their
advice is generally sound, their jurisdiction hardly ever extends beyond the
The established notational convention already applied above is to use
single quotes for making names of sentences and other expressions, for ex-
ample, names of characters, words, phrases, and sentences. These are the
so-called single-quotes-names.24 Thus ‘one plus two is three’ is a ﬁve-word
English sentence and ‘square’ is a six-letter English word, both recognized
by me, neither of which would have been recognized by Aristotle. Following
Bertrand Russell, Rudolf Carnap, Morris Cohen, Ernest Nagel, John Lyons
and others, double quotes are used in naming propositions, incoherent mes-
sages, and other meanings.25 These are the so-called double-quotes-names.
Thus, “one plus two is three” is a true proposition known both to me and
to Aristotle and “square” is a concept also well known to both.
22According to this traditional conception of proposition it does not seem coherent to
say that a proposition is true in,at,on or of a possible world, a time, a circumstance, a
situation or anything else; a proposition is either true or false. This is of course not to
say that the word could not be used in some other sense according to which it is coherent
to say, e. g. that some propositions true in this world are false in some other world,
although in this case it would be hard to imagine how such an apparently inaccessible
fact could be known. Frege, Moore, Church, and Austin are silent on these senses of
23The question arises whether incoherent messages have the same objective existence
that belongs to propositions or whether they are entirely subjective, merely mental con-
structs. This is a diﬃcult issue which may turn out to be as much a matter of convention
as of fact.
24Church (1956p, 61) says that Frege used single quotes to indicate that the quoted
material was being used autonymously; this is not using single-quotes-names. In diamet-
rical opposition, both Tarski and Quine regard a sentence using an occurrence of a quotes
name as not using the quoted expression at all. Consider the next sentence in the body
of the paper; it mentions ‘square’. Frege regarded it as using the 6-letter word ‘square’;
Tarski and Quine regarded it as using the 8-character quotes-name of the 6-letter word
but not using the 6-letter word at all.
25Bertrand Russell 1903, 53ﬀ, 1905/1967, 99; Carnap 1934/1937, 14; Cohen-Nagel
1934/1962/1993, xxii; Lyons 1977, Vol. I, x.
Sentence, Proposition, Judgment, Statement, and Fact 85
In simple cases, expressions [are used to] express meanings or senses and
they [are used to] name, or mention, entities or things. Thus the number-
word ‘three’ expresses the individual concept “three” and it names the num-
ber three, the number named ‘three’. In some cases the situation is more
complicated: the sentence ‘one plus two is three’ expresses the proposition
“one plus two is three”; but it is incoherent, even ungrammatical, to say
that it names one plus two is three. The expression ‘three’ expresses the
concept “three”. This is a pattern often safe to follow. The expression
‘three’ names the entity three. This pattern is less safe; it must be used
The expression ‘no number’ denotes no number, but ‘some number’ does
not denote some number. In fact, no number is named ‘some number’, or
for that matter ‘no number’. Of course, in a heteronymous sense, the ex-
pression ‘no number’ does not denote at all, a fortiori it does not denote
something named ‘no number’. There is nothing named ‘no number’ except
the expression itself, which is no number. In the autonymous sense of ‘no
number’, no number denotes no number, which is an expression not a num-
ber. Likewise in the autonymous sense of ‘no number’, no number denotes
nothing, thus not an expression, a number, or anything else.27
There are several other terminological systems that are in use; some may
be more appropriate in certain contexts. All must be used with care. Some
writers say that an expression expresses, connotes, or signiﬁes an intension,
a connotation, or a meaning, or a sense. And, in contrast, they say that
an expression names, refers to, or denotes an extension, a referent, or a
denotation. Some writers take “the connotation” of an expression to be its
emotional overtones and explain that it may vary from person to person
even when there is agreement on what is expressed and on what is named.
The expression ‘seven’ taken by two people to express “seven” and name
seven may connote good luck for one person and bad luck for another.
There is little chance to confuse a judgment with a sentence or with a
proposition. Each judgment is an event; it is an act done at a particular
time by a particular person. The person judging that a certain proposition
is true usually grasps the proposition before judging that it is true. I remem-
26Church (1956p, 25f.) follows Frege in holding that, in a logically perfect or formalized
language totally devoid of ambiguity, all sentences expressing true propositions name
an entity truth and all expressing false propositions name an entity falsehood. This
“discovery” by Frege was unprecedented in the entire history of logic and linguistics and,
as far as I can tell, it has yet to be widely accepted. My guess is that in the fullness of
time it will be relegated to a footnote in the history of ideas.
27Contrary to the impression given by Russell in his 1905 paper “On Denoting”, most
of the expressions he calls denoting phrases do not denote in any sense of ‘denote’ I know
of—certainly not in the sense of Church 1956p used in this paper—except when used
86 John Corcoran
ber clearly that before concluding that thirty degrees Celsius is eighty-six
Fahrenheit, I considered the proposition and wondered whether it was true.
Before doing the mental process of verifying by calculation the premise that
thirty-two plus three times eighteen is eighty-six, I considered the conclusion
as a hypothesis, consciously suspending judgment. You might have done the
same. As mentioned above, neither the proposition nor the sentence used
to express it has a time of occurrence.28 These facts are independent of the
fact that the word ‘judgment’ exhibits what is known as process/product
ambiguity—besides being used for the action or process of concluding that
a proposition is true it is also used for the result or product, for the propo-
sition concluded, the conclusion of a judging. The same ambiguity belongs
to the word ‘conclusion’.
There is little chance to confuse a judgment or conclusion with a state-
ment or assertion although they are both actions. Both require a proposition
but the judgment does not require a sentence. Perhaps you made a judg-
ment or arrived at a conclusion and then later decided which sentence or
language to use to express it. Both require an agent but no judgment can
have an audience.
When the words are not used in the senses recommended there is much
room for confusion. The same process/product ambiguity belongs to the
word ‘statement’—besides being used for the action or process of stating
that a proposition is true it is also used for the result or product, for the
proposition stated, or even the sentence used. When the words are both
used in the product sense, we have the sorry spectacle of saying correctly
that the judgment is a proposition, the statement is a proposition, and the
judgment is a statement. Even though there is nothing objectively wrong
about using ambiguous words, or even using them in multiple senses in one
and the same paragraph, nevertheless many able writers have confused their
readers by doing so.
In the above product senses of ‘judgment’ and ‘statement’ there is noth-
ing incoherent about saying that a judgment or statement is true or is false.
There are other such senses as well: the sentence ‘The statement you made
is true’ can be taken as elliptical for ‘The [proposition asserted in the] state-
ment you made is true’.
