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Lesson 1 - 09/
SENTENCE SEMANTICS
In order to have a sentence, 2 components are needed: nouns and verbs. If there’re sentences without nouns or verbs, they’re elliptical sentences, since nouns and verbs are implicit. We have this structure because:
- nouns refer to objects, things and individuals
- verbs refer to events or states, where events are something that happens, while states are represented by the fact that objects have some properties, relations And so, objects are the substratum of properties, relations, events. ARGUMENTS The objects involved in the eventuality (hypernym of states and events) denoted by the verbs are often arguments of the verb. A verb can have one or more arguments.
Impersonal verbs ( to rain) can be interpreted as zero-argument verbs. They have a
subject, but it doesn’t refer to any object. This can be questioned, but here we’ll assume it for simplicity.
The term argument comes from the fact that verbs can be seen as functions from
objects to truth-values. This will be clarified in the following. VALENCY The number of the arguments of a verb is called valency.
For instance, to walk has valency 1, to hit has valency 2, and to offer has valency 3,
etc.
Sometimes, verbs of valency 1 are called monovalent , verbs of valency 2 are called
bivalent , verbs of valency 3 are called trivalent , etc.
STATES AND VALENCY
- Some stative verbs have valency 1: Ann slept / John is blonde
- Other stative verbs have valency 2: I believe that it’s not over / John loves Ann
- Still other stative verbs have valency 3: a is the middle between b and c / This town
is 10 km from Milan
EVENTS AND VALENCY
- Some eventive verbs have valency 1: Ann ran to the shop / The dog barks every
night
- Other eventive verbs have valency 2: John reached the top of the mountain / Ann
ate some bread
- Still other eventive verbs have valency 3: John gave the book to Ann / The pupil
asked a question to the teacher
STATIVE/EVENTIVE DISTINCTION AND VALENCY
The stative/eventive distinction is independent of the valency of the verb:
- States can involve one, two or three objects
- Events can involve the same quantities of objects Stative monovalent verbs mostly predicate a (relatively static) property of an object.
However, other stative verbs, such as to stay or to sleep are not directly
interpretable as predicating such properties. However, an analysis of these verbs in such terms can be advanced.
For instance to stay can be analyzed as: to durably possess the property of being in
the point x.
By contrast, stative bivalent and trivalent verbs can be seen as predicating a certain relation between two or among three objects.
For instance, if John loves Ann, John and Ann have a certain relation. If a is the middle
between b and c, a, b and c have a certain relation.
To believe can be interpreted as a relation between an individual and a proposition. To
feel can be interpreted as a relation between an individual and a certain feeling.
This supports the interpretation of states as the possession of properties by an object or as the existence of a relation among some objects. VALENCY HIGHER THAN 3?
Perhaps there’re verbs with valency higher than 3. One might argue that to sell has
valency 4. To sell something there must be:
- a seller
- a buyer
- the thing that is sold
- the money that the buyer gives to the seller In general, it’s better not to put a limit to the number of arguments. However, verbs with valency >3 are rare. ADJUNCTS Sentences often contain NPs that are not arguments of the verbs.
Consider “Ann walked with John under the sun”: we have 3 NPs (Ann, John, the sun),
John and the sun are NPs and refer to individuals. However, they’re not arguments of
the verb. To walk has valency 1 and its sole argument is Ann.
The NPs (and the Prepositional Phrases - PP) that are not arguments are often called adjuncts. The main idea is that they’re “ optional ” and “ non-obligatory ” phrases. CORE AND ADJUNCTS Sentences can be divided into 2 parts: core and adjuncts:
- The core is the verb with its arguments
- The adjuncts are all the NPs and PPs that are not arguments of the verbs The core is the essential part of the sentence; the adjuncts are the optional part. While the valency of a verb is fixed, there’s no limit to the number of adjuncts that a sentence can have. One of the reasons why there’s no limit to the length of sentences is that the number of adjuncts is indeterminate. THE DISTINCTION BETWEEN CORE AND ADJUNCTS Sometimes it’s difficult to understand whether a NP or PP is part of the core, or it’s an adjunct. This means that it’s sometimes unclear which is the valency of the verb.
Take to cut: is the tool by which one cuts something an argument of the verb? In other
words, is this verb bivalent or trivalent? If it’s trivalent, then “ with a knife” is part of
the core: Ann is cutting the cake with a knife. Otherwise, it’s an adjunct.
