User:Philogo/TruthbearerSandbox

Sentences in languages of classical logic
In classical logic a sentence in a language is true or false under (and only under) an interpretation and is therefore a truthbearer. For example a language in the first-order predicate calculus might include one of more predicate letters and one or more individual constants and one or more variables. The interpretation of such a language would define a domain (universe of discourse); assign an element of the domain to each individual constant; assign the donation in the domain of some property to each unary (one-place predicate letter). For example if a language L consisted in the individual constant a, two unary predicate letters F and G and the variable x, then an interpretation I of L might define the Domain D as animals, assign Socrates to a,  the denotation of the property being a man to F and the denotation of the property being mortal to G. Under the interpretation I of L then Fa would be true just in case Socrates is a man, and the sentence (x)(Fx -> Gx) would be true just in case all men (in the domain) are mortal. In some text an interpretation is said to give meaning to the symbols of the language. Since Fa has the value true under some (but not all interpretations) it is not the sentence-type Fa which is said to be true but only some sentence-tokens of Fa under particular interpretations. A token of Fa without an interpretation is neither true nor false. Some sentences of a Language like L are said to be true under all interpretations of the sentence, e.g. (x)(Fx v ~Fx), such sentences being termed logical truths, but again such sentences are neither true nor false in the absence of an interpretation.

Introduction
Some distinctions and terminology as used in this article, based on Wolfram 1989, Chapter 2 Section1) follows. It should be understood that the terminology described is not always used in the ways set out, and are it is introduced solely for the purposes of discussion in this article.    Use is made of the type-token and use-mention distinctions. In grammar a sentence can be a declaration, an explanation, a question, a command. In logic a declarative sentence is considered to be a sentence that can be used to communicate truth.    Some sentences which are grammatically declarative are not logically so.

A character is a typographic character (printed or written) etc.

A word token is a pattern of characters. A word-type is an identical pattern of characters. A meaningful-word-token is a meaningful word-token. Two word-tokens which mean the same are of the same word-meaning

A sentence-token is a pattern of word-tokens. A meaningful-sentence-token is a meaningful sentence-token or a meaningful  pattern of meaningful-word-tokens. Two sentence-tokens are of the same sentence-type if they are identical patterns of word-tokens characters A declarative-sentence-token is a sentence-token which that can be used to communicate truth or convey information. A meaningful-declarative-sentence-token is a meaningful declarative-sentence-token Two meaningful-declarative-sentence-tokens are of the same meaningful-declarative-sentence-type if if they are identical patterns of word-tokens. A nonsense-declarative-sentence-token is a declarative-sentence-token which is not a meaningful-declarative-sentence-token. A meaningful-declarative-sentence-token-use occurs when and only when a meaningful-declarative-sentence-token is used declaratively.

A referring-expression is expression that can be used to pick out or refer to particular entity. A referential success is a referring-expression’s success in identifying a particular entity. A referential failure is a referring-expression’s failure to identify a particular entity. A referentially-successful-meaningful-declarative-sentence-token-use is a meaningful-declarative-sentence-token-use containing no referring-expression that fails to identify a particular entity.

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Sentences
In grammar a sentence can be a declaration, an explanation, a question, a command. In logic a declarative sentence is considered to be a sentence that can be used to communicate truth. Some sentences which are grammatically declarative are not logically so.

A: This toucan can catch a can.

B: If you have a bucket, then you have a pail.

C: I promise to be good.

D: He is grnd.

E: Are you happy?

F: Cats blows the wind

G: This stone is thinking about Vienna

H: This circle is square

I: The author of Waverly is dead

J: The author of Ivanhoe is dead

K: I am less than six foot tall

L: I am over six foot tall

M: The conductor is a bachelor

N: The conductor is married

O: The conductor is an unmarried man.

P: I'm Spartacus.

Q: I'm Spartacus.

R: Spartacus sum.

S: He's Spartacus.

T: Spartacus did not eat all his spinach in London on Feb 11th 2009.

U: The King of France is bald V: The highest prime has no factors W: Pegasus did not exist

Glossary of Terms used in this article
Some distinctions and terminology as used in this article, based on Wolfram 1989, Chapter 2 Section1) follows. It should be understood that the terminology described is not always used in the ways set out, and are it is introduced solely for the purposes of discussion in this article.    Use is made of the type-token and use-mention distinctions.

Characters
By character we will mean a typographic character (printed or written), a unit of speech, a phoneme, a series of dots and dashes (as sounds, magnetic pulses, printed or written), a flag or stick held at a certain angle, a gesture, a sign as use in sign language, a pattern or raised indentations (as in brail) etc. in other words the sort of things that are commonly described as the elements of an alphabet.

Words
A: This toucan can catch a can.

B: If you have a bucket, then you have a pail.

C: I promise to be good.

D: He is grnd.

A word-token is a pattern of characters.
 * Word-tokens

The pattern of characters A (above) contains six word-tokens

The pattern of characters D (above) contains three word-tokens

A meaningful-word-token is a meaningful word-token. grnd in D is not meaningful.
 * Meaningful-word-tokens

A word-type is an identical pattern of characters,.
 * Word-types

The pattern of characters A (above) contains five word-types (the word-token can occurring twice)


 * Word-meanings

Two word-tokens which mean the same are of the same word-meaning. Only those word-tokens which are meaningful-word-tokens can have the same meaning as another word-token. The pattern of characters A (above) contains six word-meanings.