28Absurdly, the word ‘judgment’ (French ‘jugement’) has been used in logic since at
least the early 1600s for the ﬁctitious act of simultaneously constructing a proposition,
determining that it is true, and asserting it (Arnauld and Nicole 1662, Part II, Ch. 2
and esp. Ch. 3,: Whately 1826, 57, 75–77). Whately can not conceive of a non-judged
proposition; he thinks that the copula ‘is’ or ‘is not’ “indicates the act of judgment” (1826,
57). These writers overlook the fact that every day we consider propositions not known
to be true and not known to be false. The non-judged hypothesis that the defendant is
the murderer is often grasped by the jury even before any evidence can be presented.
Sentence, Proposition, Judgment, Statement, and Fact 87
6 Sentences, Not Propositions, Are Ambiguous or
Unambiguous, Extrinsically; Propositions, Not
Sentences, Are True or Untrue, Intrinsically.
A proposition is composed not of words,
nor yet of thoughts, but of concepts.—G. E.
Moore 1899, 179.
The words ‘sentence’ and ‘proposition’ are both ambiguous. Besides the
syntactic, typographical, or morphological meaning explained above, the
word ‘sentence’ can also be used as elliptical for ‘meaningful sentence’ in
the hybrid, or composite, sense of a sentence expressing a certain one of
the messages it normally expresses. This would be a sentence meant or
taken in a certain way; in some cases, one might say a propositional sen-
tence. Likewise, besides the intensional sense explained above, the word
‘proposition’ can also be used in a hybrid sense to indicate a message or
a proposition expressed in words in a certain way. The latter would be a
proposition expressed a certain way, a sentential proposition. There is little
more than emphasis to separate the hybrid sense of ‘sentence’ from the hy-
brid sense of ‘proposition’. This is brought out by the fact that ‘sentence’ in
its hybrid sense was used by Mahoney to translate the German word ‘Satz’
as used by Frege (1884/1964 ), while ‘proposition’ in its hybrid sense was
used by Austin to translate the same passages (1884/1964 ). Frege himself
recognized an ambiguity as did Hans Kaal, the translator of the Frege corre-
spondence, who agrees with Austin. Frege’s letter to Russell of 20 October
1902 (1980, 149) contains the following passage:
Your example ... prompts me to ask the question: What is a proposition? German
logicians understand by it the expression of a thought [sc. proposition], a group of
audible or visible signs expressing a thought [sc. proposition].29 But you evidently
mean the thought [sc. proposition] itself. This is how mathematicians tend to use
the word. I prefer to follow the logicians in their usage.
In fact, Tarski often used ‘sentence’ in the sense “meaningful sentence”,
i.e., for propositional sentence. Church discussed the use of ‘proposition’
in the sense “sentential proposition”. In this sense, a proposition is “a
composite entity, sentence plus proposition” (Church 1956p, 6). But he
29I am reluctantly following the established practice of reading ‘proposition’ where
a strict and literal translation of Frege would require the grossly inappropriate word
‘thought’. However, it is clear to me that Frege’s “thoughts” are not my propositions.
Two sentences expressing the same thought need not express the same proposition. Con-
sider the following scenario. Abe called yesterday. Also yesterday, I used the sentence
‘Abe called today’ to report the fact that Abe called yesterday. The proposition expressed
using ‘today’ is not the same as the one expressed using ‘yesterday’ but the thoughts are
the same (1918, 296).
88 John Corcoran
used the word only in the abstract intensional sense of this paper, i.e. for a
combination of meanings that is either true or false. Boolos (2002, 112, 114)
used the word ‘sentence’ as a technical term only in the purely syntactic,
typographical sense stipulated here in which a sentence per se is a “syntactic
entity”, a string of characters without regard to whether it has been used
to express a message or to which meaning has been or will be assigned to
it by writers.30 He also used it casually in the hybrid sense (2002, 103).
In order to eliminate one source of confusion, slashes (virgules) can be
used to refer to composite entities. The composite word /one/ is a two-part
system composed of the string ‘one’ and the sense “one”. The number one
is denoted by the word /one/ and determined by the sense “one”. The
composite /one plus two is three/ is “a composite entity, sentence plus
(abstract) proposition” (Church 1956p, 6), that expresses or contains the
proposition that one plus two is three. As Chateaubriand (2001, 379) points
out, composites such as /one/ give rise to what has been called a semiotic
triangle: the three vertices are respectively a string, a sense, and a ref-
erent and the three sides are respectively the relations of string-to-sense,
string-to-referent, and sense-to-referent. He says that a composite expres-
sion “contains” its sense.
I used the word /Santiago/ autonymously above in this article. As al-
ready implied, the word /autonymous/ coined by Rudolf Carnap in 1934
should not be confused with the phonetically similar and older /autonomous/
which is not etymologically related (Chateaubriand 2005, 150).The slash no-
tation cannot be used unless it is clear in the context which of the various
interpretations is intended as in:
The number one is denoted by the composite word /one/.
A string per se is meaningless: nothing is denoted by the string ‘one’ ex-
cept under an interpretation by a person. In many ways it would be graceful
to use something more familiar such as italics to mention composites.31
The composite word one composed of ‘one’ and “one” denotes the number one.
In both of the two senses mentioned the word ‘proposition’ is a technical
term of logic and has been for centuries. Church wrote (1956p, 3):
30The logician Warren Goldfarb used the word ‘statement’ for something very close to
an idealized sentence-proposition (Goldfarb 2003, 5ﬀ.). Quine (1970/1986, 13–14) used
the expression ‘eternal sentence’ for this. See the comments below about Quine’s most
31The disadvantages include the ambiguity resulting from the fact that italic font has
other uses and the fact that italic can not be italicized: //one// is /one/ or /one/ and
is composed of ‘/one/’ and “/one/”.
Sentence, Proposition, Judgment, Statement, and Fact 89
In Latin, propositio was originally a translation of the Greek protasis, and seems to
have been used at ﬁrst in the sense of premiss.
In either technical sense it has no meaningful connection with a proposing
or a proposal—just as the sense of the word ‘work’ as a technical term of
physics has no meaningful connection with employment or labor. Some logi-
cians avoid using the word ‘proposition’ as a technical term but nevertheless
say many things that I would say using that word. Some simply substitute
a diﬀerent word, and not always the same diﬀerent word. Tarski sometimes
used ‘thought’ (1956/1983, 160), sometimes ‘judgment’ (1969/1993, 106),
and sometimes ‘fact’ (ibid. 104). Sometimes Tarski simply avoided the con-
cept “proposition” by studiously using the word ‘sentence’ in his favored
sense of “propositional sentence”(1986, 143–154, 1996, 119–130).
