Truth tableFrom Wikipedia, the free encyclopedia

Contents

1 Existential quantication 11.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 The empty set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 As adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Participle 52.1 Types of participle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Indo-European languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Germanic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Latin and Romance languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Hellenic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 Celtic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.5 Slavic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.6 Baltic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Semitic languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Arabic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Finno-Ugric languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.1 Finnish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Other languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5.1 Sireniki Eskimo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5.2 Esperanto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Truth table 14

i

ii CONTENTS

3.1 Unary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.1 Logical false . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 Logical identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.3 Logical negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.4 Logical true . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Binary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 Truth table for all binary logical operators . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.2 Logical conjunction (AND) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Logical disjunction (OR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.4 Logical implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.5 Logical equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.6 Exclusive disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.7 Logical NAND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.8 Logical NOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.1 Truth table for most commonly used logical operators . . . . . . . . . . . . . . . . . . . . 173.3.2 Condensed truth tables for binary operators . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.3 Truth tables in digital logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.4 Applications of truth tables in digital electronics . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Truth-bearer 204.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Sentences in natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Sentences in languages of classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.6 Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Truthmaker 285.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

CONTENTS iii

6 Universality (philosophy) 306.1 Universality in ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.2 Universality in logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.3 Universality in metaphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.4 Quotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Chapter 1

Existential quantication

Existential quantier redirects here. For the symbol conventionally used for this quantier, see Turned E."" redirects here. It is not to be confused with .

In predicate logic, an existential quantication is a type of quantier, a logical constant which is interpreted asthere exists, there is at least one, or for some.It is usually denoted by the turned E () logical operator symbol, which, when used together with a predicate variable,is called an existential quantier ("x or "(x)"). Existential quantication is distinct from universal quantication(for all), which asserts that the property or relation holds for all members of the domain.Symbols are encoded U+2203 there exists (HTML as a mathematical symbol) and U+2204 there does not exist (HTML ).

1.1 BasicsConsider a formula that states that some natural number multiplied by itself is 25.

00 = 25, or 11 = 25, or 22 = 25, or 33 = 25, and so on.

This would seem to be a logical disjunction because of the repeated use of or. However, the and so on makes thisimpossible to integrate and to interpret as a disjunction in formal logic. Instead, the statement could be rephrasedmore formally as

For some natural number n, nn = 25.

This is a single statement using existential quantication.This statement is more precise than the original one, as the phrase and so on does not necessarily include all naturalnumbers, and nothing more. Since the domain was not stated explicitly, the phrase could not be interpreted formally.In the quantied statement, on the other hand, the natural numbers are mentioned explicitly.This particular example is true, because 5 is a natural number, and when we substitute 5 for n, we produce 55 =25, which is true. It does not matter that "nn = 25 is only true for a single natural number, 5; even the existenceof a single solution is enough to prove the existential quantication true. In contrast, For some even number n, nn= 25 is false, because there are no even solutions.The domain of discourse, which species which values the variable n is allowed to take, is therefore of critical im-portance in a statements trueness or falseness. Logical conjunctions are used to restrict the domain of discourse tofulll a given predicate. For example:

For some positive odd number n, nn = 25

is logically equivalent to

1

2 CHAPTER 1. EXISTENTIAL QUANTIFICATION

For some natural number n, n is odd and nn = 25.

Here, and is the logical conjunction.In symbolic logic, "" (a backwards letter "E" in a sans-serif font) is used to indicate existential quantication.[1]Thus, if P(a, b, c) is the predicate "ab = c and N is the set of natural numbers, then

9n2NP (n; n; 25)is the (true) statement

For some natural number n, nn = 25.

Similarly, if Q(n) is the predicate "n is even, then

9n2N Q(n) ^ P (n; n; 25)is the (false) statement

For some natural number n, n is even and nn = 25.

In mathematics, the proof of a some statement may be achieved either by a constructive proof, which exhibits anobject satisfying the some statement, or by a nonconstructive proof which shows that there must be such an objectbut without exhibiting one.

1.2 Properties

1.2.1 NegationA quantied propositional function is a statement; thus, like statements, quantied functions can be negated. The :symbol is used to denote negation.For example, if P(x) is the propositional function x is between 0 and 1, then, for a domain of discourse X ofall natural numbers, the existential quantication There exists a natural number x which is between 0 and 1 issymbolically stated:

9x2XP (x)This can be demonstrated to be irrevocably false. Truthfully, it must be said, It is not the case that there is a naturalnumber x that is between 0 and 1, or, symbolically:

: 9x2XP (x)If there is no element of the domain of discourse for which the statement is true, then it must be false for all of thoseelements. That is, the negation of

9x2XP (x)is logically equivalent to For any natural number x, x is not between 0 and 1, or:

8x2X:P (x)

1.3. AS ADJOINT 3

Generally, then, the negation of a propositional function's existential quantication is a universal quantication of thatpropositional functions negation; symbolically,

: 9x2XP (x) 8x2X:P (x)A common error is stating all persons are not married (i.e. there exists no person who is married) when not allpersons are married (i.e. there exists a person who is not married) is intended:

: 9x2XP (x) 8x2X:P (x) 6 : 8x2XP (x) 9x2X:P (x)Negation is also expressible through a statement of for no, as opposed to for some":

@x2XP (x) : 9x2XP (x)Unlike the universal quantier, the existential quantier distributes over logical disjunctions:9x2XP (x) _Q(x)! (9x2XP (x) _ 9x2XQ(x))

1.2.2 Rules of InferenceA rule of inference is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inferencewhich utilize the existential quantier.Existential introduction (I) concludes that, if the propositional function is known to be true for a particular elementof the domain of discourse, then it must be true that there exists an element for which the proposition function is true.Symbolically,

P (a)! 9x2XP (x)Existential elimination, when conducted in a Fitch style deduction, proceeds by entering a new sub-derivation whilesubstituting an existentially quantied variable for a subject which does not appear within any active sub-derivation.If a conclusion can be reached within this sub-derivation in which the substituted subject does not appear, then onecan exit that sub-derivation with that conclusion. The reasoning behind existential elimination (E) is as follows: Ifit is given that there exists an element for which the proposition function is true, and if a conclusion can be reachedby giving that element an arbitrary name, that conclusion is necessarily true, as long as it does not contain the name.Symbolically, for an arbitrary c and for a proposition Q in which c does not appear:

9x2XP (x)! ((P (c)! Q)! Q)P (c) ! Q must be true for all values of c over the same domain X; else, the logic does not follow: If c is notarbitrary, and is instead a specic element of the domain of discourse, then stating P(c) might unjustiably give moreinformation about that object.

1.2.3 The empty setThe formula 9x2;P (x) is always false, regardless of P(x). This is because ; denotes the empty set, and no x of anydescription let alone an x fullling a given predicate P(x) exist in the empty set. See also vacuous truth.

1.3 As adjointMain article: Universal quantication As adjoint

In category theory and the theory of elementary topoi, the existential quantier can be understood as the left adjoint ofa functor between power sets, the inverse image functor of a function between sets; likewise, the universal quantieris the right adjoint.[2]

4 CHAPTER 1. EXISTENTIAL QUANTIFICATION

1.4 See also First-order logic List of logic symbols - for the unicode symbol Quantier variance Quantiers Uniqueness quantication

1.5 Notes[1] This symbol is also known as the existential operator. It is sometimes represented with V.

[2] Saunders Mac Lane, Ieke Moerdijk, (1992) Sheaves in Geometry and Logic Springer-Verlag. ISBN 0-387-97710-4 Seepage 58

1.6 References Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0.

Chapter 2

Participle

A participle is a form of a verb that is used in a sentence to modify a noun, noun phrase, verb or verb phrase, andthus plays a role similar to that of an adjective or adverb.[1] It is one of the types of nonnite verb forms. Its namecomes from the Latin participium,[2] a calque of Greek metoch partaking or sharing";[3] it is so named becausethe Ancient Greek and Latin participles share some of the categories of the adjective or noun (gender, number,case) and some of those of the verb (tense and voice).Participles may correspond to the active voice (active participles), where the modied noun is taken to representthe agent of the action denoted by the verb; or to the passive voice (passive participles), where the modied nounrepresents the patient (undergoer) of that action. Participles in particular languages are also often associated withcertain verbal aspects or tenses. The two types of participle in English are traditionally called the present participle(forms such as writing, singing and raising; these same forms also serve as gerunds and verbal nouns) and the pastparticiple (forms such as written, sung and raised; regular participles such as the last, as well as some irregular ones,have the same form as the nite past tense).In some languages, participles can be used in the periphrastic formation of compound verb tenses, aspects or voices.For example, one of the uses of the English present participle is to express continuous aspect (as in John is working),while the past participle can be used in expressions of perfect aspect and passive voice (as in Anne has written andBill was killed).A verb phrase based on a participle and having the function of a participle is called a participle phrase or participialphrase (participial is the adjective derived from participle). For example, looking hard at the sign and beaten by hisfather are participial phrases based respectively on an English present participle and past participle. Participial phrasesgenerally do not require an expressed grammatical subject; therefore such a verb phrase also constitutes a completeclause (one of the types of nonnite clause). As such, it may be called a participle clause or participial clause.(Occasionally a participial clause does include a subject, as in the English nominative absolute construction The kinghaving died, ... .)

