4. LANGUAGE AND LOGIC

Outline

James Fieser, UT Martin

updated 9/26/2025

This chapter explores some of the logical aspects of statements in natural languages such as English. Natural languages often lack precision, which can make it difficult to evaluate arguments clearly. Logic helps address this problem by offering tools that clarify how we speak and reason. It points out patterns in our use of language, showing how sentences can be misunderstood, and explaining how different kinds of disputes arise. When we understand how logic applies to everyday language, we are better able to explain our views and spot confusion in conversation.

A. FOUR GRAMMATICAL FORMS OF LANGUAGE

We express ourselves in sentences, where a sentence is an utterance that conveys a complete thought. In both speech and writing, sentences take on different forms depending on what the speaker aims to do, namely, state a fact, ask a question, give a command, or show emotion. Grammarians commonly designate four types of forms that our sentences take: declarative, interrogative, imperative, and exclamatory.

1. Declaratives: such and such is the case.

e.g., "the time now is 2:00"

e.g., "you are standing on my foot"

e.g., "I have a headache"

e.g., "it looks like rain"

e.g., "the window in the kitchen is broken"

e.g., "I haven t eaten since breakfast"

2. Interrogatives: questions.

e.g., "what time is it?"

e.g., "is there a reason why you are standing on my foot?"

e.g., "can you give me something for this headache?"

e.g., "is it going to rain today?"

e.g., "did you notice the kitchen window is broken?"

e.g., "can we get something to eat soon?"

3. Imperatives: commands, do X.

e.g., "get that done by 2:00"

e.g., "get off my foot"

e.g., "please give me something for this headache."

e.g., "grab an umbrella before you leave"

e.g., "fix that window today"

e.g., "make me a sandwich, please"

4. Exclamatories: exclamations.

e.g., "2:00!"

e.g., "my foot!"

e.g., "oh, my aching head!"

e.g., "rain again!"

e.g., "the window!"

e.g., "I m starving!"

B. FOUR FUNCTIONS OF LANGUAGE

Understanding the function of language helps us better analyze what speakers actually mean, which is not always revealed by the grammatical form of their sentences. The function of language is how we use sentences to accomplish different purposes, and there are four such functions: to inform, to express, to direct, and to perform. The four above forms of sentences are not identical to these four functions, and, each of the four forms can be used for several different functions. Imagine, for example, that the waiter at a restaurant was a robot which was programmed to understand the four forms of sentences, but not their functions. You then say "I d like a cup of coffee." This is in the declarative form and, for all the robot knows, you are just informing it of the fact that you would like a cup of coffee. So the robot would just stand there. But behind the declarative form of your utterance is a directive for the robot to perform some action, namely, to get you a cup of coffee. It is the implied meaning of your utterance that is central to the function of language, regardless of what form your utterance is.

1. Informative function (assertions): to communicate information, such as affirming or denying propositions, or presenting arguments. The informative function is often, but not always, in the form of declarative sentences, and here are examples in all four forms, where in each case the primary function is to convey information:

Examples in declarative form: "I have a headache", "Water boils at 100 degrees Celsius", "The meeting starts at 3:00".

Examples in interrogatory form: "Did you know that I have a headache?", "Is it true that whales are mammals?", "Have you heard that the library is closing early?"

Examples in imperative form: "Stop making my headache worse!", "Remember that we have a quiz tomorrow", "Tell her I ll be late".

Examples in exclamatory form: "Oh, my aching head!", "What a complicated system that is!", "How cold it is in here!"

2. Expressive function (emotives): to vent or arouse feelings. The expressive function is often, but not always, in the form of exclamatory sentences, and here are examples in all four forms, where in each case the primary function is to vent or arouse feelings:

Examples in declarative form: "That s too bad", "I m so happy to see you!", "I can t believe how awful that was".

Examples in interrogative form: "Can you believe that?", "Why would anyone do such a thing?", "Are you kidding me?"

Examples in imperative form: "Go to the devil", "Leave me alone", "Don t even start with me".

