Tarski's truth definition
SEMANTICS: definition which formalizes the relation between a sentence
and its °truth value. The definition
formalizes what Wittgenstein (1922)
describes as: to understand a proposition means to know what is the case if it is true.
It has the following general form:
(i) For any S in L and any v, S is true in v if and only if p.S is a structural description of a sentence of a language L, v is a circumstance which has to be met, and p are the conditions that describe how this circumstance should be met. EXAMPLE: the definition for sentence S in (i) has (partly) the form of (ii).
(ii) Snow is white is true iff Snow is whiteTarski's truth definition is crucial to a denotational °meaning theory.
Tautology
SEMANTICS: a sentence which is always true, not due to its word meanings
but to its °logical form. A tautology is
also called a °logical truth or a
°necessary truth. A sentence of
°propositional logic is a tautology
when it is true for every possible assignment of
°truth values to the
°propositional letters of that
sentence. The formula p V Neg p is a tautology of propositional logic. A sentence
of °predicate logic is a tautology when
it is true for every possible denotation of the variables and individual and
predicate constants that it contains. The formula All(x) [ P(x) V Neg P(x) ] is a
tautology of predicate logic.
LIT.
Gamut (1991).
Tautosyllabic
PHONOLOGY: belonging to the same
°syllable.
EXAMPLE: in the English word contact the first t is
tautosyllabic with a but not with n.
Template morphology
MORPHOLOGY: a term which is used for
°nonconcatenative
morphological systems in which it is difficult or impossible to analyze the
formation of complex words as the addition of affixes one by one to a stem. Rather,
in these systems a word consists of several obligatory and optional affixes, where
each obligatory affix has its own position in the string and optional affixes are
slotted into this string, at the appropriate point in the sequence. Usually one
finds °discontinuous dependencies
between affixes. Languages with template morphologies are
Navajo (Young & Morgan (1980)),
Arabic (McCarthy (1981)),
Sierra Miwok (Smith (1985)), and
Yawelmani (Archangeli (1984)).
Tense
SYNTAX/SEMANTICS: grammatical feature or category expressing a temporal
relation between the event described by the verb and the moment of utterance.
Tense has been analyzed either as a morpho-syntactic feature of
°INFL, or as a category T in its own right.
Traditional tense features are past, present and future.
LIT.
Chomsky (1981),
Guéron & Hoekstra (1988),
Hornstein (1990),
Giorgi & Pianesi (1991).
PHONOLOGY: a °feature which
characterizes vowels that are produced with a tongue body or tongue root
configuration involving a greater constriction than that found in their lax
counterparts; this greater degree of constriction is frequently accompanied by
greater length
(Halle & Clements (1983:7)).
EXAMPLE: In English, [I] in willis lax, while [i] in
wheel is tense.
Tensed-S Condition (TSC)
SYNTAX: one of the conditions on transformations proposed by Chomsky (1973).
It states that extraction out of a tensed sentence is impossible, and is formulated as
No rule can involve X,Y in the structure ...X...[a...Y...]... where a is a tensed sentenceEXAMPLE: the TSC accounts for the contrast in (i) and (ii). In (i) passivization involves NP-movement out of an infinitival (i.e. non-tensed) clause, while in (ii) the same NP is moved out of the tensed counterpart, violating the TSC.
(i) John is believed [t to be ill] (ii) *John is believed [(that) is ill] (iii) who do you believe [S' COMP [S I met t]]Wh-movement in (iii) can violate the TSC, because wh-movement involves the COMP-position which (by definition) is an °escape hatch. More recently, the TSC has been subsumed under the °binding conditions for anaphors (including NP-traces), through the definition of °governing category.
Terminative aspect
°Aspectual classes.
That-trace effect
SYNTAX: the phenomenon that the complementizer (that) cannot be
followed by a trace (except in relative clauses) in some languages (e.g. English).
Thus, in languages showing the that-t(race) effect, a subject cannot be
extracted when it follows that. This is shown by the contrast in (i) and (ii).