GRAMMATICAL CRITERION
It’s possible to propose some criteria to distinguish arguments from adjuncts. One might argue that the arguments of a verb must be referred in a sentence that contains that verb. Adjuncts need not. This is a grammatical criterion : sentences must have a certain structure. However, this criterion doesn’t work. Arguments are not
necessarily mentioned in sentences: “ What’s Ann doing? She’s eating”
Other criteria were proposed by other linguists and philosophers. The schema is this one, we have eventualities that can be distinguished into:
- states : possession or property by an individual or relations among individuals; here nothing changes
- events : here something changes; we can distinguish events into: telic events = events that have an aim, a goal, processes where there’s no goal Telic events can be distinguished into accomplishments and achievements , depending on whether they have duration in time or not: accomplishments have a duration, they last a certain amount of time, while achievements are instantaneous, they’re events that happen instantaneously. STATES
- Copula + adjective, copula + NP, to be + PP, verbs such as: to remain, to believe, etc
- They don’t usually accept the progressive form : “ I’m believing that …", “It’s
being yellow”, “I’m being in the garden”
- They’re not agentive verbs : they don’t describe any action of the subject. We don’t perform the action of being blonde or the action of being in the garden. Being agentive means that something is in progress, is changing but here nothing
changes. ( To walk is agentive, because it causes something, it has an effect)
- They’re not telic , since there’s no aim here. If an individual is in state s for the time
interval t, that individual is in s at every instant of t
- State verbs accept the modifier for a time t. Ann was blonde for more than a year: this modifier measures the duration of the state, and, in general, of the eventuality denoted by the verb
- State verbs don’t accept the modifier in a time t. Ann was blond in a year: this modifier measures the time needed to reach the aim. However, states are not telic and, thus, they have no aim
- In certain contexts, state verbs can accept the modifier “ in a time t ”. John was
tired in two minutes.
However, in these cases, the modifier doesn’t measure the time needed to reach the aim of the state (there’s no aim in being tired). Rather, the modifier measures the time needed to reach the state of being tired. The sentence means that it took two minutes to John for being tired, i.e., that the process that led John to be tired lasted two minutes. This process has an aim (the state of being tired itself). PROCESSES
To walk, to run, to talk, etc are verbs that don’t change and don’t have a goal.
The processes are events , i.e. a kind of change. However, they’re not telic. Not having an aim means: the aim is not part of the meaning of the word, there’s no aim embedded in the meaning of the process verb.
If we say to walk, in its meaning there’s no aim, while if we say “ to build a house”,
there’s an aim.
In principle, the change they denote could go on indefinitely. In theory, one could go on talking indefinitely. Compare with reading a book. This doesn’t mean that at a certain point one doesn’t stop talking. Of course, one does. But no precise aim is
included in the meaning of to talk.
Processes are usually composed of a series of small cyclic changes. For instance, walking is composed of: put a leg in front, put the other leg in front, etc. This explains why in principle they can go indefinitely: we can always add another cycle. The number of cycles is not defined in advance. Because they last over time, process verbs like state verbs accept the modifier: for
a time t
Ann walked two hours
To walk is necessary at least a complete cycle : to put a leg in front, to put the other
leg in front. Because they denote changes that last a certain amount of time, process verbs accept the progressive form.
Ann is walking —> it means that a certain process is occurring in the present
Since they’re not telic, process verbs don’t accept the modifier: in a time t.
Ann walked in two hours: again, the modifier doesn’t measure the time needed to
reach the aim of the process (there’s no aim in walking). Rather, the modifier measures the time needed to reach the start of a walking process. For example, Ann was paralyzed and she took two hours to walk again. Process can be agentive, but they don’t need to be. ACCOMPLISHMENTS
They’re not usually denoted by verbs but by verb phrases (VPs) : to build a house, to
read a book.