According to Tarski: the sentence ‘zero is even’ is true if and only if zero
is even. But that would be incoherent in the terminology of this paper.
According to this paper: the proposition “zero is even” is true if and only
if zero is even. Likewise, the sentence ‘zero is even’ is used to express a true
proposition if and only if it is used in a sense in which zero is even.
In the purely intensional sense of ‘proposition’, it is incoherent to speak of
the meaning or of the wording of a proposition—but in the hybrid sense both
make perfect sense. Likewise for ‘sentence’, in the pure syntactic sense, it is
incoherent to speak of the meaning or of the wording of a sentence—whereas
in the hybrid sense both make perfect sense. Accordingly, in the hybrid
senses it is incoherent to speak of ambiguous propositions or of ambiguous
As indicated above, in the purely syntactical sense advocated in this work,
a sentence is ambiguous if there are two or more meanings that it normally
used to express. And there is no such thing as a pair of sentences composed
of exactly the same characters in exactly the same order: a sentence is a
series of characters. However, in the hybrid sense, often but not always used
by Tarski, a sentence is ambiguous if there is another sentence composed of
exactly the same characters in exactly the same order but having a diﬀerent
meaning. And there is no such thing as one sentence having two meanings:
a sentence is a meaningful unity. A hybrid sentence is a compound unity
uniting two parts: it is a pure sentence plus one of its meanings. Many
modern logicians seem to use the word ‘sentence’ usually in the pure sense
as is suggested in the common locution that a sentence is only true or false
under an interpretation. Many philosophers follow Tarski in usually using
the word in the composite sense as suggested in the common locution that
not every true sentence is known to be true. But, there are other important
senses to be recognized.32
32Frege (1918, 292–4, passim) uses it in a sense very close to “spoken statement”. He
90 John Corcoran
The notions of sentence and proposition are both useful for discussing
various phenomena including translation and ellipsis. In order for a sentence
used to express a certain proposition to be elliptical it is suﬃcient for the
sentence to omit a word corresponding to a constituent of the proposition.
The following is elliptical under two of its normal interpretations: Russell
read Frege more than Peano. Under one interpretation, it expresses the
proposition that Russell read Frege more than Peano did [sc. read Frege].
Under another interpretation, it expresses the proposition that Russell read
Frege more than he did [sc. Russell read] Peano.
7 Tokens, Not Occurrences, Embody Types.
Occurrences, Not Tokens, Are in Types.
A string-type has instances which are string-
tokens or string-inscriptions composed of in-
stances (not occurrences) of characters; the
string-tokens are ultimately composed of
character-tokens.—John Corcoran et al. 1974,
Let us try a little thought experiment. Imagine that we have one copy of
a Greek edition of Euclid’s Elements on the left end of the table open to
Proposition 47 of Book I, now known as the Pythagorean Theorem. In a
line spreading to the right, imagine one copy each of ten perfect translations
into ten diﬀerent languages, say Arabic, Chinese, English, French, German,
Portuguese, Spanish and three other languages that you may choose. Of
course we imagine them all open to Proposition 47 of Book I. The English
translation by Thomas Heath has the following: In right-angled triangles
the square on the side subtending the right angle is equal to the squares on
the sides containing the right angle.
We have one proposition which is expressed by the Greek sentence writ-
ten by Euclid. And we have eleven sentences each in a diﬀerent language,
all of which express one and the same proposition, namely the Pythagorean
Theorem. The Pythagorean Theorem is a proposition in the preferred in-
tensional sense used above. It is made up of concepts not words, not letters,
not characters. This one proposition, called ‘the Pythagorean Theorem’, is
that in virtue of which the ten translations of Euclid’s Greek sentence are
all perfect translations not only of Euclid’s Greek sentence but also of each
other. All eleven sentences express one and the same thing—and that thing
is the proposition. In the hybrid sense, there are eleven propositions. A
says that an indicative sentence is a series of sounds (ibid. 292) that contains not only a
proposition but also the assertion of the proposition (ibid. 294).
Sentence, Proposition, Judgment, Statement, and Fact 91
person who does not recognize propositions in the intensional sense should
not say ‘the Pythagorean Theorem’ but rather ‘aPythagorean Theorem’ or
‘one of the Pythagorean Theorems (plural)’even when it is clear from the
context that English is the language being used. After all there are many
diﬀerent ways of expressing a given proposition in a given language.
If persons who knew none of the eleven languages were to carefully exam-
ine all eleven sentences expressing the Pythagorean Theorem, they would
ﬁnd nothing to indicate a common meaning. Not even one character is
found in all eleven. However, were someone to look at two copies of one and
the same translation, say the English, turned to the same page, it would
be easy to ﬁnd an exact duplicate of the sentence token located in my copy
of the Heath translation embodying a sentence expressing the Pythagorean
In one sense of the word ‘sentence’ we have two English sentences, one
in each of the two copies of the book—two concrete, visible things made
of scattered bits of dried ink on paper. In another sense there is only one
sentence, but it has been printed twice—one abstract ideal thing capable of
being printed but not itself visible. The one thing printed twice (and capable
of being printed indeﬁnitely often) is what Peirce called a type, speciﬁcally
asentence-type. The two printings he called tokens, speciﬁcally sentence-
tokens. The two sentence-tokens, in Peirce’s terminology, are embodiments
of the sentence-type. The two sentence-tokens came into existence at a
certain deﬁnite time—maybe we could ﬁnd out from the publisher the exact
day. Sometimes the year of the printing is in the front of the book. But,
when did the sentence-type, the abstract thing that there is only one of,
come into existence?
Again, the two sentence-tokens can be destroyed by a ﬁre in one library,
or maybe by two ﬁres in two libraries. But how could a sentence-type be
destroyed? If someone destroyed all of the books, would that destroy the
sentence-type? These questions are raised only to clarify the diﬀerences
between types and tokens. Before exploring the type-token ambiguity33
further, let us read Peirce’s exact words. The following is from his 1906
Monist article (1906, 504–5) quoted in Ogden-Richards 1923, 280–281.
33Many nouns and noun phrases are subject to type-token ambiguity in that they can
be used to refer to types or to tokens. See the previous paragraph. The type-token
distinction is used to explain type-token ambiguity. Incidentally, a language that makes
the type-token distinction need not have expressions that have the type-token ambiguity,
and conversely. A suﬃciently rich logically perfect metalanguage could have symbols for
types and ways of mentioning tokens of those types but, of course, have no ambiguous
expressions. A suﬃciently primitive language could suﬀer from type-token ambiguity
while lacking resources to discuss the distinction.