(i) who did you think [CP t' [C' e [IP t would win ]]] (ii) *who did you think [CP t' [C' that [IP t would win ]]]As noted, the that-t effect is not a universal phenomenon. It is absent in e.g. Dutch, as shown by the fact that the Dutch translation of (ii) is grammatical:
(iii) wie denk je [CP t' [C' dat [IP t gewonnen heeft ]]]LIT. Perlmutter (1971), Chomsky & Lasnik (1977), Chomsky (1981, 1986b), Kayne (1980), Pesetsky (1982), Taraldsen (1978).
Thematic relations
°Thematic theory.
Thematic roles
SYNTAX: semantic roles in a system which takes location and movement
through space as (some of) the key notion in the interpretation of natural language
sentences.
EXAMPLE: the sentence Don gives Ed a donut is about a relocation
or transfer of a donut (the Theme) from Don (the Source) to Ed (the Goal). In
syntactic theory thematic roles are usually referred to as
°Theta-roles with not much concern for their
content. °Thematic theory.
LIT.
Fillmore (1968),
Gruber (1965),
Jackendoff (1990).
Thematic structure
°Thematic theory.
Thematic theory
Theory about the thematic interpretation of natural language sentences. According to
Jackendoff (1990), the thematic interpretation of a sentence is the association of
its syntactic structure with thematic representations (its thematic structure, also
referred to as its °lexical
conceptual structure). Thematic representations are taken to consist of distinct
levels: thematic tiers and aspectual (action) tiers. The thematic tiers represent
the structure of the event which the sentence is about, in terms of predicates such
as THING, EVENT, PATH, GO, BE, CAUSE, TO, AT, etc. (constituting a localistic
framework), whereas the aspectual tiers represent the aspectual properties of
events in terms of predicates such as AFFECT, ACTOR and PATIENT.
°Linking rules determine what part of
thematic structure is syntactically relevant by mapping (part of the) conceptual
structure onto syntactic structure. The usual assumption is that the
°argument structure of a predicate
(verb, adjective, etc.) is the interface between thematic structure and syntactic
structure, such that some of the arguments in thematic structure correspond to
arguments in syntactic structure. Thematic interpretation is only part of the
semantic interpretation of natural language sentences, which also encompasses
quantificational, temporal and modal aspects.
LIT.
Fillmore (1968),
Gruber (1965),
Jackendoff (1983,
1990).
Thematic tier
°Thematic theory.
Theme
SEMANTICS: the object that is located or relocated in space.
EXAMPLE: the glass is a theme both in (i)a (location) and (i)b
(relocation).
(i) a The glass is on the table
b The glass fell from the table
(ii) a The glass is mine
b John gave the glass away
The interpretation of thematic roles has been generalized in terms of 'abstract
space'; then, a theme is what is in a state or in a change of state, and the
notion applies to the glass in (ii)a (state) and (ii)b (change of state)
as well.
Theme vowel
MORPHOLOGY: morphological marker which indicates which is the declensional or
conjugational class that a word belongs to.
EXAMPLE: in Ancient Greek, nouns are grouped into three declensional classes.
If a noun belongs to the first declension, the root is followed by the theme vowel
-a-, and if it belongs to the second declension the root is followed by
-o-: môus+a+n (acc.sg.), môus+a+y (nom.pl.)
'Muse', log+o+n (acc.sg.), log+o+y (nom.pl.) 'word'. Another term
is 'extension vowel'.
LIT.
Goodwin (1894),
Spencer (1991).
There-insertion
SYNTAX: operation which inserts
°expletive there in the first,
presumably the subject position of an existential sentence.
EXAMPLE: (i)a has been analyzed as being derived from (i)b by inserting
there in the empty slot '__'.
(i) a there is someone in the room
b __ is someone in the room
This construction is now usually analyzed with the expletive base-generated in
subject position and the logical subject adjoined or attached to VP. See
°CHAIN,
°Definiteness effect.