Unlike processes, accomplishments are telic. They’re changes that last an amount of time and that have an aim. This explains why they’re usually denoted by a VP (V + NP or V + PP). The NP or PP denotes an objects that gives an aim to the change and “measures” it. Building is a process, but building a house is an accomplishment: the house gives an aim to the process of building and measures this process. The state of the object or the relationship between the subject and the object tells us how close the accomplishment is to its aim. We can say how close the builder is to his/her aim by observing the state of the house. If we add a NP or a PP to a process verb, we often obtain an accomplishment verb:
To read a book / To run to the shop / To build a house
However, this is not always the case. Sometimes, an operation on an object can be
repeated indefinitely: To drive the car / To carry the water
- Being telic, these verbs accept the modifier in a time t : Paul built his house in a
year
- Being extended over time, they accept the modifier for a time t : Paul built his
house for a year
However, while:
- “in a time t” measures the time the accomplishment takes to reach its aim
- “for a time t” simply says that the activity lasts for a certain period, it’s neutral regarding the reaching of the aim
- Accomplishments are changes that have some duration in time and so they’re telic and accept the progressive form
- Achievements are telic since there’s an aim, they don’t accept the progressive form because they don’t change and there’s no duration, so we cannot say that something is happening now
Lesson 2 - 10/
THEMATIC ROLES
The arguments of verbs are not alike.
Ann bought an ice-cream: Ann and an ice-cream are the arguments of the verb to buy.
However, Ann and the ice-cream fill different roles in the event of buying: Ann is the
buyer , the ice-cream is the thing that is bought.
The different roles played by the arguments are usually called thematic roles. A thematic role is the role played by an object in the state or event denoted by the verb. We can consider thematic roles as the basis under which an object participates in a state or event. Alternatively, we can understand thematic roles as a relation between the object and the eventuality in which it participates. The thematic grid (or θ-grid = theta-gird) is a grid related to a verb that lists the thematic roles that the arguments play in the event or state denoted by that verb:
- Run —> AnnAgent is running
- Eat <Agent, Patient> —> AnnAgent is eating the ice-creamPatient
- Give <Agent, Patient, Recipient> —> AnnAgent has given the bookPatient to JohnRecipient In knowing the meaning of a verb, a speaker also knows the thematic grid of that verb. The knowledge of the thematic grid of a verb is a fundamental component of the knowledge of the meaning of that verb. One cannot know the meaning of a verb without knowing the thematic grid of that verb. However, there’re some problems regarding thematic roles:
- how many thematic roles are there?
- which are they?
- how do the thematic grid project onto the syntactic structure of the sentences? HOW MANY THEMATIC ROLES ARE THERE? It’s difficult to compile a complete list of thematic roles. The list risks to be too coarse or too fine grained. To give an idea of these difficulties, let us consider just the thematic roles that the referents of a subject can fulfill:
1. Ann ate the salad
2. John loves Ann
3. The rock, falling from the cliff, destroyed the car
4. Ann stumbled in her shoes
In eating something, Ann is playing the classical role of an agent. However, we cannot say the same of the subjects of the other sentences. In loving someone, we don’t properly perform an action. Maybe we wouldn’t like to love someone. It’s not voluntary, it’s something that happens to us. Is John performing an action? Is loving an action? For the moment, there’s a difference between the two sentences: Ann performs an action in willing do that action, while the case of John is not voluntary. The same remarks apply to the rock in 3. The rock does something, but the rock has not the will to destroy the car and it’s not voluntary. So, we wouldn’t say that the rock performs an action.
Difference between 2 and 3: John has a mind, a feeling, while the rock has no feeling. Again, if someone stumbles, s/he doesn’t do this at purpose. Stumbling is not a voluntary action. Since they don’t do actions, John, the rock and Ann are not agents in 2 and 4. However, there’re differences among the roles that these individuals fulfill:
- the rock causes something in 3. It has some effects : the destruction of the car
- John and Ann don’t do something that has effects on other things
- love is something we feel. Loving implies the existence of a mind It seems that the classification of the roles should be neither too generic nor too specific, but it’s hard to individuate an adequate level of specificity. ATTEMPTS OF SOLUTIONS: DAVID DOWTY According to Dowty, there’re just 2 thematic roles: the agent and the patient , that are defined by a set of features. The agent is:
- intentional (volitional involvement in the eventuality)
- somebody who feels, i.e. sentient (and/or percipient)
- cause of a change in another participant
- something moving in relation to the other participants
- independent ; its existence is independent of the existence of the other participants The patient has the opposite features. The typical patient:
- is not intentional
- is not sentient being
- doesn’t cause but is causally affected by other participants
- is static in relation to other participants
- its existence is dependent on that of other participants The main thesis of Dowty is that these roles are prototypical. There’re some individuals that prototypically fulfill the role of the agent or that of the patient. This means that they have all the features that characterize these roles. For instance, Ann in 1 is a prototypical agent, she is a subject who intentionally and consciously causes a certain effect. Her existence is independent of that of the salad, and she moves independently of it.