Examples in exclamatory form: "Oh my goodness!", "What a disaster!", "Unbelievable!"

3. Directive function (commands): to influence the actions or decisions of another person. The directive function is mostly in the form of interrogative and imperative sentences, but also can be in declarative an exclamatory form, and here are examples in all four forms, where in each case the primary function is to bring about some overt action.

Examples in declarative form: "You are on my foot", "The door is still open", "Your phone is ringing"

Examples in interrogative form: "Can you get off my foot?", "Could you please move your car?", "Would you mind closing the window?"

Examples in imperative form: "Get off my foot", "Turn off the lights", "Keep your voice down"

Examples in exclamatory form: "My foot!", "The lights!", "Quiet!"

4. Performative function: to carry out an action through the act of speaking itself. The clearest examples of performative utterances are those that begin with the phrase "I hereby," which signal that the speaker is doing something by speaking, such as making a promise, issuing a warning, or declaring a marriage. For example: "I hereby resign from the committee." Technically, all speech functions are performative in that they accomplish something through language. For instance, "I hereby inform you that the door is brown" performs the act of informing (informative function), and "I hereby request that you shut the door" performs the act of requesting (directive function). However, performative utterances in the narrow sense focus especially on an action that is accomplished by the words themselves, where the words do not merely describe the action but constitute the very action. The performative function is most often found in declarative sentences, but can also appear in interrogative, imperative, and exclamatory forms. In each case, the primary function is to accomplish something directly through speech.

Examples in declarative form: "I promise to return this", "I now pronounce you husband and wife", "You have the right to remain silent", "I claim this continent in the name of Lord Keg-Chugger", "I confer on you all the rights and benefits thereof", "I call firsties!"

Examples in interrogative form: "Do you take this woman to be your lawfully wedded wife?", "Shall we begin the ceremony?", "Will the court please come to order?"

Examples in imperative form: "Satan be gone!", "Let the games begin!", "Cease and desist!"

Examples in exclamatory form: "Bingo!", "Present!", "I quit!"

5. Historical Note: Logical Positivism and ethical emotivism

A movement in twentieth-century ethical theory called emotivism draws on the critical distinction between the form and function of language. It maintains that declarative ethical statements such as "John is good" in fact have no informative function, but instead have merely expressive and directive functions. That is, "John is good" merely expresses the speaker s emotional approval of John, essentially meaning "hooray for John", and it also directs others to approve of John s conduct, essentially meaning "you too should approve of John". Similarly, the declarative statement "Abortion is wrong" means "boo for abortion" (expressive function) and "you should disapprove of abortion" (directive function). The theory of emotivism had its defenders, especially A.J. Ayer, but it later declined in popularity because it failed to adequately account for the apparent factual and rational components in moral discussions. People often argue about ethical claims as though they were more than just expressions of feeling, which emotivism seemed unable to explain. Nevertheless, emotivism stands as an important warning: it reminds us not to assume that a sentence s grammatical form (such as a declarative moral claim) tells us its actual function. Ethical language, like all language, may serve multiple functions at once, and understanding those functions is essential to analyzing what is really being said.

C. THREE TYPES OF DISPUTES

Disputes, or disagreements among people, are common in all walks of life. With trivial matters, such as whether a movie was a good or a bad one, we argue for the sake of arguing, and don t particularly care whether the matter gets settled. With more serious matters, such as how to stop global warming, we might hope to find some resolution or consensus. Often we don t, though, and part of the cause is failing to distinguish between different types of disputes. Some involve facts, others are merely verbal, and yet others rest on strongly held values.