92 John Corcoran
A common mode of estimating the amount of matter in a ... printed book is to
count the number of words. There will ordinarily be about twenty ‘thes’ on a page,
and, of course, they count as twenty words. In another sense of the word ‘word’,
however, there is but one word ‘the’ in the English language; and it is impossible
that this word should lie visibly on a page, or be heard in any voice ... . Such a ...
Form, I propose to term a Type. A Single ... Object ... such as this or that word
on a single line of a single page of a single copy of a book, I will venture to call a
Token. ... In order that a Type may be used, it has to be embodied in a Token
which shall be a sign of the Type, and thereby of the object the Type signiﬁes.
In my copy of Euclid’s Elements, the concrete visible token of the sentence-
type expressing the Pythagorean Theorem contains six concrete tokens of
the word-type ‘the’, tee-aitch-ee. The sentence-token is spread out over
three lines of print with two thes on the ﬁrst line, three on the second, and
one on the third. This means that the abstract sentence-type contains six
thes, to use Peirce’s clever word. Does this mean that a sentence-type is
made up of sentence-tokens? Does this mean that something abstract, a
sentence-type, is made up of concrete embodiments, sentence-tokens? We
need another distinction here to deal with those questions and to make the
point that, although the type ‘the’ has only one occurrence of the type ‘e’,
the type ‘e’ occurs twice in the type ‘these’ and the type ‘e’ is instantiated,
betokened, or embodied (to use Peirce’s term) twice in every token of the
type ‘these’, tee-aitch-ee-ess-ee. One and the same word-type ‘the’ occurs
six times in the one sentence type expressing the Pythagorean Theorem.
Every sentence-type is made up of character-types. Typically a sentence-
type has multiple occurrences of at least one of it character types. Every
token of a sentence type has the same number of tokens of a given word-type
as the sentence-type itself has occurrences of the given word-type.
Some authors explicitly make the three-part type-token-occurrence dis-
tinction without introducing special terminology for the occurrence relation.
Lyons (1977, Vol. 1, 13–18) has a section called Type and token which dis-
cusses the three-way distinction while using the same expressions for token
and occurrence. However, the present terminology is familiar to logicians,
as pointed out in my “Meanings of word: type-occurrence-token”, Bulletin
of Symbolic Logic 11(2005) 117.
At some point in the history of logic the type-token dichotomy gave
way to the type-token-occurrence trichotomy. Given Peirce’s penchant for
trichotomies and his logical creativity, one is led to speculate, even hope,
that it was Peirce who made this discovery.34
Another diﬀerence between sentence-tokens and sentence-types is that
every sentence-token has a length in inches or centimeters, whereas the
sentence-type, being abstract, has no length—not zero length, no length—
34Substantially the same points have been made recently (Corcoran 2005 ).
Sentence, Proposition, Judgment, Statement, and Fact 93
in inches or centimeters. Of course every expression-type has a “length” in
character-occurrences. Even though the Nile is longer than the Niagara, in
inches, ‘the Nile’ is “shorter” than ‘the Niagara’, in character-occurrences.
Moreover the sentence-token may be squeezed onto one line of a book or
spread out over several lines, but these formattings do not apply to sentence-
types. Probably the most important diﬀerence is that, as Peirce implies in
the above passage, sentence-tokens are visible whereas sentence-types are
invisible—a point rarely made. Church (1956p, 8) makes a closely related
point without explicitly noting that they are invisible. From an ontological
and epistemological point of view the diﬀerence between scientiﬁc treatment
of string tokens and scientiﬁc treatment of string-types is dramatic. String
tokens are studied in physics; string types are studied in mathematics—in
the ﬁeld known as string theory (Corcoran et al. 1974, “String Theory”, J.
Sym. Logic. 39: 625–37).
Let us consider an application. Consider, e.g., the word ‘letter’. In one
sense there are exactly twenty-six letters (letter-types or ideal letters) in
the English alphabet and there are exactly four letters in the word-type
‘letter’. In another sense, there are exactly six letters (letter-repetitions or
letter-occurrences) in the word-type ‘letter’. In yet another sense, every
new inscription (act of writing or printing) of ‘letter’ brings into existence
six new letters (letter-tokens or ink-letters) and one new word that had
not previously existed. The number of letter-occurrences (occurrences of a
letter-type) in a given word-type is the same as the number of letter-tokens
(tokens of a letter-type) in a single token of the given word. Many logicians
fail to distinguish “token” from “occurrence” and a few actually confuse the
two concepts. It is almost a rule that any article or book that explicitly
mentions two of the three concepts without mentioning the third confuses
the third with one of the other two.
There is a kind of cold war of words in the literature between two camps of
logician philosophers, neither of which “oﬃcially” recognizes the type-token
ambiguity of the word ‘sentence’. Tarski may be taken as representing one
camp, Quine the other. Whenever it is relevant for Tarski to clarify his
use of the word ‘sentence’, he makes it a point to say that sentences are
“inscriptions”. He never mentions Peirce’s work in this area. He almost
never uses the word ‘type’ in Peirce’s sense and he never uses the word
‘token’. In the single place (1930/1983, 31) where he mentions “the type
of a sentence”, he says that it is “the set of all sentences [inscriptions]” in
the same shape as the sentence [token]35. This is not Peirce’s view at all.
35Although Tarski repeatedly says that shape is the determining factor, it is clear that
he has not explored the details very far. Don’t the left and right parentheses have the
same shape? How about the plus and times signs, the less-than and greater than signs,
94 John Corcoran
Tarski seems to imply repeatedly and ﬂatly that it is “not strictly correct”,
an “error” or even a “widespread error”, to use to the word ‘sentence’ in the
sense of “sentence-type” (1969/1993, 114, 1930/1983, 30, 31, 1933/1983,
Quine, however, in his last comprehensive statement of his philosophy
of logic explicitly mentioned Peirce’s work; and he used both ‘type’ and
‘token’ in Peirce’s senses. But he never mentioned the type-token ambiguity
of the word ‘sentence’: for Quine a sentence is a sentence-type-exactly the
opposite of Tarski’s usage. Let us look at Quine’s own words36 (1970/1986,
13–14): “In Peirce’s terminology, utterances and inscriptions are tokens of
the sentence or other linguistic expression concerned; and this [sentence or
other] linguistic expression is the type of those utterances and inscriptions.