Theta Criterion (theta-criterion)
SYNTAX: condition which states that at
°D-structure each
°argument is in a
°theta-position, and that each
theta-position contains an argument. If the theta-criterion is defined over
°LF it says that each theta-position is in a unique
°chain, and that each chain contains a unique
theta-position. The theta-criterion accounts for the contrasts in (i) and (ii).
(i) a it seems that John is ill
b *Bill seems that John is ill
(ii) a Bill believes that John is ill
b *it believes that John is ill
In (i)b the argument Bill is in a °theta-bar
position; in (ii)b the expletive it (a non-argument) is in a theta-position.
Originally, the theta-criterion stipulated a unique relationship between arguments
and theta-roles (rather than theta-positions), forcing the analysis in (iii)b on
John left angry: unless angry brings its own (PRO) argument,
there would be two theta-roles (assigned by left and angry,
respectively) for one argument (John). Revising the theta-criterion in
terms of theta-positions allows the analysis in (iii)a where the unique argument
John is in a unique theta-position associated with two theta-roles.
(iii) a John left angry
b John left [PRO angry]
In Brody (1993) it is argued that the theta-criterion can be dispensed with.
Theta-absorption (theta-absorption)
SYNTAX: some morpho-syntactic elements have the capacity of absorbing a
theta-role assigned by the verb (stem) to which they are attached, with the effect
that that theta-role can not be assigned to its regular position.
EXAMPLE: °Passive morphology
is an example, and °clitics have been argued to
absorb a theta-role as well.
LIT.
Jaeggli (1986a),
Spencer (1991).
Theta-bar position (theta'- or theta-bar
position)
°Theta-position.
Theta-government (theta-government)
SYNTAX: the configuration in which a
°governor
°theta-marks the phrase that it governs,
formally defined as in (i).
(i) a theta-governs b iff a governs b and a theta-marks bLIT. Chomsky (1986b).
Theta-grid (theta-grid)
SYNTAX: the lexical specification of the
°thematic properties of a
°predicate. Originally (cf. Williams 1981b),
a theta-grid is conceived of as an unordered list of theta-roles with only an
indication of their status as being external (through underlining) or internal
roles. More recently (cf. Grimshaw 1990), the theta-grid of a predicate is equated
with its °argument structure, which not
only specifies an hierarchy among the theta-roles (through bracketing), but also
their status as being a direct or indirect role, optional or obligatory, etc.
LIT.
Williams (1981b),
Grimshaw (1990),
Roca (1992).
Theta identification (theta-identification)
SYNTAX/SEMANTICS: process by which two
°arguments are identified.
EXAMPLE: in the nice man the arguments of the predicates
nice and man are identified resulting in the interpretation 'x
is nice & y is a man & x = y' where the 'x = y' part represents the identification.
LIT.
Higginbotham (1985).
Theta-marking (theta-marking)
°Theta-role assignment.
Theta-position (theta-position)
SYNTAX: an °A-position to which a
°theta-role is assigned.
EXAMPLE: in John buys the apples both the arguments John
and the apples are in a theta-position being theta-marked as
°Agent and °Theme.
But the verb seems in (i) does not assign a theta-role to its grammatical
subject John. Therefore, the matrix subject position, although an A-position,
is not a theta-position.
(i) John seems [ti to have left]The positions in (i) containing John and its °trace ti are both A-positions, but only the position of the trace is a theta-position. An A-position to which no theta-role is assigned (such as the one in (i) which is taken by John) is called a theta-bar position (theta'-position). Another name for theta-bar position is non-theta position.
Theta-role (theta-role)
SYNTAX: the semantic relationship of an
°argument with the predicate it is an argument
of is expressed through the assignment of a role by the predicate to the argument,
in conformity with the °theta-criterion.
Different theta-roles have different labels, such as
°Agent and °Theme.
EXAMPLE: in Ken sent me the argument Ken is the Agent of
the predicate sent me, and the argument me is the Theme of the
predicate sent. Other theta-roles that have been distinguished are
°Goal, °Source,
°Experiencer.