By contrast, in the sentence “ John built his house”: house is a prototypical patient, it
doesn’t cause anything, it’s not intentional and it’s not sentient. Its existence is dependent on that of John and doesn’t move independently of him.
- There’re some individuals that have some but not all the features of the prototypical agent. They’re non-prototypical agents
- There’re some individuals that have some but not all features of the prototypical patient. They’re non-prototypical patients
- There’re some individuals that have both the features of the agent and of the patient. They’re in the middle between the category of the agent and that of the patient
rule 2):
The torpedo sank the ship
In this case, only 2 arguments of the verb are mentioned. Nemo is not mentioned. However, the rules are the same. We apply rule 2, which states that the argument having the greatest number of features of the agent is lexicalized as the subject and the argument having the greatest number of features of the patient is lexicalized as the object.
The ship sank
In this case, only one argument is mentioned. Nemo and the torpedo are not mentioned. However, the rules are the same. We apply rule 1, which says that the sole argument of the verb is lexicalized as the subject.
Reduction is not always possible : Ann ate the salad vs The salad ate
Sometimes the arguments of a bivalent verb have both some features of the agent and some features of the patient. In these cases, which arguments is projected onto the subject and which onto the object can depend on the verb and on the language.
John fears thunders
Thunders scares John
Thunder —> cause, movement (but not intentional or feeling) John —> feeling (but he’s causally affected by the other participant)
I like chocolate ice-cream vs Mi piace il gelato al cioccolato
The 2 arguments behave in different ways. English and Italian project the two arguments differently (the Italian amare, on the contrary, behaves as the English to love or to like). I —> feeling (but I’m casually affected by the other participant) Chocolate ice-cream —> cause (but not intentional, not feeling)
The difference with fear is that in this case the feeling can be appreciated by the
psychological subject, while fearing is usually unwelcome. ATTEMPTS OF SOLUTIONS: TANYA REINHART Tanya Reinhart distinguishes more thematic roles, 4 roles, it’s a more fine grained distinction and it’s more rigid since it’s not prototypical. Tanya Reinhart takes a different approach: she believes that thematic roles are characterized by 2 fundamental traits: mental and causal (m and c):
- mental implies both intention and feeling
- causal is the same as in Dowty A role has these traits or not. Thus, we have 4 possible combinations and 4 different roles:
1. mental and causal [+m, +c], agent
2. mental but not causal [+m, -c], sentient
3. causal but not mental [-m, +c], instrument
4. neither mental nor causal [-m, -c], patient
With respect to Dowty:
- Reinhart restricts the relevant features to only 2 : mental and cause. Other features, such as movement or independent existence, are not considered
- Reinhart adds new roles ; her classification is finer grained than Dowty’s. What Dowty considered as intermediate cases between the agent and the patient are seen by Reinhart as new roles different from both the agent and the patient
Therefore, Reinhart’s classification is more rigid than Dowty’s. It’s a classification based on traits , not a prototypical classification.