In Frege’s terminology, truth and falsity are the two truth-values. Succinctly,
then, an eternal sentence is a sentence whose tokens all have the same truth-
value [i.e. are all true or all false].”
In the recommended senses, both sentences and propositions are time-
less. In contrast, each inscription or embodiment of a sentence comes into
existence at a unique time and, in the fullness of time, will become illeg-
ible. Likewise, each thought of a proposition comes into existence as it is
being thought of and goes out of existence as soon as the person thinking
it turns to the next thought. Euclid and each of his translators thought of
the Pythagorean Theorem, one and the same proposition which is timeless.
But Euclid’s thought of the Pythagorean Theorem was long defunct before
his translators’ thoughts of it came into being.
As I wrote this paper I reread an English translation of the Pythagorean
Theorem and, despite its awkwardness and distracting ambiguity, I managed
to have a thought of the Theorem. My thought was private; your belief
that it ever existed is based on “mere hearsay”. Some people are inclined
to suppose that a particular thought of the Pythagorean Theorem is to the
abstract proposition as a concrete inscription of a sentence expressing it is
to the abstract sentence. There might be a sense in which this is true. But
even so, it would be a stretch to conclude that a thought of a proposition
is an embodiment of the proposition. A thought of a proposition seems to
be an action, a kind of performance, more akin to an act of inscribing than
to a static inscription.
the vee and the caret, the zee and the en? Besides, there are arbitrarily “long” sentence-
types but there is evidently a limit to the total number of character-tokens ever made in
the history of writing. Of course, every argument against any given attempt to reduce
types to tokens can be answered by an ever more elaborate “epicyclical construction”.
36The italicizations are Quine’s and the bracketed interpolations are mine.
Sentence, Proposition, Judgment, Statement, and Fact 95
8 Sentential Functions: Roles Sentences Play
A proposition grasped by a person for the ﬁrst
time can be put in a form of words which
will be understood by someone to whom the
proposition is completely new. This would
be impossible were we not able to distinguish
parts in the proposition corresponding to the
parts in the sentence, so that the structure of
the sentence serves as a model for the struc-
ture of the proposition.—Frege 1979, 1923.
The topic of sentential functions—the functions, roles, or uses of sentences—
is one of the most interesting subjects in logic. A sentence can be mentioned.
A sentence can be used to make a statement. A sentence can be used to make
a promise, to make a request, to give an order, to refuse an order, to make
an apology, to make a prediction, to ask a question, to insult, to inform, to
deceive, to retract a previous statement. A sentence can be used ironically
to make a statement that contradicts what would be “expected”. Plato tells
us that Cephalus, then quite old, “welcomed” the decline in his appetite.
Continuing, Plato adds that Socrates “admired” Cephalus for saying so
(Republic I, 329). A sentence can be used poetically or ﬁctionally where
the speaker is not responsible for the accuracy of what is expressed, but
only for its poetic or literary qualities. The “pedagogical license” enjoyed
by the teacher is warranted by the fact that the teacher’s role demands
combining the role of witness responsible for accuracy with the role of poet
or storyteller responsible for something else (Corcoran 1999 ).
In fact, of all the topics in logic suitable for discussion with an audience
of logicians and non-logicians, this topic, despite its attractiveness, even its
fascination, is hardly ever discussed. Consequently, the literature of logic
and of non-logic abounds in confusions and fallacies that would be corrected
by a small amount of concentrated attention to this topic.
The three-word sentence ‘Ten is ﬁfty’, as peculiar as it may be, is in
many ways typical. Of course, ten is not ﬁfty. The number ten is only one-
ﬁfth of ﬁfty. But, ten degrees Celsius equals ﬁfty degrees Fahrenheit. The
sentence ‘Ten is ﬁfty’ can be used to express a true proposition concerning
temperature conversions and to express a false proposition about numbers.
It is exactly the same with corresponding sentences in other languages. We
could just as easily consider the Portuguese sentence ‘Dez ´e cinq¨uenta’, the
Spanish ‘Diez es cincuenta’, the German ‘Zehn ist f¨unfzig’, or the French
‘Dix est cinquante’.
The English sentence mentioned above can also be used to express many
other propositions, some true and some false; and it can be used to express
96 John Corcoran
incoherent messages. Thus when someone has used the sentence to make
an utterance in writing, we cannot, without further investigation, conclude
that a true proposition or a false proposition was stated.
Examples of other propositions that can be expressed with this sentence
can be discovered by imagining diﬀerent contexts in which it might be writ-
ten. The sentence ‘ten is ﬁfty’ might be used to answer a question about
the price of handkerchiefs; it might mean that the price of ten handkerchiefs
is ﬁfty cents, or ﬁfty dollars, or ﬁfty pounds, or ﬁfty euros. In such cases
the grammarian would explain that the three-word sentence is elliptical for
an eight-word sentence by writing, e.g.: ‘[the price of] ten [handkerchiefs] is
ﬁfty [cents]’—using the brackets to indicate “restored ellipsis”. It is only in
a logically perfect language that grammatical structure of sentences models
logical structure of the respective propositions expressed.
In order to determine the message, if any, that a given inscription of a
sentence was intended to convey, it is often necessary to consider the context,
including the discourse preceding the inscription in time and sometimes
the past history of communication among the participants. Some interpret
Frege as having puzzled his readers by saying that a word did not have
meaning in isolation but only as part of a sentence. He asserted that the
word gets its meaning from the sentence, and he denied that the sentence
gets its meaning from the meanings of the words in it.37 For years this
was regarded as an extreme viewpoint.38 Now we see to the contrary that
Frege did not go far enough. Using the same exaggerated and elliptical style
that Frege used, we can say that no sentence has meaning in isolation but
only in the context of a communication—among several people, between
two people, or by a single person, perhaps making a memorandum. We can
aﬃrm that determining what was meant by a sentence in a given inscription
requires grasping the context and we can deny that it is suﬃcient to consider
37In his Grundlagen der Arithmetik Frege wrote: “... we must always consider a
complete sentence. Only in [the context of] the latter do words really have a meaning. ...
It is enough if the sentence as a whole has a sense; by means of this its parts also receive
their content.”(Frege 1884/1964,§60, tr. M. Mahoney). Later writers have followed
him on this issue, not always giving due acknowledgement. Over a quarter century later
Wittgenstein wrote: “Only sentences make sense; only in the context of a sentence does
a name signify anything.”(Wittgenstein 1921/1998,§3.3 tr. D. Kolak).