1. Factual dispute: disagreeing about whether some event happened, or something is true. These can be resolved by determining the facts, such as through scientific inquiry or reliable historical accounts. Sometimes, however, factual disputes cannot easily be resolved when the facts are not accessible, such as a dispute about a scientific theory that cannot yet be tested, or a dispute about a past event where critical facts are lost to time.

e.g., (a) "Illinois is east of Indiana"; (b) "no it is west of Indiana"

e.g., (a) "We were told to be there at 3:00"; (b) "no, it was 4:00"

e.g., (a) "Mom cooked chicken last night; (b) "no, she cooked snake"

e.g., (a) "Oswald acted alone when killing JFK"; (b) "no he was part of a conspiracy" (disagreement over facts that are inaccessible)

2. Verbal (terminological) dispute: disagreement about the use of an ambiguous term; the dispute can be resolved by clarifying the ambiguous term.

e.g., (a) "Bob lives far away; it took two hours to walk to his home"; (b) "no, Bob lives close; it took ten minutes to drive there" (verbal disagreement over the term "far away" regarding walking vs. driving)

e.g., (a) "Joe is an excellent student; he takes a lively interest in everything and asks intelligent questions in class"; (b) "no, Joe is not an excellent student; he never gets his assignments in on time" (verbal disagreement over the term "excellent student" and its criteria)

e.g., (a) "All humans are created equal"; (b) "no, we all have unique DNA" (verbal disagreement over the term "equal" and its political vs. biological meanings)

e.g., (a) "I bought a new house"; (b) "no, it was built 20 years ago" (verbal disagreement over the term "new" regarding time of acquisition vs. time of construction)

e.g., From William James: a hunter chases a squirrel around a tree, but the two are always on opposite sides: they both go around the tree, but does the hunter go around the squirrel? James s answer: "If by going around you mean going east, south, west, north, east, etc., then the man certainly goes around the squirrel. But if by 'going around' you mean going from side to back to side to belly to side, etc. then the man does not go around the squirrel."

e.g. (a) "Joe is tall, much more so than his father;" (b) "no, Joe is not tall; his father is exceptionally short" (verbal disagreement over the term "tall" and whether it is relative to his Joe's father or an average person)

3. Value dispute: disagreement about major value assumptions and fundamental world-life outlooks that are largely unprovable, such as in religion, ethics, aesthetics, political ideology, economic ideology, and metaphysics. Disputes in these areas often appear to be factual in nature, but they typically cannot be resolved by establishing empirical facts or clarifying terminology. The school of Logical Positivism, discussed above, describes such value statements as pseudo-propositions, which do not involve genuine facts about the world.

e.g., (a) "Lou Reed was a musical visionary"; (b) "no, he was just an untalented rebel" (disagreement in value over musical aesthetics)

e.g., (a) "The pay for this job is reasonable"; (b) "no, the pay for this job is exploitive" (disagreement in value over economic fairness)

e.g., (a) "My parents are in heaven"; (b) "no they re not because there is no heaven" (disagreement in value about the existence of heaven)

e.g., (a) "Beth and Jill just got married;" (b) "no they didn t, marriage is only between a man and a woman." (disagreement in value about the nature of true marriage)

e.g., (a) "The President s welfare policies will reduce poverty;" (b) "no, they will destroy our capitalist economy and ultimately put us all in poverty" (disagreement in value about welfare and capitalism)

e.g., (a) "I freely chose to major in business"; (b) "no, your parents nagged you into it, and that s not a free choice" (disagreement in value about free will)

e.g., (a) "My dog is demon possessed" (b) "dude, you watch too many horror movies; everyone knows that demons only possess people" (disagreement in value about demon possession)

4. Combination disputes: disputes that have some mixture of factual, verbal, and value disagreement.

e.g., (a) "You were fired because of your poor work performance"; (b) "no, my work performance was fine, I was fired because my supervisor didn t like me" (disagreement in terminology over "work performance", and disagreement in fact over why he was fired)

e.g., (a) "I went to the God Boy concert last night and it was heaven!" (b) "as I ve argued with you before, heaven doesn t exist, so it follows that you didn t go to the concert, which is just as well since those religious bands are awful" (factual disagreement over whether he went to the concert, verbal disagreement over the word "heaven", and value disagreement over the quality of religious bands)

D. DEFINITIONS

Many arguments or disputes rest on the definitions of key terms, and whether a given definition is a good or bad one. The task of defining a word is a complex one and involves differing purposes and techniques.