1. Ann ate the salad
2. John loves Ann
3. The rock, falling from the cliff, destroyed the car
4. Ann stumbled in her shoes
- Ann in 1 is an agent, she has both the mental and the causal traits [+m, +c]
- John in 2 is a sentient, he’s [+m, -c]
- The rock in 3 is an instrument, it’s [-m, +c]
- Ann in 4 is a patient, she is [-m, -c] Ann in 4 might seem to be a problematic case: she causes nothing, but she has mental activities. However, Reinhart explains that -m means that no mental activity or intentionality is required for participating as patient to the eventuality denoted by the verb. Ann has mental activities and is an intentional being. However, to stumble no intentionally and no mental activity are required. So, Ann can be classified as the patient of the eventuality even though potentially she’s an intentional being and even through she has feelings. The projection rules are similar to Dowty’s:
- If the verb is monovalent, the unique argument is lexicalized as subject independently of its traits
- If the verb is bivalent and one argument is [+m, +c] and the other argument is [-m, -c], then the agent is lexicalized as the subject and the patient as the direct object
- When the arguments are [+m, -c] (sentient) and [-m, +c] (instrument), we have more complex cases. In these cases, there’s no fixed rule, and which argument is projected onto the subject and which onto the object (or indirect object or PP) depends on the verb and on the language
The letter worried Paul
Paul worried about the letter
Both these sentences are grammatical. Since Paul is a sentient and the letter is an instrument, potentially they can both be lexicalized as subjects. Sometimes, the structure depends on the verb:
Ann enjoyed the music / The music amused Ann
The water filled the swimming pool / The swimming pool filled with water
Probably, both the arguments are [-m, -c]. The water doesn’t directly cause the filling because the water is passive and directed to the swimming pool by a pump. In this
case, it’s also unfixed which argument must be the subject. Again, with fill both
arguments can be subjects. The treatment of reduction is similar to Dowty’s. When less arguments of the verb are mentioned, then it’s as if the verb has less arguments and the subsequent rules apply. For instance:
The wind opened the door / The door opened
strictly related to the knowledge of the world. This is one of the central tenets of cognitive semantics. A frame is a structure of concepts that grasps a certain portion of the world. One of Fillmore’s preferred examples is the frame that grasps a commercial transaction. The frame outlines the basic elements of the transaction and their relationships. For instance, in a commercial transaction at least 4 elements are involved:
1. The seller
2. The buyer
3. The money
4. The good
A frame describes the situation, the objects involved, the roles they play and what happens. Of course, the roles are connected to the thematic roles but are more specific. A frame can be interpreted as a conceptual representation of the scene in which something happens. The scene includes what happens: the eventuality and the individuals that participate to that eventuality. In the scene, the relationships among the individuals that participate to the eventuality are clarified and displayed. A frame can be seen as a structure of connected concepts that describe the scene in which the eventuality occurs. Therefore, the frame is a sketched representation of a portion of the world. Words, in particular verbs, evoke frames. Without knowing the frame evoked by a verb, a speaker cannot know the meaning of the verb.
A frame can be evoked by more than one verb. For example, sell, buy and pay evoke
the same frame: the frame that outlines a commercial transaction. However, these verbs highlight some aspects of the frame and overshadow others. Of course, the role of the buyer has something in common with thematic roles in general and with the role of the agent in particular. However, it’s clearly more specific. The role of the agent concerns a very high number of verbs. By contrast, the role of the buyer concerns specific verbs: the verbs that evoke the frame of commercial transactions. Likewise, the elements of a frame and their roles have something in common with thematic grids. Again, however, the grid of a frame is much more specific than its thematic grid. The concept of frame is intended to describe the complete meaning of a verb. By contrast, the concept of thematic grid hasn’t this aim. Another example of frame is that schematizes the “judging scene”. The elements of this frame are:
- the judge, the person who forms and expresses some sort of judgment on other people or on a situation
- the defendant, a person whose behavior is relevant for the judge to make a judgment
- the situation, the situation about which it’s relevant to make a judgment
This frame is also evoked by several verbs like to accuse, to criticize and to blame.
Lesson 3 - 30/
FRAMENET
FrameNet is a computational resource based on the concept of frame. The project is based at the International Computer Science Institute (ICSI) in California and it was founded by Charles Fillmore in 1997 and directed by him until his death in
An entry of FrameNet consists of various elements:
- A definition of the frame, which describes the scene
- A sentence, in which the various elements of the definition are present
- A set of core frame elements (FE)
- A set of non-core FE
- Frame relations
- Lexical units DEFINITION The first element of an entry of FrameNet is a definition : it provides a textual description of the frame and, thus, of all involved elements.
For example, the definition of the frame connected to sell is: These are words
describing basic commercial transactions involving a buyer and a seller exchanging money and goods, taking the perspective of the seller. The words vary individually in the patterns of frame element realization they allow. For example, the typical patterns for SELL: SELLER sells GOODS to BUYER for MONEY EXAMPLE The second element of an entry of FrameNet is an example sentence that contains the words referring to the individuals mentioned in the definition. These words are colored differently for a better reference. For instance, the example sentence of sell is: The verb is the unique word in capital letters. The sentence refers to the seller, to the good, to the buyer and to the money, which are the individuals mentioned in the definition. The seller is marked in red, the buyer in blue, the good in green, etc. FRAME ELEMENTS Frame elements are what we have called arguments and adjuncts:
- arguments are called core FEs
- adjuncts are called non-core FEs So also here in FrameNet, we have a verb which denotes an event or a state and some arguments or adjuncts denoted by the NPs in the sentence. The arguments play different roles, but the difference is that here we have a more fine grained classification of roles.