2.1 Types of participleParticiples are often identied with a particular tense, as with the English present participle and past participle (seeunder English below). However, this is often a matter of convention; present participles are not necessarily associatedwith the expression of present time, or past participles necessarily with past time.Participles may also be identied with a particular voice: active or passive. Some languages (such as Latin andRussian) have distinct participles for active and passive uses. In English the present participle is essentially an activeparticiple, while the past participle has both active and passive uses. The following examples illustrate this:

I saw John eating his dinner. (eating is an active participle; the modied noun John is understood as the agent)

I have eaten my dinner. (perfect construction; eaten is an active participle here)

The sh was eaten by lions. (here eaten is a passive participle; the sh is understood as the patient, i.e. toundergo the action)

5

6 CHAPTER 2. PARTICIPLE

A distinction is also sometimes made between adjectival participles and adverbial participles. An adverbial par-ticiple (or a participial phrase/clause based on such a participle) plays the role of an adverbial (adverb phrase) inthe sentence in which it appears, whereas an adjectival participle (or a participial phrase/clause based on one) playsthe role of an adjective phrase. Some languages have dierent forms for the two types of participle; such languagesinclude Russian[4] and other Slavic languages, Hungarian, and many Eskimo languages, such as Sireniki,[5] which hasa sophisticated participle system. Details can be found in the sections below or in the articles on the grammars ofspecic languages.Some descriptive grammars treat adverbial and adjectival participles as distinct lexical categories, while others includethem both in a single category of participles.[4][6] Sometimes dierent names are used; adverbial participles in certainlanguages may be called converbs, gerunds or gerundives (although this is not consistent with the meanings of theterms gerund or gerundive as normally applied to English or Latin), or transgressives.Sometimes adjectival participles come to be used as pure adjectives, without any verbal characteristics (deverbaladjectives). They then no longer take objects or other modiers typical of verbs, possibly taking instead modiersthat are typical of adjectives, such as the English word very. The dierence is illustrated by the following examples:

The subject interesting him at the moment is Greek history. Greek history is an interesting subject.

In the rst sentence interesting is used as a true participle; it acts as a verb, taking the object him, and forming theparticipial phrase interesting him at the moment, which then serves as an adjective phrase modifying the noun subject.However, in the second sentence interesting has become a pure adjective; it stands in an adjectives typical positionbefore the noun, it can no longer take an object, and it could be accompanied by typical adjective modiers such asvery or quite (or in this case the prex un-). Similar examples are "interested people, a frightened rabbit, "fallenleaves, "meat-eating animals.

2.2 Indo-European languages

2.2.1 Germanic languagesEnglish

In Old English, past participles of Germanic strong verbs were marked with a ge- prex, as were most strong andweak past participles in Dutch and German today, and often by a vowel change in the stem. Those of weak verbswere marked by the ending -d, with or without an epenthetic vowel before it. Modern English past participles derivefrom these forms (although the ge- prex, which became y- in Middle English, has now been lost).Old English present participles were marked with an ending in -ende (or -iende for verbs whose innitives ended in-ian). In Middle English, various forms were used in dierent regions: -ende (southwest, southeast, Midlands), -inde(southwest, southeast), -and (north), -inge (southeast). The last is the one that became standard, falling together withthe sux -ing used to form verbal nouns. See -ing (etymology).Modern English verbs, then, have two participles:

The present participle, also sometimes called the active, imperfect, or progressive participle, takes the ending-ing. It is identical in form to the gerund (and verbal noun); the term present participle is sometimes used toinclude the gerund, and the term gerundparticiple is also used.

The past participle, also sometimes called the passive or perfect participle, is identical to the past tense form(ending in -ed) in the case of regular verbs, but takes various forms in the case of irregular verbs, such as sung,written, put, gone, etc.

Details of participle formation can be found under English verbs and List of English irregular verbs.The present participle, or participial phrases (clauses) formed from it, are used as follows:

to form the progressive (continuous) aspect: Jim was sleeping.

2.2. INDO-EUROPEAN LANGUAGES 7

as an adjective phrase modifying a noun phrase: The man sitting over there is my uncle. adverbially, the subject being understood to be the same as that of the main clause: Looking at the plans, I

gradually came to see where the problem lay. He shot the man, killing him.

similarly, but with a dierent subject, placed before the participle (the nominative absolute construction): Heand I having reconciled our dierences, the project then proceeded smoothly.

more generally as a clause or sentencemodier: Broadly speaking, the project was successful. (See also danglingparticiple.)

Past participles, or participial phrases (clauses) formed from them, are used as follows:

to form the perfect aspect: The chicken has eaten. to form the passive voice: The chicken was eaten. as an adjective phrase: The chicken eaten by the children was contaminated. (See also reduced relative clause.) adverbially: Seen from this perspective, the problem presents no easy solution. in a nominative absolute construction, with a subject: The task nished, we returned home.

Both types of participles are also often used as pure adjectives (see Types of participles above). Here present partici-ples are used in their active sense (an exciting adventure, i.e. one that excites), while past participles are usually usedpassively (the attached les, i.e. those that have been attached), although those formed from intransitive verbs maysometimes be used with active meaning (our fallen comrades, i.e. those who have fallen). Some such adjectivesalso form adverbs, such as interestingly and excitedly.The gerund is distinct from the present participle in that it (or rather the verb phrase it forms) acts as a noun ratherthan an adjective or adverb: I like sleeping'"; "Sleeping is not allowed. There is also a pure verbal noun with thesame form (the breaking of ones vows is not to be taken lightly). For more on the distinctions between these usesof the -ing verb form, see -ing: uses.For more details on uses of participles and other parts of verbs in English, see Uses of English verb forms, includingthe sections on the present participle and past participle.

2.2.2 Latin and Romance languagesLatin

Main article: Latin conjugation: Participles

Latin has three or four participles:

present active participle: present stem + -ns (gen. -ntis); e.g. amns he who loves perfect passive participle: participial stem + -us, -a, -um; e.g. amtus he who is loved future active participle: participial stem + -rus, -ra, -rum; e.g. amtrus about to love gerundive (sometimes considered the future passive participle): present stem + -(e)ndus, -(e)nda, -(e)ndum;e.g. amandus he who is to be loved

French

There are two basic participles:

Present active participle: formed by dropping the -ons of the nous form of the present tense of a verb (exceptwith tre) and then adding ant: marchant walking, tant being

8 CHAPTER 2. PARTICIPLE

Past participle: formation varies according to verb group: vendu sold, mis placed, march walked,t been, and fait done. The sense of the past participle is passive as an adjective and in most verbalconstructions with avoir, but active in verbal constructions with "tre, in reexive constructions, and withsome intransitive verbs.[7]

Compound participles are possible:

Present perfect participle: ayant appel having called, tant mort being dead

Passive perfect participle: tant vendu being sold, having been sold

Usage:

Present participles are used as qualiers as in un insecte volant" (a ying insect) and in some other contexts.They are never used to form tenses. The present participle is used in subordinate clauses, usually with en: Jemarche, en parlant.

Past participles are used as qualiers for nouns: la table casse" (the broken table); to form compound tensessuch as the perfect Vous avez dit" (you have said) and to form the passive voice: il a t tu" (he/it has beenkilled).