38In view of the strangeness and implausibility of this view, not to mention the fact
that neither Frege nor Wittgenstein gives a scintilla of evidence for it, we should not
be surprised to ﬁnd it being ignored by practicing linguists. For example, John Lyons
does not mention it in his comprehensive, two-volume treatise Semantics. In fact, like
many other semanticists, he takes a kind of diametrical opposite view to be obvious. He
writes: “[It] ... is obvious enough ... that the meaning of a ... sentence is a product of
the meanings of the words of which it is composed.” (Lyons 1977, Vol. I, 4). Also see
the above quote from Frege 1923.
Sentence, Proposition, Judgment, Statement, and Fact 97
only the sentence per se.39
Frege tells us “never to ask for the meaning of a word in isolation”
(1884/1959, Intro., X). But he thinks that it is coherent to inquire as
to the meaning of a sentence-type. If this paper has been successful, we
have learned that we should never ask for the meaning or meanings of any
expression-type, we should never ask “what does it mean to say such-and-
such?” Rather, we should ask “what did so-and-so mean when writing
such-and-such in thus-and-so text?”
9 Propositions Imply; Statements Implicate:
Propositions Have Implications; Statements Have
The fact that a person deﬁnes an expression in
a certain sense, in and of itself, is no evidence
that the person uses the expression in that
sense.—Frango Nabrasa, 2001.
The proposition “Socrates taught Plato, who taught Aristotle” implies the
proposition “Plato, who was taught by Socrates, taught Aristotle”, which
in turn implies the proposition “Plato taught Aristotle”. But it does not
imply the proposition “Socrates taught Aristotle”. The relation in question
is not transitive–even if it were, there would still be no implication.40 Nor
does it imply “Corcoran believes that Plato taught Aristotle”.
Let us compare the above proposition with the statement I will now make.
Socrates taught Plato, who taught Aristotle.
39Similar points have been made before by Quine, who says that the sentence per se is
not what carries the truth or falsity but rather each token, more accurately each “event
of utterance” (1970/1986, 13). Quine does not reveal it, but his view was anticipated by
his teacher C. I. Lewis over thirty years earlier. Lewis wrote: “it is ... the word as read,
the visual impression received, which arouses in the mind of the reader a corresponding
meaning, and it is only when a mark ... gives rise to meaning that it is operating as
a symbol” (1932, 311). Michael Scanlan observed that Lewis never follows up on or
develops this interesting idea (personal communication). Other philosophers have made
points similar to the one just attributed to Frege, Lewis, and Quine. But in many cases
their works are so labored or so cryptic that it is hard to be conﬁdent that this point
has even been grasped, let alone articulated. It is important to note that this thesis
does not exclude the proposition that diﬀerent tokens of the same sentence have shared
40I am using ‘implies’ in the information-containment sense as elaborated in Corcoran
1998 ; it is the second of twelve senses listed in Corcoran 1973. In many cases, the
information in a proposition involving a relational concept that determines a transitive
relation does not contain the information that the relation is transitive. “One precedes
two” does not imply “Every number preceding a given number precedes every number the
given number precedes”, which is logically equivalent to transitivity of number-theoretic
precedence. Thinking otherwise is fallacious, perhaps the fallacy of premise-smuggling.
98 John Corcoran
The last statement I made implicates “Corcoran believes that Socrates
taught Plato, who taught Aristotle”. Moreover, it implicates “Corcoran
believes that Plato taught Aristotle”. Every statement implicates every
proposition implied by a proposition it implicates. It also implicates the
proposition that I, Corcoran, understand the sentence / Socrates taught
Plato, who taught Aristotle/. It also implicates that I believe that you
understand the sentence.
As a rule, propositions about other people do not imply informative
propositions about me. As a rule, statements implicate that the speaker
believes the proposition stated, that the speaker understands the composite
sentence uttered, and that the speaker believes that the audience under-
stands it also.
The implications of a given proposition are the propositions whose infor-
mation content is included in that of the given proposition.41 Clearly, the
information that Corcoran believes something is not contained in the infor-
mation that Socrates taught Plato, who taught Aristotle. The implicatures
of a given statement are the propositions it implicates; they include many of
the propositions to which the speaker is morally committed by making the
statement. Every proposition implies its implications and every statement
implicates its implicatures. Propositions do not have implicatures.
According to the above conventions “imply” and “implicate” require non-
human subjects: the former a proposition; the latter a statement. However,
it is natural to used both elliptically so that human subjects are available.
A person can be said to imply the propositions implied by a proposition
the person stated; a person can be said to implicate a propositions if it is
implicated by a statement the person made.42
It took years to ﬁgure out satisfactory ways to say what a proposition’s
implications are. We should not forget that both ‘implication’ and ‘impli-
cature’ are ambiguous and that some of their meanings are vague. Maybe
someone knows a satisfactory explanation of what a statement’s implica-
tures are (Borchert 1996, 225); I do not. Roughly, perhaps, we can say that
a statement also implicates what the audience can take the statement as
evidence for—assuming that the speaker is sincere, sane, responsible, etc.
41This traditional view is elaborated in Corcoran 1998.
42For these important conventions, which correspond to observed usage, I am indebted
to the late Kenneth Barber (Barber and Corcoran 2009 ). However, this terminology
is not universally accepted. What I call a person’s implicatures (singular implicature)
others call a person’s implicata (singular implicatum)—which is far too stuﬀy for my
taste. Moreover, some restrict implicatures to persons only without allowing coherent
mention of the implicatures of a statement, contrary to the usage found here and elsewhere
(e. g., Borchert 1996, 225). I thank my colleague David Braun for many helpful points
including the need for this footnote.
Sentence, Proposition, Judgment, Statement, and Fact 99
A statement of a proposition is made in a language, by a speaker, to
an audience, at a time and place. My statements today were all made
in English using English sentences that can be used tomorrow to make
other statements with other implicatures. A statement (aﬃrms / denies)
the propositions (implied / contradicted) by the proposition it states. If
we deﬁne the implications of a statement to be the implications of the
proposition it states, we can say that a statement aﬃrms exactly what it
implies and it denies exactly the propositions whose negations it implies. I
do not recommend this deﬁnition which would burden the word ‘implication’
with yet another meaning having another range of applicability.
Normally, statements do not implicate all they aﬃrm. My statement
“Socrates taught Plato, who taught Aristotle” aﬃrms but does not implicate
the proposition “Plato taught Aristotle”. Moreover, even when we can
be conﬁdent that the speaker is being sincere, sane, responsible, etc., the
statement need not be evidence for its aﬃrmations. A statement of the
axioms and deﬁnitions of arithmetic aﬃrms the Goldbach Hypothesis or it
aﬃrms the Goldbach Hypothesis’s negation.