- Precedes : the precedes relation captures a temporal order that holds between subframes of a complex scenario. For instance, in the criminal process frame, the frame Arrest precedes the frame Trial LEXICAL UNITS The lexical units are the words that evoke a certain frame. For instance, the words
auction (noun) , auction (verb) , retail, retailer, sale, sell, seller, vend, vendor all evoke
the frame of commercial transactions. These words foreground some elements of the frame and background other elements. An entry of FrameNet lists the lexical units that evoke a certain frame, and gives some information on their status (when they were created, the ID of the annotator who created them, the date of creation, etc) SEMANTIC ROLE LABELLING FrameNet is an example of semantic role labelling. Texts can be annotated at various levels: morphological, syntactic, semantic:
- at the morphological level : the words of a text are assigned to their canonical form and information about their inflection is added
- at the syntactic level : the annotation concerns the syntactic structure of the sentences of a text and the syntactic role of the words of a text
- at the semantic level : the annotation concerns the meanings of the words of a text. This can be done in many ways. One of them is to add information about the thematic roles (or more specific roles) of the referents of NPs
LOGIC
WHY LOGIC?
Logic studies correct reasoning. A reasoning is correct when preserves the truth (if the premises of the reasoning - i.e. the argument - are true, the conclusion is also true). Formal logic studies the procedures by which the verification of the correctness of reasoning can be rendered automatic. It regiments the sentences that constitute the premises and conclusions of our reasoning by organizing them into fixed schemas. It then identifies the valid schema of inference. In this way, formal logic has constituted the first form of the automatization of thought. For this reason, those who developed programming languages created them based on the formal languages of logic. It’s therefore useful for computational linguists to have some knowledge of logic and formal semantics. COMPOSITIONAL SEMANTICS In order to sketch a semantics of sentences, we must assign meanings to the words of the sentence and then compose these meanings into more and more complex meanings. Natural languages are compositional : the meanings of larger constituents depend on the meanings of smaller constituents. This composition follows the syntactic structure of sentences. When two constituents syntactically merge, their meanings compose with each other.
For instance, it’s because a noun and adjective syntactically merge that we must compose the meanings of the noun and of the adjective. Meanings must be assigned to the leaves (words) of the syntactic tree. Any time two constituents of the tree merge and a larger constituent is created, the meanings of the two constituents must be combined into a more complex meaning that depends on the meanings of the two constituents. The semantic structure mirrors the syntactic structure. This means that there’s a morphism between the syntactic structure and the semantic structure. This morphism is at least a homomorphism : that is, if M is the function that maps a string onto its meaning, R is the syntactic composition of two strings s1 and s2, S is their semantic composition, we have: M(s 1 R s 2 ) = M(s 1 ) S M(s 2 ) The homomorphism presupposes that a given semantic type is assigned to any syntactic constituent of a certain kind. It’s because nouns and adjectives have a certain kind of meaning that they can be combined. In other words, the meanings of nouns and adjectives must be such that their composition is possible. In principle, we should always be able to combine the meanings of nouns and adjectives. Therefore, in principle, the meanings of nouns and adjectives must be combinable. This is so because the type of meaning of nouns is combinable with the type of meaning of adjectives. Homomorphism implies that the same semantic operation is always associated to a given syntactic operation. Adjectives have a certain kind of meaning. Likewise, nouns have a certain kind of meaning. Therefore, when A and N merge, the semantic operation of combination of their meanings is always of the same kind. It’s the combination between a certain kind of meaning with another kind of meaning. Therefore, it’s the same kind of combination. If the semiotic triangle is a valid schema, the task of compositional semantic is:
- to connect a type of concept and a type of objects to any syntactic type (syntactic category)
- to connect a composition of concepts and a composition of objects to any syntactic composition Since this task doesn’t envisage to connect single meanings to single words (this is the task of lexicography), but types of meanings to syntactic types, the meanings connected to syntactic types will necessarily be abstract. LOGIC, MATHEMATICS, SEMANTICS
• By definition, a set is always included in itself because a set has all its members.
Therefore, A ⊆ A is true by definition
• If A includes B, then A is a superset of B and B is a subset of A.