1. Parts of definitions

Definiendum (di-fin-ee-EN-dum): the term to be defined

Definiens (di-fin-ee-ENS): the definition

e.g., "triangle (defiendum) means three sides (definiens)]

2. Purposes of Definitions

a. Lexical: designates meaning by showing how English-speaking people use a word as based on statistical usage.

e.g., "mountain" means a large mass of earth or rock rising to a considerable height above the surrounding country

e.g., "robot" means a machine capable of carrying out a complex series of actions automatically, especially one programmable by a computer.

e.g., "cat" means a small domesticated carnivorous mammal with soft fur, a short snout, and retractile claws.

e.g., "microphone" is an instrument for converting sound waves into electrical energy variations, which may then be amplified, transmitted, or recorded

b. Precising: designates meaning by clarifying vague ordinary usage with a more precise meaning.

e.g., "moral right" is a justified constraint upon how others may act

e.g., "poor" means earning less than $15,000 a year

e.g., "crime" is an action or omission that constitutes an offense that may be prosecuted by the state and is punishable by law

c. Theoretical: designates meaning by indicating how a word is used in some presupposed theory; these are even narrower than precising definitions, and often refer to a single theory

e.g., "heat" is a form of energy in a body involving the irregular motion of its molecules

e.g., "Bourgeoisie" for Marx means the capitalist class who own most of society's wealth and means of production

d. Stipulative: designates meaning by proposing a new meaning of a term; these typically indicate that the new definition is for a specific or limited purpose.

e.g., "G factor" for the purposes of this experiment means intelligence

e.g., by "guest" I mean anyone who I formally invite to my exclusive party

e.g., the word "sinner", for our cult, means anyone who has not recited the Sacred Creed of Credulity

e. Persuasive definition: designates meaning with the purpose of influencing attitudes.

e.g., "abortion" is the murdering of the innocent unborn children.

e.g., "Politics" is derived from the words "poly" meaning "many", and "tics" meaning "blood-sucking parasites."

e.g., "faith" means believing something that you know is not true.

e.g., "blonde jokes" refer to jokes that are easy enough for men to understand.

e.g., "committee" means a body that keeps minutes and wastes hours.

e.g., "student loan" refers to a mystical transaction in which students subject themselves to life-long financial slavery.

3. Techniques of Definition

a. Extensional (denotative): designates meaning by indicating members that belong to a class of objects.

Ostensive (demonstrative): designates meaning by demonstrating an object, such as by pointing to something.

e.g., that object over there is a desk

e.g., look at where I m pointing, that object is a computer chip

e.g., look directly at me, what you see is an eyeball

Limitations: cannot tell what exactly is referred to

Enumerative: designates meaning by listing objects that belong to a class.

e.g., "planet" means Mercury, Venus, Earth, Mars, etc.

e.g., "romance language" means Spanish, Portuguese, French, Italian, Romanian, etc.

e.g., "robot" means things like R2-D2, C-3PO, Optimus Prime, etc.

Limitations: lists may be incomplete, two lists might be identical

b. Intensional (connotative): designates meaning by indicating a set of characteristics agreed to be common to all objects being defined.

Synonymous: designates meaning by indicating how two words have the same meaning.

e.g., "voracious" means insatiable

e.g., "hat" means headwear

e.g., "boast" means brag

Genus and difference: designates meaning by first indicating a class that an object belongs to, and second a distinguishing feature of that object.

e.g., "triangle" means polygon with three sides

e.g., "planet" means a celestial body moving in an elliptical orbit around a star

e.g., "beauty" means a combination of qualities, such as shape, color, or form, that pleases the aesthetic senses, especially the sight

Etymological: designates meaning by explaining the historical origin of a word.

e.g., "philosophy" means love of wisdom in the original Greek

e.g., "robot" comes from the Czech word "robota" meaning forced labor

e.g., "psychology" means study of the soul in the original Greek

Functional: designates meaning by indicating the purpose of an object; these are often genus and difference definitions, with the function indicated in the difference.

e.g., "hammer" is an instrument made for pounding nails into a solid surface

e.g., "telescope" is an instrument made for making distant objects appear closer

e.g., "picture frame" is an instrument made to hold and display photographs and other two-dimensional representations

Operational: designates meaning by specifying a repeatable procedure, often through some test.

e.g., "genius" means a score above 140 on the Stanford Binet intelligence scale

e.g., "profound hearing loss" means the inability to hear sounds lower than 95 dB

e.g., "antique" means over 100 years old

4. Five Rules of Good Lexical Definitions

a. Should state the essential attributes of the object, not accidental attributes.