Spanish

In Spanish, the present or active participle (participio activo or participio de presente) of a verb is traditionally formedwith one of the suxes -ante, -ente or -iente, but modern grammar does not consider it a verbal form any longer, asthey become adjectives or nouns on their own: e.g. amante loving or lover, viviente living or live.The continuous is constructed much as in English, using a conjugated form of estar (to be) plus the gerundio (some-times called a verbal adverb or adverbial participle as it does not decline) with the suxes -ando (for -ar verbs),-iendo (for both -er and -ir verbs whose stems end in consonants), or -yendo (for both -er and -ir verbs whose stemsend in vowels): for example, estar haciendo means to be doing (haciendo being the gerundio of hacer, to do), andthere are related constructions such as seguir haciendo meaning to keep doing (seguir being to continue).The past participle (participio pasado or pasivo) is regularly formed with one of the suxes -ado, -ido, but severalverbs have an irregular form ending in -to (e.g. escrito, visto), or -cho (e.g. dicho, hecho). The past participle is usedgenerally as an adjective meaning a nished action, and it is variable in gender and number in these uses; and also itis used to form the compound tenses (as in English) in which it is indeclinable. Some examples:

As an adjective

las cartas escritas the written letters

To form compound tenses

Ha escrito una carta. "(She, he, it) Has written a letter.

2.2.3 Hellenic languages

Ancient Greek

Main article: Ancient Greek grammar: Participle

The Ancient Greek participle shares in the properties of adjectives and verbs. Like an adjective, it changes form forgender, case, and number. Like a verb, it has tense and voice, is modied by adverbs, and can take verb arguments,including an object.[8]

2.2. INDO-EUROPEAN LANGUAGES 9

There is a form of the participle for every combination of tense (present, aorist, perfect, future) and voice (active, mid-dle, passive). All participles are based on the stems of the corresponding tenses. Here are the masculine nominativesingular forms for a thematic and an athematic verb:Like an adjective, it can modify a noun, and can be used to embed one thought into another.

he who intends to be a good general must have a great deal of ability and knowledge,

In the example, the participial phrase , literally the one going to be a good general, is usedto embed the idea he will be a good general within the main verb.The participle is very widely used in ancient Greek, especially in prose.

2.2.4 Celtic languages

Welsh

In Welsh, the eect of a participle in the active voice is constructed by yn followed by the innitive form (for thepresent participle) and wedi followed by the innitive form (for the past participle). There is no mutation in eithercase. In the passive voice, participles are usually replaced by a compound phrase such as wedi cael ei/eu (having gothis/her/their ...ing) in contemporary Welsh and by the impersonal form in classical Welsh.

2.2.5 Slavic languages

Polish

The Polish word for participle is imiesw (pl.: imiesowy). There are four types of imiesowy in two classes:Adjectival participle (imiesw przymiotnikowy)

active adjectival participle (imiesw przymiotnikowy czynny): robicy - doing, one who does

passive adjectival participle (imiesw przymiotnikowy bierny): robiony - being done (can only be formedo transitive verbs)

Adverbial participle (imiesw przyswkowy)

present adverbial participle (imiesw przyswkowy wspczesny): robic - doing, while doing

perfect adverbial participle (imiesw przyswkowy uprzedni): zrobiwszy - having done (formed in virtuallyall cases o verbs in their perfective forms, here denoted by the prex z-)

Dangling participleDue to the distinction between adjectival and adverbial participles, in Polish it is practically impossible to make adangling participle mistake in the classical English meaning of the term. For instance, in the sentence:I have found them hiding in the closet.it is unclear, whether I or them is hiding in the closet. In Polish there is a clear distinction:

Znalazem ich, chowajc si w szae. - chowajc is a present adverbial participle regarding the subject (I)

Znalazem ich chowajcych si w szae - chowajcych is an active adjectival participle regarding the object(them)

10 CHAPTER 2. PARTICIPLE

Russian

Verb: [sl.t] (to hear, imperfective aspect)Present active: [sl..j] hearing, who hearsPresent passive: [sl..mj] being heard, that is heard, audiblePast active: [sl.f.j] who heard, who was hearingPast passive: [sl.n.nj] that was heard, that was being heardAdverbial present active: [sl.] "(while) hearingAdverbial past active: [sl.f] having been hearingVerb: [sl.t] (to hear, perfective aspect)Past active: [sl.f.j] who has heardPast passive: [sl.n.nj] that has been heardAdverbial past active: [sl.f] having heardFuture participles formed from perfective verbs are technically possible, though not considered a part of standardlanguage.[9]

Bulgarian

Verb: pravja (to do, imperfective aspect)Present active: pravetPast active aorist: pravilPast active imperfect: pravel (only used in verbal constructions)Past passive: pravenAdverbial present active: pravejkiVerb: napravja (to do, perfective aspect)Past active aorist: napravilPast active imperfect: napravel (only used in verbal constructions)Past passive: napravenParticiples are adjectives formed as verbs

Macedonian

Macedonian completely lost or transformed the participles of the Common Slavic, unlike the other Slavic languages.The following is noted:[10]

present active participle: it transformed into verbal adverb; present passive participle: there are some isolated cases of present passive participle (or remnants of it), suchas the word [lakom] (greedy);

past active participle: there is only one word (remnant) of the past active participle, which is the word [biv] (former). However, this word is often substituted with the word [poraneen] (former);

past passive participle: transformed into verbal adjective (it behaves like normal adjective); resultative participle: transformed into verbal l-form ( -). It is not a participle since it doesn'tfunction attributively.

2.2.6 Baltic languagesLithuanian

Among Indo-European languages, the Lithuanian language is unique for having thirteen dierent participial formsof the verb, that can be grouped into ve when accounting for inection by tense. Some of these are also inected bygender and case. For example, the verb eiti (to go, to walk) has the active participle forms eins/einantis (going,

2.3. SEMITIC LANGUAGES 11

walking, present tense), js (past tense), eisis (future tense), eidavs (past frequentative tense), the passive partici-ple forms einamas (being walked, present tense), eitas (walked past tense), eisimas (future tense), the adverbialparticiples einant (while [he, dierent subject] is walking present tense), jus (past tense), eisiant (future tense),eidavus (past frequentative tense), the semi-participle eidamas (while [he, the same subject] is going, walking) andthe participle of necessity eitinas (that which needs to be walked). The active, passive and the semi- participles areinected by gender and the active, passive and necessity ones are inected by case.

2.3 Semitic languages

2.3.1 ArabicMain article: Arabic verbs: Participle

The Arabic verb has two participles: an active participle ( ) and a passive participle ( ),and the form of the participle is predictable by inspection of the dictionary form of the verb. These participles areinected for gender, number and case, but not person. Arabic participles are employed syntactically in a varietyof ways: as nouns, as adjectives or even as verbs. Their uses vary across varieties of Arabic. In general the activeparticiple describes a property of the syntactic subject of the verb fromwhich it is derived, whilst the passive participlesdescribes the object. For example, from the verb kataba, the active participle is ktib and the passiveparticiple is maktb . Roughly these translate to writing and written respectively. However, they havedierent, derived lexical uses. ktib is further lexicalized as writer, author and maktb as letter.In Classical Arabic these participles do not participate in verbal constructions with auxiliaries the same way as theirEnglish counterparts do, and rarely take on a verbal meaning in a sentence (a notable exception being participlesderived from motion verbs as well as participles in Qur'anic Arabic). In certain dialects of Arabic however, it is muchmore common for the participles, especially the active participle, to have verbal force in the sentence. For example,in dialects of the Levant, the active participle is a structure which describes the state of the syntactic subject after theaction of the verb from which it is derived has taken place. kil, the active participle of akala (to eat), describesones state after having eaten something. Therefore it can be used in analogous way to the English present perfect(for example, An kil meaning I have eaten, I have just eaten or I have already eaten). Other verbs,such as ra (to go) give a participle (ryi ) which has a progressive (is going...) meaning. The exacttense or continuity of these participles is therefore determined by the nature of the specic verb (especially its lexicalaspect and its transitivity) and the syntactic/semantic context of the utterance. What ties them all together is that theydescribe the subject of the verb from which they are derived. The passive participles in certain dialects can be usedas a sort of passive voice, but more often than not, are used in their various lexicalized senses as adjectives or nouns.

2.4 Finno-Ugric languages

2.4.1 FinnishVerb: tehd (to do)Present active: tekev(doing)Present passive: tehtv(doable)Past active: tehnyt (has done)Past passive: tehty(been done)Agent participle (passive): tekem (done by...)Negative participle: tekemtn (undone)

2.5 Other languages

2.5.1 Sireniki EskimoSireniki Eskimo language, an extinct EskimoAleut language, has separate sets of adverbial participles and adjectival

12 CHAPTER 2. PARTICIPLE

participles. Dierent from in English, adverbial participles are conjugated to reect the person and number of theirimplicit subjects; hence, while in English a sentence like If I were a marksman, I would kill walruses requires twofull clauses (in order to distinguish the two verbs dierent subjects), in Sireniki Eskimo one of these may be replacedwith an adverbial participle (since its conjugation will indicate the subject).