A statement is self-denying (SD) if it denies one of its own implica-
tures (or equivalently, implicates a proposition it denies). Examples of SD
propositions are: a statement to the eﬀect that the speaker is not making
a statement or is (using / not using) a word that in fact (is not / is) being
used. A statement is self-aﬃrming (SA) if it aﬃrms one of its own implica-
tures (or, equivalently, it implicates a proposition it aﬃrms). Examples of
SA statements are: a statement to the eﬀect that the speaker is making a
statement or is (addressing / not addressing) an audience that in fact (is / is
not) being addressed. A statement is (correct / incorrect) if (every / some)
proposition it aﬃrms or implicates is (true / false). Every SD statement
is incorrect. Every statement of a contradiction is SD. Nevertheless, it is
not the case that every SD statement aﬃrms a contradiction: the propo-
sition stated in a statement to eﬀect that the speaker disbelieves a certain
one of the statement’s own aﬃrmations need not contradict itself. As has
been noted in discussions of “Moore’s Paradox”, such statements can be
made using sentences such as ‘Abe died but I disbelieve it’. Not every SA
statement is correct. In fact, every SA lie is incorrect. A half-truth is a
statement implicating a falsehood but aﬃrming only truths.
100 John Corcoran
The proposition must be distinguished from
the sentence, the combination of words or
signs though which it is expressed; from the
fact, the actual complex situation whose ex-
istence renders it true or false; and from the
judgment, which aﬃrms or denies the propo-
sition the proposition.—Eaton 1931, 12. 43
The ﬁve ambiguous words—sentence, proposition, judgment, statement,
and fact—have received distinct recommended meanings that are needed
for any suﬃciently full discussion of logical phenomena such as self-denying
and self-aﬃrming statements. This paper has sketched a conceptual frame-
work necessary for discussing issues that have concerned logicians over the
years. I hope that most objective and informed logicians will agree that this
framework is indispensable for comprehensive and comparative treatment
of the history of logic. A historian inadvertently or willfully ignorant of
any given one of these ﬁve concepts will be unable to give a full and fair
treatment of a historical ﬁgure that uses it. I believe that appreciation of
the place of this framework in logical thought will reveal how crude, evasive,
and incomplete many important logic papers are. I hope that this awareness
does not lessen the credit attributed to the authors of those works. If this
paper succeeds, every future logic book should be aﬀected.
This paper clariﬁes, qualiﬁes, and—in only a few cases I am glad to say—
retracts views I previously expressed. I started writing it in 2003 partly as
an independent philosophical preface to my 1973 Dialogos article “Meanings
of Implication”. As I became more and more clear about the content of this
paper, my dissatisfaction with 1973 paper grew. There is still much to like
about that paper despite the deﬁciencies in its conceptual framework made
evident by this paper. The ﬁrst published fruits of my rethinking are in
“Sentential Functions: the Functions of Sentences” in Corcoran, Griﬃn, et
For bringing errors and omissions to my attention, for useful suggestions,
and for other help, I gladly acknowledge the following scholars : G. Fulugo-
nio (Argentina), O. Chateaubriand, F. Nabrasa, J. da Silva, and L. Weber
43Eaton seems to be combining the statement with the judgment. This is in keeping
with other logicians who call Frege’s “judgment-stroke” the “assertion sign” (Eaton 1931,
369–374). But, the situation is actually much worse, this sign, which resembles a counter-
clockwise rotated tee, is often taken to indicate a logically cogent judgment—a judgment
based entirely on “laws of logic”, a logical inference.
Sentence, Proposition, Judgment, Statement, and Fact 101
(Brazil); D. Hitchcock and J. Van Evra (Canada); R. Torretti (Chile); K.
Miettinen (Finland and USA); B. Smith (Germany and USA); H. Masoud
(Iran), C. Penco (Italy), A. Visser (Netherlands); R. Santos (Portugal); J.
Sag¨uillo (Spain); J. Gasser (Switzerland); I. Grattan-Guinness (UK); W.
Abler, R. Barnes, D. Braun, D. Brewer, B. Decker, J. Foran, J. Gould, C.
Guignon, I. Hamid, F. Hansen, P. Hare, L. Jacuzzo, C. Jongsma, J. Kearns,
J. Miller, M. Moore, M. Mulhern, M. Murphey, S. Nambiar, S. Newberry,
P. Penner, A. Preus, M. Scanlan, K. Shockley, T. Tracy, J. Yu, and J. Zec-
cardi (USA); and others. Earlier versions were presented at the Buﬀalo
Logic Colloquium, the University of Buﬀalo Philosophy Colloquium, Can-
isius College, the University of Santiago de Compostela, the University of
South Florida, and the 14th Latin-American Symposium on Mathematical
Logic in Paraty, Brazil.
My greatest debt is to Professor Jos´e Miguel Sag¨uillo, Catedr´atico de
L´ogica of the University of Santiago de Compostela, Spain. He discussed
these ideas with me over many years and he inspired me to write this arti-
cle. He has done more than any other person toward clarifying for me the
synergistic relations between the teaching of logic and the epistemology of
 Arnauld, A. and P. Nicole. 1662. Port Royal Logic. Tr. and ed. J. Buroker. 1996.
Cambridge: Cambridge UP.
 Aristotle. Aristotle’s Categories and De Interpretatione. Tr. J. Ackrill. 1963. Oxford:
 Aristotle. Nicomachean Ethics. Tr. T. Irwin. 1999. Indianapolis: Hackett.
 Audi, R., Ed. 1995/1999. The Cambridge Dictionary of Philosophy. Second edition.
Cambridge: Cambridge University Press.
 Austin, J.L. 1950. Truth. Proceedings of the Aristotelian Society. Supp. Vol. 24.
Reprinted Austin 1961.
 Austin, J.L. 1961. Philosophical Papers. Ed. J. Urmson and G. Warnock. Oxford:
 Barber, K. and J. Corcoran. 2009. Agent and Premise Implication. Bulletin of Sym-
bolic Logic 15: 235
 Beaney, M., ed. 1997. The Frege Reader. Oxford: Blackwell.
 Bochenski, I., A. Church, N. Goodman. 1956. The Problem of Universals. Notre
Dame, IN: Notre Dame UP.
 Boolos, G., J. Burgess, and R. Jeﬀrey. 2002. Computability and logic. Cambridge:
 Borchert, D. , ed. 1996. Encyclopedia of Philosophy. New York: Simon & Schuster.