Theta-role assignment (theta-role
assignment)
SYNTAX: assignment by a °predicate of
a °theta-role to an
°argument.
Theta-theory (theta-theory)
SYNTAX: theory about °theta-roles.
The major principle of theta-theory is the
°theta-criterion.
LIT.
Chomsky (1981,
1986a).
Third Construction
SYNTAX: construction typical of Dutch and German in which part of an
infinitival complement appears to be
°extraposed.
EXAMPLE: in (Dutch) (i)a the infinitival complement has been extraposed in
toto, but in (i)b, the so-called third construction, part of the complement (de
prijs) is in situ.
(i) a Kees heeft geweigerd [de prijs in ontvangst te nemen]
b Kees heeft de prijs geweigerd [in ontvangst te nemen]
Kees has the prize refused in acceptance to take
'Kees has refused to accept the prize'
The construction in (i)b can be analyzed as a case of
°remnant extraposition: some element(s) - de
prijs in (i)b - is (are) removed (by
°scrambling) from the infinitival complement,
before the remnant of it is extraposed. In many cases, the remnant will only contain
the infinitival verb and the result looks like a
°verb raising construction. The distinguishing
feature is the impossibility of
°IPP in case of a third
construction, and the fact that IPP is obligatory in case of verb raising (cf. (iii).
(ii) Jan heeft een boek geprobeerd te lezen Jan has a book tried to read (iii) Jan heeft een boek proberen te lezen 'Jan has tried to read a book'The third construction cannot be equated with °VP-Raising.
Though-movement
SYNTAX: movement which looks like
°topicalization but is not, because the
preposed phrase is separated from the rest of the sentence by the adverb though.
EXAMPLE: though-movement derives (i)b from (i)a.
(i) a Though I think she is pretty, I don't like her.
b Pretty though I think she is, I don't like her.
Though-movement is easy to confuse with
°tough-movement.
Tier
PHONOLOGY/MORPHOLOGY: a level of representation where particular information
is decoded.
EXAMPLE: McCarthy (1981) has argued that the representation of the Arabic
form katab 'write, perfect active' consists of three independent tiers:
(a) a tier where the root ktb 'write' is represented, (b) a tier where the
inflectional morpheme a is represented, and (c) a tier where the binyam I
information CVCVC is decoded (= CV tier). Another term is plane.
SYNTAX/SEMANTICS: °Thematic theory.
LIT.
Goldsmith (1976,
1990),
McCarthy (1981),
Clements & Keyser (1983),
Spencer (1991).
Tier Conflation (TC)
MORPHOLOGY/PHONOLOGY: a process by which multilinear representations are
linearized. For the Arabic form katab 'write-perfect active' this entails
the following change:
a
/ \
C V C V C TC C V C V C
| | | | | | | |
k t b k a t a b
LIT.
McCarthy (1981,
1986),
Spencer (1991).
T-model
Model of grammar prevalent in
°Principles and Parameters theory,
which has the general structure in (i):
(i) DS | SS / \ PF LFIt is usually assumed that, in English, DS (°D-structure) is generated by °rewrite rules, or is projected from the rules of °Xbar-theory, and obeys the conditions of °theta-theory and the °Extended Projection Principle. SS (°S-structure) is derived from DS by the repeated application of °affect alpha (e.g. °NP-movement and °Wh-movement), and must meet the demands of °Case theory, and possibly °Binding Theory. LF (°Logical Form) is derived from SS through the application of affect alpha (e.g °QR, Wh-raising (°Wh-in-situ)), and is regarded as the interface with the conceptual system; possibly, LF obeys Binding Theory and is the locus of gamma-checking (°gamma-marking). °PF is derived from SS and is considered the interface with the articulatory-perceptual system. Both PF and LF are subject to the principle of °Full interpretation. The division of labor among the three syntactic levels of representation (DS, SS, LF) is subject to debate, and may vary across languages.