2.5.2 EsperantoMain article: Esperanto grammar: Participles

Esperanto has 6 dierent participle conjugations; active and passive for past, present and future. The participles areformed as follows:For example, a falonta botelo is a bottle which will fall. A falanta botelo is one that is falling through the air. After ithits the oor, it is a falinta botelo. These examples use the active participles, but the usage of the passive participlesis similar. A cake that is going to be divided is a dividota kuko. When it is in the process of being divided, it is adividata kuko. Having been cut, it is now a dividita kuko.These participles can be used in conjunction with the verb to be, esti, forming 18 compound tenses (9 active and 9passive). However, this soon becomes complicated and often unnecessary, and is only frequently used when rigoroustranslation of English is required. An example of this would be la knabo estos instruita, or, the boy will have beentaught. This example sentence is then in the future anterior.When the sux -o is used, instead of -a, then the participle refers to a person. A mananto is someone who is eating.A maninto is someone who ate. A manonto is someone who will eat. Also, a manito is someone who was eaten,a manato is someone who is being eaten, and a manoto is someone who will be eaten.These rules hold true to all verbs, and there are no exceptions.An informal addition to these six are the participles for conditional forms, which use -unt-. The active participles arethe only ones generally used. For example, a komencunto is a person who would (have) begun. A parolunto issomeone who would (have) spoken.

2.6 See also Attributive verb Gerund Grammar Hanging participle Nonnite verb Transgressive (linguistics)

2.7 Notes[1] What is a participle? in Glossary of linguistic terms at SIL International.

[2] participium. Charlton T. Lewis and Charles Short. A Latin Dictionary on Perseus Project.

[3] . Liddell, Henry George; Scott, Robert; A GreekEnglish Lexicon at the Perseus Project

[4] The Russian Participles. Part of An Interactive On-line Reference Grammar Russian by Dr. Robert Beard.

[5] Menovshchikov, G.A.: Language of Sireniki Eskimos. Phonetics, morphology, texts and vocabulary. Academy of Sciencesof the USSR, Moscow Leningrad, 1964. Original data: .. : . , , . . . , 1964

[6] . Kiss, Katalin; Kiefer Ferenc - Siptr Pter (2003). j magyar nyelvtan (in Hungarian) (3. kiads ed.). Budapest: OsirisKiad.

2.8. REFERENCES 13

[7] Maurice Grevisse, Le Bon Usage, 10th edition, 776.

[8] Smyth, A Greek Grammar for Colleges, section 2039.

[9] Shagal (Krapivina), Future participles in Russian: Expanding the participial paradigm

[10] Macedonian Grammar, Victor Friedman

2.8 References Participles from the American Heritage Book of English Usage (1996).

2.9 External links List of English simple past and past participle verb forms from myenglishteacher.net Ernest De Witt Burton: Moods and Tenses of New Testament Greek. The adverbial participle.

Chapter 3

Truth table

A truth table is a mathematical table used in logicspecically in connection with Boolean algebra, boolean func-tions, and propositional calculusto compute the functional values of logical expressions on each of their functionalarguments, that is, on each combination of values taken by their logical variables (Enderton, 2001). In particular,truth tables can be used to tell whether a propositional expression is true for all legitimate input values, that is, logicallyvalid.Practically, a truth table is composed of one column for each input variable (for example, A and B), and one nalcolumn for all of the possible results of the logical operation that the table is meant to represent (for example, A XORB). Each row of the truth table therefore contains one possible conguration of the input variables (for instance, A=trueB=false), and the result of the operation for those values. See the examples below for further clarication. LudwigWittgenstein is often credited with their invention in the Tractatus Logico-Philosophicus,[1] though they appeared atleast a year earlier in a paper on propositional logic by Emil Leon Post.[2]

3.1 Unary operationsThere are 4 unary operations:

3.1.1 Logical false

3.1.2 Logical identity

Logical identity is an operation on one logical value, typically the value of a proposition, that produces a value of trueif its operand is true and a value of false if its operand is false.The truth table for the logical identity operator is as follows:

3.1.3 Logical negation

Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value oftrue if its operand is false and a value of false if its operand is true.The truth table for NOT p (also written as p, Np, Fpq, or ~p) is as follows:

3.1.4 Logical true

3.2 Binary operationsThere are 16 possible truth functions of two binary variables :

14

3.2. BINARY OPERATIONS 15

3.2.1 Truth table for all binary logical operators

Here is a truth table giving denitions of all 16 of the possible truth functions of two binary variables (P and Q arethus boolean variables: information about notation may be found in Bocheski (1959), Enderton (2001), and Quine(1982); for details about the operators see the Key below):where T = true and F = false. The Com row indicates whether an operator, op, is commutative - P op Q = Q op P.The L id row shows the operators left identities if it has any - values I such that I op Q = Q. The R id row showsthe operators right identities if it has any - values I such that P op I = P.[note 1]

The four combinations of input values for p, q, are read by row from the table above. The output function for each p,q combination, can be read, by row, from the table.Key:The key is oriented by column, rather than row. There are four columns rather than four rows, to display the fourcombinations of p, q, as input.p: T T F Fq: T F T FThere are 16 rows in this key, one row for each binary function of the two binary variables, p, q. For example, inrow 2 of this Key, the value of Converse nonimplication ('8 ') is solely T, for the column denoted by the uniquecombination p=F, q=T; while in row 2, the value of that '8 ' operation is F for the three remaining columns of p, q.The output row for8 is thus2: F F T Fand the 16-row[3] key isLogical operators can also be visualized using Venn diagrams.

3.2.2 Logical conjunction (AND)

Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces avalue of true if both of its operands are true.The truth table for p AND q (also written as p q, Kpq, p & q, or p q) is as follows:In ordinary language terms, if both p and q are true, then the conjunction p q is true. For all other assignments oflogical values to p and to q the conjunction p q is false.It can also be said that if p, then p q is q, otherwise p q is p.

3.2.3 Logical disjunction (OR)

Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces avalue of true if at least one of its operands is true.The truth table for p OR q (also written as p q, Apq, p || q, or p + q) is as follows:Stated in English, if p, then p q is p, otherwise p q is q.

3.2.4 Logical implication

Logical implication or the material conditional are both associated with an operation on two logical values, typicallythe values of two propositions, that produces a value of false just in the singular case the rst operand is true and thesecond operand is false.The truth table associated with the material conditional if p then q (symbolized as p q) and the logical implicationp implies q (symbolized as p q, or Cpq) is as follows:It may also be useful to note that p q is equivalent to p q.

16 CHAPTER 3. TRUTH TABLE

3.2.5 Logical equality

Logical equality (also known as biconditional) is an operation on two logical values, typically the values of twopropositions, that produces a value of true if both operands are false or both operands are true.The truth table for p XNOR q (also written as p q, Epq, p = q, or p q) is as follows:So p EQ q is true if p and q have the same truth value (both true or both false), and false if they have dierent truthvalues.

3.2.6 Exclusive disjunction

Exclusive disjunction is an operation on two logical values, typically the values of two propositions, that produces avalue of true if one but not both of its operands is true.The truth table for p XOR q (also written as p q, Jpq, or p q) is as follows:For two propositions, XOR can also be written as (p q) (p q).

3.2.7 Logical NAND

The logical NAND is an operation on two logical values, typically the values of two propositions, that produces avalue of false if both of its operands are true. In other words, it produces a value of true if at least one of its operandsis false.The truth table for p NAND q (also written as p q, Dpq, or p | q) is as follows:It is frequently useful to express a logical operation as a compound operation, that is, as an operation that is built up orcomposed from other operations. Many such compositions are possible, depending on the operations that are takenas basic or primitive and the operations that are taken as composite or derivative.In the case of logical NAND, it is clearly expressible as a compound of NOT and AND.The negation of a conjunction: (p q), and the disjunction of negations: (p) (q) can be tabulated as follows:

3.2.8 Logical NOR

The logical NOR is an operation on two logical values, typically the values of two propositions, that produces a valueof true if both of its operands are false. In other words, it produces a value of false if at least one of its operands istrue. is also known as the Peirce arrow after its inventor, Charles Sanders Peirce, and is a Sole sucient operator.The truth table for p NOR q (also written as p q, Xpq, (p q)) is as follows:The negation of a disjunction (p q), and the conjunction of negations (p) (q) can be tabulated as follows:Inspection of the tabular derivations for NAND and NOR, under each assignment of logical values to the functionalarguments p and q, produces the identical patterns of functional values for (p q) as for (p) (q), and for (p q)as for (p) (q). Thus the rst and second expressions in each pair are logically equivalent, and may be substitutedfor each other in all contexts that pertain solely to their logical values.This equivalence is one of De Morgans laws.