 Carnap, R. 1934/1937. The Logical Syntax of Language. Tr. A. Smeaton. London:
Routledge & Kegan Paul.
 Chateaubriand, O. 2001. Logical Forms. Part I. Campinas: CLE Unicamp.
 Chateaubriand, O. 2005. Logical Forms. Part II. Campinas: CLE Unicamp.
 Church, A. 1956i. Introduction to Mathematical Logic. Princeton: Princeton UP.
 Church, A. 1956p. Propositions and Sentences. Bochenski et al. 1956. Notre Dame,
IN: Notre Dame UP.
102 John Corcoran
 Cohen, M., and E. Nagel. 1934/1962/1993. Introduction to Logic. Indianapolis: Hack-
 Corcoran, J. 1973. Meanings of Implication. Dialogos 9:59-76. Spanish translation by
Jos´e M. Sag¨uillo, “Signiﬁcados de la Implicaci´on”, Agora 5 (1985) 279–294, updated
reprint in Hughes 1993.
 Corcoran, J., ed. 1974. Ancient Logic and its Modern Interpretations. Reidel: Dor-
 Corcoran, J. 1998. Information-theoretic logic, in Truth in Perspective edited by
C. Mart´ınez, U. Rivas, L. Villegas-Forero. Aldershot, England: Ashgate Publishing
 Corcoran, J. 1999. Critical thinking and pedagogical license. Manuscrito. XXII:
 Corcoran, J. 2004. Sentential Functions: the Functions of Sentences. In Corcoran,
Griﬃn, et al. 2004.
 Corcoran, J. 2005. Meanings of word: type-occurrence-token. Bulletin of Symbolic
Logic 11 : 117.
 Corcoran, J. 2006e. An Essay on Knowledge and Belief. The International Journal
of Decision Ethics. II.2, 125–144.
 Corcoran, J. 2006s. Schemata: the Concept of Schema in the History of Logic.
Bulletin of Symbolic Logic 12 : 219–240.
 Corcoran, J., W. Frank, and M. Maloney. 1974. String Theory. Journal of Symbolic
Logic 39 : 625–637.
 Corcoran, J., J. Griﬃn, et al. 2004. Discursos da Investidura de D. John Corcoran y
D. James Griﬃn como Doutores Honoris Causa. Santiago: Universidade de Santiago
 Eaton, R. 1931. General Logic. New York: Charles Scribner’s Sons.
 Frege, G. 1884/1959. The Foundations of Arithmetic. Breslau: Koebner. Tr. J. L.
Austin. Oxford: Basil Blackwell.
 Frege, G. 1884/1964. The Foundations of Arithmetic. Breslau: Koebner. Excerpted
tr. M. Mahoney in Eds. P. Benacerraf and H. Putnam. Philosophy of Mathematics.
Cambridge: Cambridge UP.
 Frege, G. 1918/1956. The Thought: a Logical Inquiry. Trans. A. and M. Quinton.
Mind 65 (1956) 289–311.
 Frege, G. 1919/1952. Negation. Trans. P. Geach. Reprinted in Geach-Black
1952/1966 : 117–135.
 Frege, G. 1979. Posthumous Writings. Ed. H. Hermes et al. Trs. P. Long and R.
White. Chicago: University of Chicago Press.
 Frege, G. 1980. Philosophical and Mathematical Correspondence. Chicago: Univer-
sity of Chicago Press.
 Frege, G. 1997. The Frege Reader. Ed. M. Beaney. Oxford: Blackwell.
 Geach, P. and M. Black, trs. 1952/1966. Translations from the Philosophical Writings
of Gottlob Frege. Oxford: Basil Blackwell.
 Goldfarb, W. 2003. Deductive Logic. Indianapolis: Hackett.
 Grice, P. 1989. Studies in the Way of Words. Cambridge, MA: Harvard UP.
 Hilbert, D. and W. Ackermann. 1928/1938/1950. Principles of Mathematical Logic.
Providence, RI: American Mathematical Society.
 Hughes, R., ed. 1993. Philosophical Companion to First-order Logic. Indianapolis:
 Kretzmann, N. 1974. Aristotle on Spoken Sound Signiﬁcant by Convention. In Cor-
 Lewis, C. I. and C. H. Langford. 1932/1959. Symbolic Logic. New York: Century.
Reprinted New York: Dover.
 Luce, R. 1950. Notes. In Hilbert and Ackermann 1928/1938/1950.
 Lyons, J. 1977. Semantics. 2 Vols. Cambridge: Cambridge UP.
 Moore, G. E. 1899. The Nature of Judgment. Mind. n. s. 8: 176–193.
Sentence, Proposition, Judgment, Statement, and Fact 103
 Ogden, C. K. and Richards, I.A. 1923. The Meaning of Meaning. Reprint of eighth
edition. Harcourt: New York.
 Peirce, C. 1906. Prolegomena to an Apology for Pragmaticism. Monist 16 : 492–546.
 Peirce, C.S. 1933. Collected Papers of Charles Sanders Peirce. Eds. C. Hartshorne
and P. Weiss. Cambridge: Harvard UP.
 Plato. Complete Works. Ed. J.Cooper. Indianapolis: Hackett.
 Quine, W. 1945. On the Logic of Quantiﬁcation. Journal of Symbolic Logic 10 : 1–12.
 Quine, W. 1951. Mathematical Logic. Cambridge MA: Harvard UP.
 Quine, W. 1970/1986. Philosophy of logic. Cambridge MA: Harvard UP.
 Russell, B. 1903. The Principles of Mathematics. Cambridge: Cambridge UP.
 Russell, B. 1905. On denoting. Mind, n. s. 14: 479-493. Reprinted in Copi and Gould
 Shapiro, S., ed. 1996. The Limits of Logic. Aldershot, UK: Dartmouth.
 Tarski, A. 1969/1993. Truth and proof. Scientiﬁc American. June 1969. Reprinted
 Tarski, A. 1956/1983. Logic, Semantics, Metamathematics. Trans. J. H. Woodger.
 Tarski, A. 1986. What are Logical Notions? History and Philosophy of Logic 7:143–
154. Reprinted in Shapiro 1996.
 Whately, R. 1826. Elements of Logic. Ed. P. Dessi. 1988. Bologna: Editrice CLLEB.
 Wittgenstein, L. 1921/1998. Tractatus Logico-Philosophicus. Tr. D. Kolak. Mountain
View, CA: Mayﬁeld.
Department of Philosophy
University of Buﬀalo
Buﬀalo, NY 14260-4150 USA