Tone
PHONOLOGY: In many, so-called tone languages, a difference in pitch is
used to distinguish lexical items. In such languages tone is a
°distinctive feature. The most
common opposition is that between a high and a low tone. But also three- or
four-way tonal distinctions occur in languages. A high tone is usually indicated
with an acute (té) and a low tone with a grave (tè).
EXAMPLE: in Margi we find the following lexical items:
(i) shú 'tail'
shù 'to dry up'
Also we may find so-called contour tones, i.e. a high and a low tone realized on
the same vowel. We distinguish between a rising tone (te) and a falling tone
(tê). Contour-tones are often analysed as complex tones.
See °tone stability.
Tone stability
PHONOLOGY: a phenomenon found in
°tone languages. A segment can be deleted
while leaving behind the specification of tone height. This specification is found
on a neighboring segment, often creating a
°contour tone. In Margi the two
°morphemes /ngè/ and /yá/ are
realized as [ngya]. The vowel [e] is deleted, leaving behind the low tone. This low
tone settles on the remaining vowel and forms a contour tone with the original tone
of the second vowel.
LIT.
Dell, Hirst & Vergnaud (1984),
Goldsmith (1976).
Topic
That part of the utterance about which new information is given.
SYNTAX: it is assumed that the topic may, or, in certain languages, must
be placed at the beginning of the sentence. In some languages, the topic may also
be provided with a specific morphological marking.
EXAMPLE: consider the following data from Japanese:
(i) a Taroo-wa kaeru-o koros-i-ta
Taroo-TOP frog-OBJ killed
'Taroo killed the frog'
b Kaeru-wa Taroo-ga koros-i-ta
frog-TOP taroo-SUBJ killed
'The frog Taroo killed'
In (i)a the subject Taroo is marked for TOPIC by wa; in (i)b
Taroo is marked for SUBJECT by ga.
Topic-comment-distinction
SEMANTICS: the division of a sentence into the thing that is talked about
(the topic) and what is said about it (the comment). In simple
sentences this distinction coincides with the distinction between
°subject and
°predicate, but the topic-comment-distinction
can also be expressed by °topicalization,
°left-dislocation and
°cleft-constructions, or by
°prosodic means. Some authors identify this
distinction with the distinction between focus and presupposition or theme and rheme.
LIT.
Jackendoff (1972).
Topicalization
SYNTAX: the preposing of some topical constituent.
EXAMPLE: in (i) to Mary is topicalized.
(i) To Mary John has given many presents.In °Verb Second languages topicalization is often believed to occur in all declarative main clauses. The following examples are from German:
(ii) a Johann hat Maria viele Geschenke gegeben
Johann has Maria many presents given
b Maria hat Johann viele Geschenke gegeben
c Viele Geschenke hat Johann Maria gegeben
In the 'neutral' case, the subject - Johann - is topicalized, as in (ii)a.
If another constituent is topicalized, such as the indirect object Maria in
(ii)b or the direct object viele Geschenke in (ii)c, it receives emphatic
stress.
Tough-movement
SYNTAX: the kind of movement that has been assumed in the derivation of
sentences such as (i), which are headed by predicates like tough and
easy.
(i) John is easy to pleaseIf the adjective easy assigns no °external theta-role, one is led (by the °theta-criterion) to assume that the subject John has been moved out of the object position ei of please:
(ii) *Johni is easy [CP PRO to please ei]However, such an analysis is excluded, either as a case of °improper movement, or as a °binding violation. As a solution it has been proposed that John in (i) is an argument of the adjective easy, and that the CP is a modifying adjunct:
(iii) Johni is easy [CP Oi [PRO to please ei]]The object of to please, then, is a variable bound by the °empty operator O which is coindexed with John. Easy to confuse with °though-movement though it is, it is different.
Trace theory
SYNTAX: theory about traces left by movement. This theory assumes that
if an element X has been moved in the course of a derivation, it has left a
°trace in its original position.