3.3 Applications

Truth tables can be used to prove many other logical equivalences. For example, consider the following truth table:This demonstrates the fact that p q is logically equivalent to p q.

3.3. APPLICATIONS 17

3.3.1 Truth table for most commonly used logical operatorsHere is a truth table giving denitions of the most commonly used 6 of the 16 possible truth functions of 2 binaryvariables (P,Q are thus boolean variables):Key:

T = true, F = false^ = AND (logical conjunction)_ = OR (logical disjunction)_ = XOR (exclusive or)^ = XNOR (exclusive nor)! = conditional if-then = conditional "(then)-if

() biconditional or if-and-only-if is logically equivalent to ^ : XNOR (exclusive nor).

Logical operators can also be visualized using Venn diagrams.

3.3.2 Condensed truth tables for binary operatorsFor binary operators, a condensed form of truth table is also used, where the row headings and the column headingsspecify the operands and the table cells specify the result. For example Boolean logic uses this condensed truth tablenotation:This notation is useful especially if the operations are commutative, although one can additionally specify that therows are the rst operand and the columns are the second operand. This condensed notation is particularly usefulin discussing multi-valued extensions of logic, as it signicantly cuts down on combinatoric explosion of the numberof rows otherwise needed. It also provides for quickly recognizable characteristic shape of the distribution of thevalues in the table which can assist the reader in grasping the rules more quickly.

3.3.3 Truth tables in digital logicTruth tables are also used to specify the functionality of hardware look-up tables (LUTs) in digital logic circuitry.For an n-input LUT, the truth table will have 2^n values (or rows in the above tabular format), completely specifyinga boolean function for the LUT. By representing each boolean value as a bit in a binary number, truth table valuescan be eciently encoded as integer values in electronic design automation (EDA) software. For example, a 32-bitinteger can encode the truth table for a LUT with up to 5 inputs.When using an integer representation of a truth table, the output value of the LUT can be obtained by calculating abit index k based on the input values of the LUT, in which case the LUTs output value is the kth bit of the integer.For example, to evaluate the output value of a LUT given an array of n boolean input values, the bit index of the truthtables output value can be computed as follows: if the ith input is true, let Vi = 1, else let Vi = 0. Then the kth bit ofthe binary representation of the truth table is the LUTs output value, where k = V0*2^0 + V1*2^1 + V2*2^2 + ...+ Vn*2^n.Truth tables are a simple and straightforward way to encode boolean functions, however given the exponential growthin size as the number of inputs increase, they are not suitable for functions with a large number of inputs. Otherrepresentations which are more memory ecient are text equations and binary decision diagrams.

3.3.4 Applications of truth tables in digital electronicsIn digital electronics and computer science (elds of applied logic engineering and mathematics), truth tables can beused to reduce basic boolean operations to simple correlations of inputs to outputs, without the use of logic gates orcode. For example, a binary addition can be represented with the truth table:

18 CHAPTER 3. TRUTH TABLE

A B | C R 1 1 | 1 0 1 0 | 0 1 0 1 | 0 1 0 0 | 0 0 where A = First Operand B = Second Operand C = Carry R = ResultThis truth table is read left to right:

Value pair (A,B) equals value pair (C,R). Or for this example, A plus B equal result R, with the Carry C.

Note that this table does not describe the logic operations necessary to implement this operation, rather it simplyspecies the function of inputs to output values.With respect to the result, this example may be arithmetically viewed as modulo 2 binary addition, and as logicallyequivalent to the exclusive-or (exclusive disjunction) binary logic operation.In this case it can be used for only very simple inputs and outputs, such as 1s and 0s. However, if the number of typesof values one can have on the inputs increases, the size of the truth table will increase.For instance, in an addition operation, one needs two operands, A and B. Each can have one of two values, zero orone. The number of combinations of these two values is 22, or four. So the result is four possible outputs of C andR. If one were to use base 3, the size would increase to 33, or nine possible outputs.The rst addition example above is called a half-adder. A full-adder is when the carry from the previous operationis provided as input to the next adder. Thus, a truth table of eight rows would be needed to describe a full adder'slogic:A B C* | C R 0 0 0 | 0 0 0 1 0 | 0 1 1 0 0 | 0 1 1 1 0 | 1 0 0 0 1 | 0 1 0 1 1 | 1 0 1 0 1 | 1 0 1 1 1 | 1 1 Same as previous,but.. C* = Carry from previous adder

3.4 HistoryIrving Anellis has done the research to show that C.S. Peirce appears to be the earliest logician (in 1893) to devise atruth table matrix. From the summary of his paper:

In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russells1912 lecture on The Philosophy of Logical Atomism truth table matrices. The matrix for negation isRussells, alongside of which is the matrix for material implication in the hand of Ludwig Wittgenstein.It is shown that an unpublished manuscript identied as composed by Peirce in 1893 includes a truthtable matrix that is equivalent to the matrix for material implication discovered by John Shosky. Anunpublished manuscript by Peirce identied as having been composed in 188384 in connection withthe composition of Peirces On the Algebra of Logic: A Contribution to the Philosophy of Notationthat appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truthtable for the conditional.

3.5 Notes[1] The operators here with equal left and right identities (XOR, AND, XNOR, andOR) are also commutativemonoids because

they are also associative. While this distinction may be irrelevant in a simple discussion of logic, it can be quite importantin more advanced mathematics. For example, in category theory an enriched category is described as a base categoryenriched over a monoid, and any of these operators can be used for enrichment.

3.6 See also Boolean domain Boolean-valued function Espresso heuristic logic minimizer Excitation table

3.7. REFERENCES 19

First-order logic Functional completeness Karnaugh maps Logic gate Logical connective Logical graph Method of analytic tableaux Propositional calculus Truth function

3.7 References[1] Georg Henrik von Wright (1955). Ludwig Wittgenstein, A Biographical Sketch. The Philosophical Review 64 (4): 527

545 (p. 532, note 9). JSTOR 2182631.

[2] Emil Post (July 1921). Introduction to a general theory of elementary propositions. American Journal of Mathematics43 (3): 163185. JSTOR 2370324.

[3] Ludwig Wittgenstein (1922) Tractatus Logico-Philosophicus Proposition 5.101

3.8 Further reading Bocheski, Jzef Maria (1959), A Prcis of Mathematical Logic, translated from the French and German edi-tions by Otto Bird, Dordrecht, South Holland: D. Reidel.

Enderton, H. (2001). A Mathematical Introduction to Logic, second edition, New York: Harcourt AcademicPress. ISBN 0-12-238452-0

Quine, W.V. (1982), Methods of Logic, 4th edition, Cambridge, MA: Harvard University Press.

3.9 External links Hazewinkel, Michiel, ed. (2001), Truth table, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Truth Tables, Tautologies, and Logical Equivalence PEIRCE'S TRUTH-FUNCTIONAL ANALYSIS AND THE ORIGIN OF TRUTH TABLES by Irving H.Anellis

Converting truth tables into Boolean expressions

Chapter 4

Truth-bearer

This article is about a term used in philosophy, logic and philosophy of logic.