If A⊆B and B⊆A, then A=B. In this case, every element of A is an element of B, and
every element of B is an element of A. Therefore, A and B have the same members
• If B is included in A, but it’s not identical to A, then B is a proper subset of A. Their
relation is written as B ⊂ A. If B is included in A but it’s identical to A, B is an
improper subset of A. Therefore, every set is an improper subset of itself
• If B⊆ A and B≠A, then B ⊂ A.
The set of cats is a subset of the set of animals, the set of regular polygons is a subset of the set of polygons, etc. The set of cats is an improper subset of the set of cats and a proper subset of the set of animals. We can define some operations on sets. An operation is a function that takes one or more sets as input and gives a set as output. Union is an operation that takes two (or more) sets as input and gives as a result the set that contains all the elements of these sets.
This operation is indicated by ∪. For instance, A ∪ B denotes the union of set A and of
set B, that is the set that contains all the elements of A and B.
For example, {a, b, c} ∪ {d, e} = {a, b, c, d, e}.
{a, b} ∪ {b} = {a, b} because {a, b, b} = {a, b}.
Intersection is another operation, that takes two (or more) sets as input and gives as output the set of all elements that are contained in all these sets. The intersection of two sets can be empty.
This operation is indicated by the symbol ∩. Then, A ∩ B indicates the intersection
between A and B. For instance, {a, b, c} ∩ {a, b, d} = {a, b}
• Union and intersection are commutative operations, that is A ∪ B = B ∪ A and A ∩ B
= B ∩ A
• Union and intersection are also associative operations:
- (A ∪ B) ∪ C = A ∪ (B ∪ C)
- (A ∩ B) ∩ C = A ∩ (B ∩ C)
Complement is an operation that takes a set as input and gives a result all the elements that don’t belong that set.
The complement of A is indicated by A’ or by Ā. The absolute complement of A is the
set of all elements of the universe that aren’t in A. If the set of these elements is
indicated by U , then A’ denotes the set of all elements of U that don’t belong to A.
The relative complement of set A with respect to another set B is the set of all
elements of B that don’t belong to A. This set can be indicated by B - A.
(A’)’ = A
SINGLETON AND EMPTY SETS
Mathematicians have introduced two types of special sets: singletons and the empty set :
• a singleton is a set that contains just one element. For instance, {b} is a
singleton.
b ≠{b} because b is an element while {b} is a set. A singleton is a strange
mathematical entity
• the empty set is the set that doesn’t contain any element. There’re as many
singletons as the elements of U, but there’s only one empty set because sets are
defined by their elements. The empty set is indicated by ∅. It’s included in any other set. Therefore, for any A,
it’s true that ∅⊆ A. This can be counterintuitive, but it follows from the definition:
any element of ∅ is included in A because ∅ has no element
EMPTY SET: PROPERTIES
SETS WITH MANY ELEMENTS
When a set has many or infinite members (for instance, the set of all men or the set of natural numbers), listing all its members can become inconvenient or impossible. So, it’s useful to have another method to represent a set. The members of the set are
identified by a property that they share. {x: x is a man} denotes the set of elements
x such that x is a man, that is the set of all things that are men, that is the set of all men. DEFINITIONS OF SET We have 2 methods for representing a set: the enumerative and the “semantic” method. The enumerative one is an extensional definition, the second one is an intensional definition.
The second one defines a subset of U. In {x: x is a man}, x varies on the elements of
U.
The definition characterizes a set as a selection from a larger set (U), by specifying a condition, that is a property that discriminates the elements of the subsets from the elements of the absolute complement of the subset.
Lesson 4 - 01/
FUNCTION
Another important mathematical notion used by formal semanticists is that of function. This notion is more complicate than that of set, but it’s useful because the composition of the meanings of two syntactic nodes is interpreted through it. The basic idea is that every time two syntactic nodes combine into a new node, from the semantic point of view, a function takes an argument and gives a value. In other words, when two nodes combine, one of them can be interpreted as an operator and the other one can be interpreted as an operand. Since the result of the function is unique, it can be seen as the result of the composition of meanings. ORDERED SETS A set is ordered if it’s a set and there’s an order among the elements of the set. An ordered set is defined by its elements and their position within the order.
A notation for ordered sets is: <a, b, c …>
The ordered pair <a, b> differs from the set {a, b} because, while in sets the order of
the elements is irrelevant, in ordered pairs the order matters. Then, while: {a, b}