Good examples (have essential attributes)

e.g., "man" is a rational animal

e.g., "wealth" is the measure of the value of all of the assets of worth owned by a person or entity

e.g., "triangle" is a three-sided rectilinear figure

e.g., "school" is an institution for education

Bad examples (have only accidental attributes)

e.g., "man" is a cooking animal

e.g., "wealth" is the power to purchase fine clothing

e.g., "magnesium" is a metal capable of ignition

e.g., "Tree" is a vegetable organism, having roots, branches, leaves, etc.

b. Must not be circular.

e.g., "man" is the sole occupier of the class of human beings

e.g., "wealth" is that which makes one wealthy

e.g., "bed" means an object suitable for bedding

e.g., "sleep deprivation" is the condition of not having enough sleep

e.g., "virtue" means acting virtuously

c. Must be neither too broad nor too narrow.

Too broad

e.g., "man" is a featherless biped (includes plucked chickens)

e.g., "wealth" is that which has value to an individual (includes non-economic goods like sleep)

e.g., "spelt" is a kind of food (includes all food)

e.g., "eloquence" is the power of influencing the feelings by speech or writing (includes threats or profanity)

Too narrow

e.g., "man" is a civilized rational animal (excludes uncivilized and irrational humans)

e.g., "wealth" consists of money (excludes non-monetary economic value)

e.g., "shoe" is a leather covering for the foot (excludes canvass shoes)

e.g., "logic" is a machine for combating fallacy (excludes formal symbolic logic)

d. Must not be ambiguous or figurative.

e.g., "man" is the only animal who blushes or needs to

e.g., "wealth" consists of having few wants

e.g., "net" refers to a reticulated fabric made with interstitial vacuities

e.g., "soul" is the first entelechy of the natural body

e.g., "bread" is the staff of life

e.g., "life" is a continuous adjustment of internal relations to external relations

e.g., "architecture" is frozen music

e.g., "body" is the visible garment of the soul.

e. Should not be negative where it can be affirmative.

e.g., "man" is neither angel nor beast but stands between the two

e.g., "wealth" is the lack of poverty

e.g., "couch" is not a bed and not a chair

e.g., "Turanian language" is one which is neither Aryan nor Semitic

E. PHILOSOPHY OF LANGUAGE (bonus material, may or may not be on the test)

Philosophers since ancient Greece had theorized about the nature of language, but the subject of "the philosophy of language" as we know it today emerged in the early twentieth-century. Much of this is intertwined with concepts in contemporary logic.

1. Theories of Meaning

The major issue within the philosophy of language is the relationship between language and meaning: how it is that verbal utterances or scribbles on paper convey meaning that we understand. It is important to recognize that "meaning" is not the same as "truth": something is true when it reflects reality as it is, whereas something is meaningful when it is simply understandable. Take the sentence, "the Statue of Liberty is in New York City": this is both meaningful and true. However, the sentence "the Statue of Liberty is in Washington DC" is meaningful, yet is false. Meaningfulness, then, is the broader concept, and an adequate theory of meaning will need to account for both true and false statements. We will look at nine such theories of meaning. The starting point for all nine of these theories is a fundamental distinction between a sign (the linguistic expression or utterance) and the thing signified (what that expression means or picks out).

a. Ideational theory of meaning (John Locke): the meaning of a linguistic utterance resides in the ideas within the speaker's mind.

e.g., "the cat is on the table" reflects the mental image I have of a particular cat being on a particular table.