A truth-bearer is an entity that is said to be either true or false and nothing else. The thesis that some things aretrue while others are false has led to dierent theories about the nature of these entities. Since there is divergence ofopinion on the matter, the term truth-bearer is used to be neutral among the various theories. Truth-bearer candidatesinclude propositions, sentences, sentence-tokens, statements, concepts, beliefs, thoughts, intuitions, utterances, andjudgements but dierent authors exclude one or more of these, deny their existence, argue that they are true only ina derivative sense, assert or assume that the terms are synonymous,[1] or seek to avoid addressing their distinction ordo not clarify it.[2]

4.1 Introduction

Some distinctions and terminology as used in this article, based on Wolfram 1989[3] Chapter 2 Section1) follow. Itshould be understood that the terminology described is not always used in the ways set out, and it is introduced solely forthe purposes of discussion in this article. Use is made of the typetoken and usemention distinctions. Reection onoccurrences of numerals might be helpful.[4] 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[nb 1] is a typographic character (printed or written) etc.A word token[nb 2] is a pattern of characters. A word-type[nb 3] is an identical pattern of characters. Ameaningful-word-token[nb 4] is a meaningful word-token. Two word-tokens which mean the same are of the same word-meaning[nb 5]

A sentence-token[nb 6] is a pattern of word-tokens. A meaningful-sentence-token[nb 7] is a meaningful sentence-token or a meaningful pattern of meaningful-word-tokens. Two sentence-tokens are of the same sentence-type if theyare identical patterns of word-tokens characters[nb 8] A declarative-sentence-token is a sentence-token which thatcan be used to communicate truth or convey information.[5] A meaningful-declarative-sentence-token is a mean-ingful declarative-sentence-token[nb 9] Two meaningful-declarative-sentence-tokens are of the same meaningful-declarative-sentence-type[nb 10] if they are identical patterns of word-tokens. A nonsense-declarative-sentence-token[nb 11] is a declarative-sentence-token which is not a meaningful-declarative-sentence-token. A meaningful-declarative-sentence-token-use[nb 12] occurs when and only when a meaningful-declarative-sentence-token is useddeclaratively.A referring-expression[nb 13] is expression that can be used to pick out or refer to particular entity. A referentialsuccess[nb 14] is a referring-expressions success in identifying a particular entity. A referential failure[nb 15] is areferring-expressions failure to identify a particular entity. A referentially-successful-meaningful-declarative-sentence-token-use[nb 16] is ameaningful-declarative-sentence-token-use containing no referring-expression that failsto identify a particular entity.

20

4.2. SENTENCES IN NATURAL LANGUAGES 21

4.2 Sentences in natural languagesAs Aristotle pointed out, since some sentences are questions, commands, or meaningless, not all can be truth-bearers.If in the proposal What makes the sentence Snow is white true is the fact that snow is white it is assumed thatsentences like Snow is white are truth-bearers, then it would be more clearly stated as What makes the meaningful-declarative-sentence Snow is white true is the fact that snow is white.Theory 1a:

All and only meaningful-declarative-sentence-types[nb 17]) are truth-bearers

Criticisms of Theory 1aSome meaningful-declarative-sentence-types will be both truth and false, contrary to our denition of truth-bearer,e.g. (i) the liar-paradox sentences such as This sentence is false. (see Fisher 2008[6]) (ii) Time, place and persondependent sentences e.g. It is noon. This is London, I'm Spartacus.Anyone may ..ascribe truth and falsity to the deterministic propositional signs we here call utterances. But if he takesthis line, he must, like Leibniz, recognise that truth cannot be an aair solely of actual utterances, since it makes senseto talk of the discovery of previously un-formulated truths. (Kneale, W&M (1962)[7])Revision to Theory 1a, by making a distinction between type and token.To escape the time, place and person dependent criticism the theory can be revised, making use or the typetokendistinction,[8] as followsTheory 1b:

All and only meaningful-declarative-sentence-tokens are truth-bearers

Quine argued that the primary truth-bearers are utterances [nb 18]

Having now recognised in a general way that what are true are sentences, we must turn to certainrenements. What are best seen as primarily true or false are not sentences but events of utterances.If a man utters the words 'It is raining' in the rain, or the words 'I am hungry' while hungry, his verbalperformance counts as true. Obviously one utterance of a sentence may be true and another utteranceof the same sentence be false.

QUINE 1970[9] page 13Criticisms of Theory 1b(i) Theory 1b prevents sentences which are meaningful-declarative-sentence-types from being truth-bearers. If allmeaningful-declarative-sentence-types typographically identical to The whole is greater than the part are true thenit surely follow that the meaningful-declarative-sentence-type The whole is greater than the part is true (just asall meaningful-declarative-sentence-tokens typographically identical to The whole is greater than the part are En-glish entails the meaningful-declarative-sentence-types The whole is greater than the part is English) (ii) Somemeaningful-declarative-sentences-tokens will be both truth and false, or neither, contrary to our denition of truth-bearer. E.g. A token, t, of the meaningful-declarative-sentence-type P: I'm Spartacus, written on a placard. Thetoken t would be true when used by Spartacus, false when used by Bertrand Russell, neither true nor false whenmentioned by Spartacus or when being neither used nor mentioned.Theory 1b.1

Allmeaningful-declarative-sentence-token-uses are truth-bearers; somemeaningful-declarative-sentence-types are truth-bearers

To allow that at least some meaningful-declarative-sentence-types can be truth-bearers Quine allowed so-called eter-nal sentences[nb 19] to be truth-bearers.

In Peircess terminology, utterances and inscriptions are tokens of the sentence or other linguisticexpression concerned; and this linguistic expression is the type of those utterances and inscriptions. In

22 CHAPTER 4. TRUTH-BEARER

Freges terminology, truth and falsity are the two truth values. Succinctly then, an eternal sentence is asentence whose tokens have the same truth values.... What are best regarded as true and false are notpropositions but sentence tokens, or sentences if they are eternal

Quine 1970[9] pages 1314Theory 1c

All and only meaningful-declarative-sentence-token-uses are truth-bearers

Arguments for Theory 1cBy respecting the usemention Theory 1c avoids criticism (ii) of Theory 1b.Criticisms of Theory 1c(i) Theory 1c does not avoid criticism (i) of Theory 1b. (ii) meaningful-declarative-sentence-token-uses are events(located in particular positions in time and space) and entail a user. This implies that (a) nothing (no truth-bearer)exists and hence nothing (no truth-bearer) is true (of false) anytime anywhere (b) nothing (no truth-bearer) existsand hence nothing (no truth-bearer) is true (of false) in the absence of a user. This implies that (a) nothing wastrue before the evolution of users capable of using meaningful-declarative-sentence-tokens and (b) nothing is true (orfalse) accept when being used (asserted) by a user. Intuitively the truth (or falsity) of The tree continues to be in thequad continues in the absence of an agent to asset it.Referential Failure A problem of some antiquity is the status of sentences such as U: The King of France is baldV: The highest prime has no factors W: Pegasus did not exist Such sentences purport to refer to entitles which donot exist (or do not always exist). They are said to suer from referential failure. We are obliged to choose either (a)That they are not truth-bearers and consequently neither true nor false or (b) That they are truth-bearers and per seare either true of false.Theory 1d

All and only referentially-successful-meaningful-declarative-sentence-token-uses are truth-bearers.

Theory 1d takes option (a) above by declaring that meaningful-declarative-sentence-token-uses that fail referentiallyare not truth-bearers.Theory 1e

All referentially-successful-meaningful-declarative-sentence-token-uses are truth-bearers; somemeaningful-declarative-sentence-types are truth-bearers

Arguments for Theory 1eTheory 1e has the same advantages as Theory 1d. Theory 1e allows for the existence of truth-bearers (i.e., meaningful-declarative-sentence-types) in the absence of users and between uses. If for any x, where x is a use of a referentiallysuccessful token of a meaningful-declarative-sentence-type y x is a truth-bearer then y is a truth-bearer otherwise yis not a truth bearer. E.g. If all uses of all referentially successful tokens of the meaningful-declarative-sentence-typeThe whole is greater than the part are truth-bearers (i.e. true or false) then the meaningful-declarative-sentence-typeThe whole is greater than the part is a truth-bearer. If some but not all uses of some referentially successful tokensof the meaningful-declarative-sentence-type I am Spartacus are true then the meaningful-declarative-sentence-typeI am Spartacus is not a truth-bearer.Criticisms of Theory 1eTheory 1e makes implicit use of the concept of an agent or user capable of using (i.e. asserting) a referentially-successful-meaningful-declarative-sentence-token. Although Theory 1e does not depend on the actual existence(now, in the past or in the future) of such users, it does depend on the possibility and cogency of their existence.Consequently the concept of truth-bearer under Theory 1e is dependent upon giving an account of the concept of auser. In so far as referentially-successful-meaningful-declarative-sentence-tokens are particulars (locatable in timeand space) the denition of truth-bearer just in terms of referentially-successful-meaningful-declarative-sentenceis attractive to those who are (or would like to be) nominalists. The introduction of use and users threatens theintroduction of intentions, attitudes, minds &c. as less-than=welcome ontological baggage

4.3. SENTENCES IN LANGUAGES OF CLASSICAL LOGIC 23

4.3 Sentences in languages of classical logicIn classical logic a sentence in a language is true or false under (and only under) an interpretation and is therefore atruth-bearer. For example a language in the rst-order predicate calculus might include one ofmore predicate symbolsand one or more individual constants and one or more variables. The interpretation of such a language would denea domain (universe of discourse); assign an element of the domain to each individual constant; assign the donation inthe domain of some property to each unary (one-place) predicate symbol.[10]

For example if a language L consisted in the individual constant a, two unary predicate letters F andG and the variablex, then an interpretation I of L might dene the Domain D as animals, assign Socrates to a, the denotation of theproperty being a man to F and the denotation of the property being mortal to G. Under the interpretation I of L thenFa would be true if, and only if Socrates is a man, and the sentence (Fx Gx) would be true if, and only if all men(in the domain) are mortal. In some texts 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 betrue but only some sentence-tokens of Fa under particular interpretations. A token of Fa without an interpretationis neither true nor false. Some sentences of a Language like L are said to be true under all interpretations of thesentence, e.g. (Fx Fx), such sentences being termed logical truths, but again such sentences are neither true norfalse in the absence of an interpretation.