EXAMPLE: in (i) the NP John is moved while leaving a trace t,
indicating its °d-structure position.
(i) Johni seems [ti to have left]Since °theta-marking occurs at d-structure, it is possible to determine the thematic role of the moved NP via its trace. The concept of a trace is crucial to the theory of movement and to °bounding theory, because a trace can be treated as an °empty category. See °NP-trace, °wh-trace.
Transfer
PHONOLOGY/MORPHOLOGY: a process proposed in Clements (1985) to account for
°reduplication. Clements assumes that a
reduplication morpheme is not prefixed or suffixed to a base, but simply parafixed:
the CV skeleton of the base and the affix are parallel. Next, the skeleton of the
parafix is associated to the skeleton of the base, and then the relevant melodic
information of the base is transferred to the parafix.
LIT.
Clements (1985),
McCarthy & Prince (1986)),
Spencer (1991).
Transformation
°Transformational rule.
Transformational rule
SYNTAX: rule that transforms syntactic structure.
°Deletion,
°Insertion, and
°movement are instances of transformational
rules. °S-structure is derived from
°d-structure by means of transformations,
and °Logical Form is derived from S-structure in a
similar way. See °affect alpha.
LIT.
Van Riemsdijk & Williams
(1986).
Transitive verb
MORPHOLOGY/SYNTAX: verb which has to be accompanied by a direct object.
EXAMPLE: an example is the English verb hit which must be
accompanied by a direct object (*he hits vs. he hits the ball).
Transparent context
°Opaque context.
Tree of numbers
SEMANTICS: in a sentence of the form [S [NP D CN] VP] the set A of
entities denoted by the common noun CN can be divided into a
°subset with elements that belong to the set B of
entities represented by VP, and a subset with elements that don't belong to that set,
i.e. A intersect B and A - B, respectively. In a domain with n dogs, the dogs can be
divided over these two subsets in n+1 ways, each of which is represented by an
ordered pair x,y where x = |A intersect B| and y = |A - B|. The tree of numbers is
a complete representation of all these pairs of numbers for each possible size of A:
(i) |A|=0 0,0
|A|=1 1,0 0,1
|A|=2 2,0 1,1 0,2
|A|=3 3,0 2,1 1,2 0,3
|A|=4 4,0 3,1 2,2 1,3 0,4
|A|=5 5,0 4,1 3,2 2,3 1,4 0,5
... ...
The meaning of a determiner D can be represented as a subset of a tree of numbers.
The determiner every, for example corresponds to the x,0 pairs on each row:
(ii) |A|=0 +
|A|=1 + -
|A|=2 + - -
|A|=3 + - - -
... ...
Many properties of determiners (like
°upward and
°downward monotonicity) and
relations between determiners (like negation) can be clarified in the tree of numbers.
Tree structure
SYNTAX: a tree structure is a graph, which comprises a set of points,
called nodes, connected by branches (represented by solid lines). Any given pair
of nodes contained in the same tree will be related by one of two different types
of relation, namely either by °dominance or by
°precedence. A tree structure has only one
top node.
(i) A
/ \
B C
| / \
| D E
| | |
b d e
EXAMPLE: in (i) the node labelled A dominates all other nodes. Node C
dominates D, but D does not dominate C. Node B precedes nodes D and E, as well
as the nodes d and e. The nodes at the bottom of each complete tree structure
(here in lower case) are called terminal nodes; other nodes are called non-terminal.
Each node carries a label. Non-terminal nodes carry category labels; A, B, C, D, E
in figure (i). Terminal nodes, unless they are empty, are labelled with an appropriate
lexical item (a word), viz. b, d, e in (i). Nodes can be branching or non-branching.
For example, node C in (i) branches into nodes D and E; node B is a non-branching
node. Tree structures are used as a representation of the constituent structure of
natural language expressions. Thus, the tree in (ii) represents the structure of
the sentence John may eat apples.