Although it contains only five word-types, the two occurrences of the word-token can have different meanings.

On the assumption that bucket and pail mean the same, the pattern of characters B (above) contains ten word-tokens, seven word-types, and six word-meanings.

Sentences
In grammar a sentence can be a declaration, an explanation, a question, a command. In logic a declarative sentence is considered to be a sentence that can be used to communicate truth. Some sentences which are grammatically declarative are not logically so.

A: This toucan can catch a can.

B: If you have a bucket, then you have a pail.

C: I promise to be good.

D: He is grnd.

E: Are you happy?

F: Cats blows the wind

G: This stone is thinking about Vienna

H: This circle is square

I: The author of Waverly is dead

J: The author of Ivanhoe is dead

K: I am less than six foot tall

L: I am over six foot tall

M: The conductor is a bachelor

N: The conductor is married

O: The conductor is an unmarried man.

P: I'm Spartacus.

Q: I'm Spartacus.

R: Spartacus sum.

S: He's Spartacus.

T: Spartacus did not eat all his spinach in London on Feb 11th 2009.

U: The King of France is bald V: The highest prime has no factors W: Pegasus did not exist


 * Meaningful Declarative-sentences

A sentence-token is a pattern of word-tokens.
 * Sentence-tokens

The pattern of characters D (above) is a sentence-token because grnd is a word-token (albeit not a meaningful word-token.)

A meaningful-sentence-token is a meaningful  sentence-token or a meaningful  pattern of meaningful-word-tokens.
 * Meaningful-sentence-tokens

The pattern of characters D (above) is not a sentence-token because grnd is not a meaningful word-token.

Two sentence-tokens are of the same sentence-type if they are identical patterns of word-tokens characters, e.g. the sentence-tokens P and Q above are of the same sentence-type.
 * Sentence-types

A declarative-sentence-token is a sentence-token which that can be used to communicate truth or convey information. The pattern of characters E (above) is not a declarative-sentence-token because it interrogative not declarative.
 * Declarative-sentence-tokens

A meaningful-declarative-sentence-token is a meaningful declarative-sentence-token.
 * Meaningful-declarative-sentence-tokens

The pattern of characters F (above) is not a meaningful-declarative-sentence-token  because it is grammatically ill-formed

The pattern of characters G (above) is not a meaningful-declarative-sentence-token  because thinking cannot be predicated of a stone

The pattern of characters H (above) is not a meaningful-declarative-sentence-token  because it is internally inconsistent

The pattern of characters D (above) is not a meaningful-declarative-sentence-token  because it contains a word-token (grnd) which is not a meaningful-word-token

Two meaningful-declarative-sentence-tokens are of the same meaningful-declarative-sentence-type  if if they are identical patterns of word-tokens characters, e.g. the sentence-tokens P and Q above are of the same meaningful-declarative-sentence-type. In other words a sentence-type is a meaningful-declarative-sentence-type if all tokens of which are meaningful-declarative-sentence-tokens
 * Meaningful-declarative-sentence-types

A nonsense-declarative-sentence-token is a declarative-sentence-token which is not a meaningful-declarative-sentence-token.
 * Nonsense-declarative-sentence-token

The patterns of characters F, G & H above are nonsense-declarative-sentence-tokens because they are declarative-sentence-tokens but not meaningful-declarative-sentence-tokens. The pattern of characters D (above) is not a nonsense-declarative-sentence-token because it is not a declarative-sentence-token because it contains a word-token (grnd) which is not a meaningful-word-token.

A meaningful-declarative-sentence-token-use occurs when and only when a meaningful-declarative-sentence-token is used declaratively, rather than, say, mentioned.
 * Meaningful-declarative-sentence-token-uses

The pattern of characters J (above) is  a meaningful-declarative-sentence-token but, in all probability, it has never be used declaratively and thus there have been no meaningful-declarative-sentence-token-uses of J.  A meaningful-declarative-sentence-token can be used zero to many times. Two meaningful-declarative-sentence-tokens-uses of the same meaningful-declarative-sentence-type are identical if and only if they are identical events in time and space with identical users.


 * referring-expression An expression that can be used to pick out or refer to particular entity, such as definite descriptions and proper names


 * referential success a referring-expression’s success in identifying a particular entity OR a meaningful-declarative-sentence-token-use’s containing one or more referring-expression all of which succeed in identifying a particular entity


 * referential failure a referring-expression’s failure to identify a particular entity is referentially successful OR a meaningful-declarative-sentence-token-use’s containing one or more referring-expression that fail to identify a particular entity


 * Referentially-successful-meaningful-declarative-sentence-token-use A meaningful-declarative-sentence-token-use containing no referring-expression that fails to identify a particular entity.   A use of a token of the meaningful-declarative-sentence-type ‘U: The King of France is bald’  is a referentially-successful-meaningful-declarative-sentence-token-use if (and only if) the embedded  referring-expression ‘The King of France’ is referentially successful.  No use of a token of the meaningful-declarative-sentence-type ‘V: The highest prime has no factors’ is a  referentially-successful-meaningful-declarative-sentence-token-use since the embedded  referring-expression ‘The highest prime’ is always a referential failure.

REFS