4.4 PropositionsMany authors[11] use the term proposition as truth-bearers. There is no single denition or usage.[12][13] Sometimesit is used to mean ameaningful declarative sentence itself; sometimes it is used to mean the meaning of a meaningfuldeclarative sentence.[14] This provides two possible denitions for the purposes of discussion as belowTheory 2a:

All and only meaningful-declarative-sentences are propositions

Theory 2b:

Ameaningful-declarative-sentence-token expresses a proposition; twomeaningful-declarative-sentence-tokens which have the samemeaning express the same proposition; twomeaningful-declarative-sentence-tokens with dierent meanings express dierent propositions.

(cf Wolfram 1989,[3] p. 21)Proposition is not always used in one or other of these ways.Criticisms of Theory 2a.

If all and only meaningful-declarative-sentences are propositions, as advanced by Theory 2a, then the terms aresynonymous andwe can just as well speak of themeaningful-declarative-sentences themselves as the trutbearers- there is no distinct concept of proposition to consider, and the term proposition is literally redundant.

Criticisms of Theory 2b

Theory 2b entails that if all meaningful-declarative-sentence-tokens typographically identical to say, I amSpartacus have the same meaning then they (i) express the same proposition (ii) that proposition is both trueand false,[15] contrary to the denition of truth-bearer.

The concept of a proposition in this theory rests upon the concept of meaning as applied to meaningful-declarative-sentences, in a word synonymy among meaningful-declarative-sentence s. Quine 1970 argues thatthe concept of synonymy among meaningful-declarative-sentences cannot be sustained or made clear, conse-quently the concepts of propositions and meanings of sentences are, in eect, vacuous and superuous[16]

see also Willard Van Orman Quine, Proposition, The Russell-Myhill Antinomy, also known as the Principles of Math-ematics Appendix B Paradox

24 CHAPTER 4. TRUTH-BEARER

see also Internet Encycypedia of Philosophy Propositions are abstract entities; they do not exist in space and time. Theyare sometimes said to be timeless, eternal, or omnitemporal entities. Terminology aside, the essential point is thatpropositions are not concrete (or material) objects. Nor, for that matter, are they mental entities; they are not thoughtsas Frege had suggested in the nineteenth century. The theory that propositions are the bearers of truth-values also hasbeen criticized. Nominalists object to the abstract character of propositions. Another complaint is that its not sucientlyclear when we have a case of the same propositions as opposed to similar propositions. This is much like the complaintthat we cant determine when two sentences have exactly the same meaning. The relationship between sentences andpropositions is a serious philosophical problem.

4.5 StatementsMany authors consider statements as truth-bearers, though as with the term proposition there is divergence in de-nition and usage of that term. Sometimes 'statements are taken to be meaningful-declarative-sentences; sometimesthey are thought to be what is asserted by a meaningful-declarative-sentence. It is not always clear in which sense theword is used. This provides two possible denitions for the purposes of discussion as below.A particular concept of a statement was introduced by Strawson in the 1950s.,[17][18][19]

Consider the following:

I: The author of Waverley is dead J: The author of Ivanhoe is dead

K: I am less than six feet tall L: I am over six feet tall

M: The conductor is a bachelor N: The conductor is married

On the assumption that the same personwroteWaverley and Ivanhoe, the two distinct patterns of characters (meaningful-declarative-sentences) I and J make the same statement but express dierent propositions.The pairs of meaningful-declarative-sentences (K, L) & (M, N) have dierent meanings, but they are not necessarilycontradictory, since K & L may have been asserted by dierent people, and M & N may have been asserted aboutdierent conductors.What these examples show is that we cannot identify that which is true or false (the statement) with the sentence used inmaking it; for the same sentence may be used to make dierent statements, some of them true and some of them false.(Strawson, P.F. (1952)[19])This suggests:

Two meaningful-declarative-sentence-tokens which say the same thing of the same object(s) make the samestatement.

Theory 3a

All and only statements are meaningful-declarative-sentences.

Theory 3b

All and only meaningful-declarative-sentences can be used to make statements

Statement is not always used in one or other of these ways.Arguments for Theory 3a

4.6. THOUGHTS 25

All and only statements are meaningful-declarative-sentences. is either a stipulative denition or a descriptivedenition. If the former, the stipulation is useful or it is not; if the latter, either the descriptive denitioncorrectly describes English usage or it does not. In either case no arguments, as such, are applicable

Criticisms of Theory 3a

If the term statement is synonymous with the term meaningful-declarative-sentence, then the applicable criti-cisms are the same as those outlined under sentence below

If all and only meaningful-declarative-sentences are statements, as advanced by Theory 3a, then the terms aresynonymous and we can just as well speak of the meaningful-declarative-sentences themselves as the truth-bearers there is no distinct concept of statement to consider, and the term statement is literally redundant.

Arguments for Theory 3b

4.6 ThoughtsFrege (1919) argued that an indicative sentence in which we communicate or state something, contains both a thoughtand an assertion, it expresses the thought, and the thought is the sense of the sentence.[20]

4.7 Notes[1] Character A character is 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 ag or stick held at a certain angle, a gesture, a sign as use in signlanguage, a pattern or raised indentations (as in brail) etc. in other words the sort of things that are commonly describedas the elements of an alphabet.

[2] Word-token A word-token is a pattern of characters.The pattern of characters A This toucan can catch a can contains six word-tokensThe pattern of characters D He is grnd contains three word-tokens

[3] Word-type A word-type is an identical pattern of characters, .The pattern of characters A: This toucan can catch a can. contains ve word-types (the word-token can occurring twice)

[4] Meaningful-word-token Ameaningful-word-token is a meaningful word-token. grnd in D He is grnd. is not meaningful..

[5] Word-meaning Two word-tokens which mean the same are of the same word-meaning. Only those word-tokens whichare meaningful-word-tokens can have the same meaning as another word-token. The pattern of characters A: This toucancan catch a can. contains six word-meanings.Although it contains only ve word-types, the two occurrences of the word-token can have dierent meanings.On the assumption that bucket and pail mean the same, the pattern of characters B: If you have a bucket, then you have apail contains ten word-tokens, seven word-types, and six word-meanings.

[6] Sentence-token A sentence-token is a pattern of word-tokens.The pattern of characters D: He is grnd is a sentence-token because grnd is a word-token (albeit not a meaningful word-token.)

[7] Meaningful-sentence-token A meaningful-sentence-token is a meaningful sentence-token or a meaningful pattern ofmeaningful-word-tokens.The pattern of characters D: He is grnd is not a sentence-token because grnd is not a meaningful word-token.

[8] Sentence-type Two sentence-tokens are of the same sentence-type if they are identical patterns of word-tokens characters,e.g. the sentence-tokens P: I'm Spartacus and Q: I'm Spartacus are of the same sentence-type.

[9] Meaningful-declarative-sentence-tokensAmeaningful-declarative-sentence-token is ameaningful declarative-sentence-token.The pattern of characters F: Cats blows the wind is not a meaningful-declarative-sentence-token because it is grammaticallyill-formedThe pattern of characters G: This stone is thinking about Vienna is not a meaningful-declarative-sentence-token becausethinking cannot be predicated of a stone

26 CHAPTER 4. TRUTH-BEARER

The pattern of characters H: This circle is square is not a meaningful-declarative-sentence-token because it is internallyinconsistentThe pattern of characters D: He is grnd is not a meaningful-declarative-sentence-token because it contains a word-token(grnd) which is not a meaningful-word-token

[10] Meaningful-declarative-sentence-typesTwomeaningful-declarative-sentence-tokens are of the samemeaningful-declarative-sentence-type if they are identical patterns of word-tokens characters, e.g. the sentence-tokens P: I'm Spartacus and Q:I'm Spartacus 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

[11] Nonsense-declarative-sentence-token A nonsense-declarative-sentence-token is a declarative-sentence-token which isnot a meaningful-declarative-sentence-token.The patterns of characters F: Cats blows the wind, G: This stone is thinking about Vienna and H: This circle is squareare nonsense-declarative-sentence-tokens because they are declarative-sentence-tokens but not meaningful-declarative-sentence-tokens. The pattern of characters D: He is grnd is not a nonsense-declarative-sentence-token because it is nota declarative-sentence-token because it contains a word-token (grnd) which is not a meaningful-word-token.