Diagram: utterance > mental image

Criticism: Some meaningful words have no particular mental image or content associated with them, e.g., "is", "of", "and".

b. Referential theory of meaning (John Stuart Mill): the meaning of a linguistic utterance is how it stands for things in the world.

e.g., "the cat is on the table" stands for an actual cat being on an actual table.

Diagram: utterance > the world

Criticism: some meaningful words do not denote actual objects, e.g., "Easter Bunny", "The present King of France". Further, they cannot explain why co-referential terms differ in cognitive impact (e.g. "the morning star" vs. "the evening star" which both denote the planet Venus)

c. Sense Reference theory of meaning (Gottlieb Frege): the meaning of a linguistic expression has two components (1) a sense (the mode of presentation, or way of thinking of the referent) and (2) a reference (the object in the world that the expression picks out).

e.g., "the cat is on the table" stands for the thought of a cat on a table (sense), and the actual cat-on-table situation (reference).

e.g., "I saw batman today" stands for the superhero in Gotham City who dresses like a bat (sense), and the person Bruce Wayne (reference), even if you didn't know Batman's true identity.

e.g., "the morning star is the evening star", stands for the bright point in the sky seen at dawn, and the bright point in the sky seen at dusk (sense), and the planet Venus (reference). Suppose you mistakenly think the morning star and the evening star are different objects and you say, The morning star is shining, but the evening star is not shining. Here you are applying the predicate is shining in two different senses (the bright point seen at dawn versus bright point seen at dusk ) even though, unbeknown to you, both of those senses actually pick out the same reference, Venus.

Diagram: utterance (sign) → sense → reference-object

Criticism: The notion of "sense" is obscure. Is it a mental image in each speaker s mind? Is it merely a linguistic rule that connects the speaker's mind with the actual world"? Is it a non-physical abstract object (a Platonic entity) hovering outside both the speaker's mind and the actual world?

d. Proposition theory of meaning (Bertrand Russell): the meaning of a linguistic utterance, when logically clarified, is an atomic proposition (a mind-independent entity that is either true or false) which corresponds to a single fact about the world.

e.g., "the cat is on the table" expresses the atomic proposition that a particular cat occupies a particular table.

Diagram: utterance > logical paraphrase > atomic proposition > atomic fact

Criticism: atomic propositions are metaphysically questionable, since they must exist eternally and be universally graspable by any rational being.

e. Verification theory of meaning (logical positivism): the meaning of a linguistic utterance is the future experience that shows its truth (verification condition; epistemic account of meaning).

e.g., "the cat is on the table" is meaningful since we can verify it by petting the cat on the table.

e.g., "Everything, including all measuring devices, has just doubled in size" is meaningless since its truth cannot be verified.

Diagram: utterance > future verifying experience

Criticism: the theory applies only to propositional utterances (expressions that state facts), not non-propositional utterances like questions, commands, expressions of feelings, jokes, ceremonial language.

f. Language-use (game) theory of meaning (Ludwig Wittgenstein): the meaning of a linguistic utterance is fixed by its role in a rule-governed language game, which in turn is embedded in a broader form of life (social practice). The rules of language games parallel the rules of leisure games like chess.

e.g., the cat is on the table belongs to the language game of reporting an animal s location in a domestic setting (e.g. answering where is the cat? ), which is itself grounded in household practices around pets.

Diagram: utterance > language game > form of life (social practice)

Criticism: merely knowing the rules of the game doesn t ensure genuine understanding. For example, we can follow ritual-like rules, such as a religious liturgy, without grasping their meaning.

g. Communicative intention theory of meaning (Paul Grice): the meaning of a linguistic utterance is the psychological intention of the speaker ("speaker-meaning"; by uttering x, S meant that P) the meaning of an utterance consists of (1) its sentence‐meaning (the literal content encoded by words) and (2) its speaker‐meaning (the psychological intention to induce a response in the hearer).

e.g., the cat is on the table literally says that a cat occupies a place on a table; the speaker intends you to form the belief that a certain cat is on that table.