(ii) IP
/|
NP I'
| |\
| I VP
John | |
| V'
may |\
V \
| NP
eat |
|
apples
The nodes with the category labels NP, IP, VP, I', V', V, I in (ii) are non-terminal
nodes. The words John, may, eat, apples are the terminal nodes in this tree.
Originally, tree structures were °generated by
°phrase structure rules, or by the
transformational rules that map the distinct levels of representation
(°d-structure,
°s-structure,
°LF,
°PF) onto each other. All binary relations (such as
°c-command,
°sisterhood) are defined over trees.
Another term for tree structure is Phrase marker or P-marker (although a Phrase
marker is formally different from a tree).
Truncation
MORPHOLOGY: a morphological operation by which one morpheme is deleted if
it is internal to another suffix, in the following general manner:
(i) Truncation
[[ base + A]X + B]Y
1 2 3 -> 1 0 3
where X and Y are major lexical categories, and A and B are affixes
As a consequence of this operation, the two affixes in question cannot co-occur.
Truncation is an alternative to °affix
substitution, and is proposed for similar reasons. Both types of operation are
necessary in a °word-based
morphology, since one often finds regularly derived words which are semantically
transparent and formed with productive affixes, although on the surface they do not
appear to have been derived from words, but from morphemes. The English suffix
-ee attaches to transitive verbs (employ:employee, pay:payee).
Although pairs such as nomin+ate:nomin+ee, evacu+ate:evacu+ee are
semantically related, the nouns ending in -ee lack the verbal suffix
-ate, and if it is assumed that word formation rules can only take words
as their base these forms are problematic. Aronoff (1976) solves this problem by
allowing for a truncation rule that deletes -ate if it is followed by
-ee, as in (ii):
(ii) [[ root +ate]V +ee]N
1 2 3 -> 1 0 3
LIT.
Aronoff (1976),
Scalise (1984),
Spencer (1991).
Truth condition (definition)
°Truth value.
Truth functional connective
°Connective.
Truth table
SEMANTICS: the device by which the truth conditions of a complex
propositional formula can be represented. By means of truth tables it is possible
to define the °connectives of
°propositional logic. See
°conjunction,
°disjunction,
°negation,
°equivalence and
°implication.
LIT.
Gamut (1991).
Truth value
SEMANTICS: the property that is assigned to sentences (or propositions or
formulas) in truth-conditional semantics. A sentence can be true (also 1 or T) or
false (also 0 or F) in a two-valued °logic, but
there are more truth-values in more-valued logics. Truth conditions (or truth
definitions) specify in which circumstances a proposition is true, relative to a
°model, i.e., an assignment of values (a valuation)
to the basic expressions that constitute the proposition.
LIT.
Gamut (1991).
Type logic
SEMANTICS: logical system based on Russell's theory of types. Every
expression of a type-logical language belongs to a particular type indicating
the set-theoretical denotation of that expression. There are two basic types,
the type e (from entity) and the type t (from
°truth value). The formulas of
°predicate logic and
°propositional logic are expressions
of type t in type logic, denoting truth values; the
°individual constants of predicate
logic are expressions of type e in type logic, denoting individuals. All other
expressions in type-logic are functional, i.e. they take an expression of type a
as their argument and yield an expression of type b, which is indicated in their
type as follows: <a,b>. The one-place predicates of predicate logic are of type
<e,t> in type logic, denoting a function from entities to truth-values, which is
another way to define a set. Two-place predicates are of type <e,<e,t>>.
Type logic also allows functions of higher order. Noun modifiers can be treated as
expressions of type <<e,t>,<e,t>>, mapping a set into a set. NPs
are of type <<e,t>,t>, i.e. functions from sets to truth values, or
equivalently, sets of sets. Determiners are relations between sets:
<<e,t>,<<e,t>,t>>. In combination with lambda-abstraction,
type logic is a very powerful logic for semantic representation. It has been
fruitfully applied in °Montague Grammar.
LIT.
Gamut (1991),
Montague (1974).