[12] Meaningful-declarative-sentence-token-use A meaningful-declarative-sentence-token-use occurs when and only whena meaningful-declarative-sentence-token is used declaratively, rather than, say, mentioned.The pattern of characters T: Spartacus did not eat all his spinach in London on Feb 11th 2009 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 T. 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 areidentical events in time and space with identical users.

[13] Referring-expression An expression that can be used to pick out or refer to particular entity, such as denite descriptionsand proper names

[14] Referential success a referring-expressions success in identifying a particular entityOR ameaningful-declarative-sentence-token-uses containing one or more referring-expression all of which succeed in identifying a particular entity

[15] Referential failure a referring-expressions failure to identify a particular entity is referentially successful OR ameaningful-declarative-sentence-token-uses containing one or more referring-expression that fail to identify a particular entity.

[16] Referentially-successful-meaningful-declarative-sentence-token-useAmeaningful-declarative-sentence-token-use con-taining 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 themeaningful-declarative-sentence-type V: The highest prime has no factors other than itself and 1 is not a referentially-successful-meaningful-declarative-sentence-token-use since the embedded referring-expression The highest prime is alwaysa referential failure.

[17] Meaningful-declarative-sentence-typesTwomeaningful-declarative-sentence-tokens are of the samemeaningful-declarative-sentence-type if they are identical pat-terns 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 its tokens of are meaningful-declarative-sentence-tokens

[18] Utterance: The term utterance is frequently used to mean meaningful-declarative-sentence-token. See e.g. Grice,Mean-ing, 1957 http://semantics.uchicago.edu/kennedy/classes/f09/semprag1/grice57.pdf

[19] Eternal Sentence: A sentence that stays forever true, or forever false, independently of any special circumstances underwhich they happen to be uttered or written. More exactly, a meaningful-declarative-sentence-type whose tokens have thesame truth values. E.g. The whole is greater than the part is an eternal sentence, It is raining is not an eternal sentence butIt rains in Boston, Mass., on July 15, 1968 is an eternal sentence

4.8 References[1] e.g.

In symbolic logic, a statement (also called a proposition) is a complete declarative sentence, which is either true orfalse. Vignette 17 Logic, Truth and Language

A statement is just that; it is a declaration about somethinganythinga declaration which can be evaluated aseither true or false. I am reading this sentence is a statement, and if indeed you have looked at it and comprehendedits meaning, then it is safe to say that that statement can be evaluated as true.Fundamental Logic Concepts: Statement

4.9. EXTERNAL LINKS 27

[2] e.g. * Some philosophers claim that declarative sentences of natural language have underlying logical forms and that theseforms are displayed by formulas of a formal language. Other writers hold that (successful) declarative sentences expresspropositions; and formulas of formal languages somehow display the forms of these propositions. Shapiro, Stewart (2008).Edward N. Zalta, ed. Classical Logic in The Stanford Encyclopedia of Philosophy (Fall 2008 ed.).

[3] Wolfram, Sybil (1989). Philosophical Logic. Routledge, London and New York. ISBN 0-415-02317-3.

[4] Occurrences of numerals

[5] name=declarative-sentence-token group="nb"> Declarative-sentence-token A declarative-sentence-token is a sentence-token which that can be used to communicate truth or convey information.The pattern of characters E: Are you happy? is not a declarative-sentence-token because it interrogative not declarative.

[6] Fisher (2008). Philosophy of Logic. ISBN 0-495-00888-5.

[7] Kneale, W&M (1962). The development of logic. Oxford. ISBN 0-19-824183-6. page 593

[8] see Wolfram, Sybil (1989) generally on the application of typetoken distinction

[9] QUINE, W.V. (1970). Philosophy of Logic. Prentice Hall. ISBN 0-13-663625-X.

[10] See also 'First-order logic#Semantics

[11] e.g. Russell,Wittgenstein, and Stanford Encyclopedia of PhilosophyURL= http://plato.stanford.edu/entries/facts/#FacPro:By proposition, we shall mean truth-bearer, and remain neutral as to whether truth-bearers are sentences, statements, be-liefs or abstract objects expressed by sentences, for instanceexcept in section 2.4.1.

[12] McGrath, Matthew, Propositions, The Stanford (Fall 2008 Edition), EdwardN. Zalta (ed.), URL = ."The term proposition has a broad use in contemporary philosophy. It isused to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other propositionalattitudes (i.e., what is believed, doubted, etc.), the referents of that-clauses, and the meanings of sentences.

[13] Mark, Richard. Propositions. On one use of the term, propositions are objects of assertion, what successful uses ofdeclarative sentences say. As such, they determine truth-values and truth conditions. On a second, they are the objectsof certain psychological states (such as belief and wonder) ascribed with verbs that take sentential complements (such asbelieve and wonder ). On a third use, they are what are (or could be) named by the complements of such verbs. Manyassume that propositions in one sense are propositions in the others.

[14] Philosophers tolerance towards propositions has been encouraged partly by ambiguity in the term 'proposition'. The termoften is used simply for the sentences themselves, declarative sentences; and then some writers who do use the term formeanings of sentences are careless about the distinction between sentences and their meanings Quine 1970, p. 2

[15] i.e. when expressed by a token-meaningful-declarative-sentence made by Spartacus, and when expressed by somebodyother than Spartacus

[16] Philosophers who favor propositions have said that propositions are needed because truth only of propositions, not ofsentences [read meaningful-declarative-sentences Ed], is intelligible. An unsympathetic answer is that we can explain truthof sentences to be propositional in their own terms: sentences are true whose meanings are true propositions. Any failureof intelligibilty here is already his own fault. Quine 1970 page 10

[17] Strawson, PF (1950). On referring. Mind 9. reprinted in Strawson 1971 and elsewhere

[18] Strawson, PF (1957). Propositions, Concepts and Logical Truths. The Philosophical Quarterly 7. reprinted in Strawson,P.F. (1971). Logico-Linguistic Papers. Methuen. ISBN 0-416-09010-9.

[19] Strawson, P.F. (1952). Introduction to Logical Theory. Methuen: London. p. 4. ISBN 0-416-68220-0.

[20] Frege (1919) Die Gedanke trans AM and Marcelle Quinton in Frege, G (1956). The thought: A logical Enquiry. Mind65. reprinted in Strawson 1967.

4.9 External links Stanford Encyclopedia of Philosophy:

Truth; 2.1 Sentences as truth-bearers; Glanzberg, Michael The Correspondence Theory of Truth; 2. Truthbearers and Truthmakers; David, Marian

Chapter 5

Truthmaker

A truthmaker for a truthbearer is that entity in virtue of which the truthbearer is true. Philosophers have speculatedon the question whether every truthbearer requires a truthmaker. Parmenides' classic claim that what does not existcannot be thought about has been read as a claim that every truthbearer must have a truthmaker, since otherwisethe truthbearer is not about anything. A falsemaker for a proposition is that existent reality in virtue of which thatproposition is false, assuming it is false.In Truth-Makers (1984), Mulligan, Simons and Smith introduced the truth-maker idea as a contribution to thecorrespondence theory of truth. Logically atomic empirical sentences such as John kissed Mary have truthmakers,typically events or tropes corresponding to the main verbs of the sentences in question. Mulligan et al. exploreextensions of this idea to sentences of other sorts, but the do not embrace any position of truthmaker maximalism,according to which every truthbearer has a truthmaker.This maximalist position leads to philosophical diculties, such as the question of what the truthmaker for an ethical,modal or mathematical truthbearer could be. Of course someone who is deeply enough committed to truthmakersand who simultaneously doubts that a truthmaker could be found for a certain kind of truthbearer will simply denythat that truthbearer could be true. Those who nd the Parmenidean insight suciently compelling often take it to bea particularly enlightening metaphysical pursuit to search for truthmakers of