Diagram:

1. utterance > sentence‐meaning

2. utterance + context + cooperative principle > speaker‐meaning (intention)

Criticism: Grice's view of speaker-meaning leads to an infinite-regress of intentions. To mean P you must intend that your hearer recognize your intention to convey P, which in turn seems to require that you intend that your hearer recognize your intention to intend that they recognize your intention.

h. Possible world theory of meaning (Saul Kripke): the meaning of a linguistic utterance is its relation to the truth values they take in all possible worlds ("possible world semantics").

e.g., "the cat is on the table" means it is true that the cat is on the table in one possible world, but not in another where both worlds are possibilities within the same universe.

Diagram: utterance > proposition in logical form > set of possible worlds where that proposition is true

Criticism: this commits one to a multiverse of possible worlds where the worlds are strange metaphysical objects, similar to the strangeness of propositions. They are not simply metaphors, but seem to be actual alternative worlds that do not exist.

i. Pragmatic-use theory of meaning (Rorty): meaning is whatever linguistic practices a community finds useful for dealing with shared concerns. Language is a tool, not a mirror of reality.

e.g., "the cat is on the table" means a cluster of practical effects in one's communal routines, such as alerting someone to move breakable items from the table.

e.g., we hold to justice for all means a network of practices and descriptions we employ when discussing fairness; it doesn t pick out some abstract Platonic Form.

Diagram: utterance → shared communal practice/use → useful outcome

Criticism: this approach abandons any direct link between words and facts about the world, risking the view that meaning is nothing but conventionally agreed-upon habits.

2. Other Concepts and Terms in the Philosophy of Language.

This section outlines the technical vocabulary that used is by philosophers of language when attempting a precise analysis of language. These distinctions give us the tools to avoid confusion when we break down sentences and evaluate their logical structure.

a. Type-token distinction

Type: the abstract sentence or expression (e.g. the sentence The cat is on the table , or the single word "cat"). Type captures what is common in all instances of the same sentence whenever it appears.

Token: a particular instance or utterance of that type (e.g. you writing or saying "the cat is on the table" on a specific occasion). Token allows us to talk about specific occurrences within particular contexts.

b. Syntax vs. Semantics vs. Pragmatics

Syntax: the rules for building well-formed expressions (grammar, formation rules).

Semantics: the literal, context-invariant content of expressions (what they mean in principle).

Pragmatics: how context and speaker intentions enrich or modify literal meaning. One example is the pragmatic nature of indexical resolution, where indexical are certain words that do not carry full content on their own and need context to pick out who or what they refer to (e.g., "put that object 'here'). Another example is the pragmatic nature of conversations, which involve extra layers of meaning that a speaker conveys indirectly (e.g., some of the cookies are gone usually implies not all of the cookies are gone ).

c. Lexical meaning vs. Sentence meaning

Lexical meaning: the meaning of an individual word or short phrase (e.g. cat, on the table ).

Sentence meaning: the meaning that arises when words combine, such as in a full sentence, including how their parts contribute (compositionality (e.g., "the cat is on the table").

d. Sense vs. Reference (Frege)

Reference: the object or entity picked out by an expression (e.g., Venus).

Sense: the manner of presentation or way of thinking that lets us grasp that reference (e.g., bright object seen at dawn ).

e. Extension vs. Intension

Extension: the set of all objects a general term applies to (e.g., all individual triangles).

Intension: the criteria or concept that determine membership (e.g., three-sided, three-angled figure ).

f. Terms: Singular, General, Indexicals

Singular terms: designate particular individuals, subdivided into (a) proper names, e.g., Joe , and (b) definite descriptions, e.g., the man with three eyes .

General terms: apply to many objects; subdivide into (a) count-terms: discrete items, e.g., chair, apple , and (b) mass-terms: undifferentiated stuff, e.g., water, sand .

Indexicals: shift reference with context ( I, here, now ).

g. Logical Form and Compositionality

Logical form: the formal structure that reveals how parts of a sentence combine in meaning. For example, in the sentence Every student skipped class, the logical form is this: ∀x {Student(x) → ∃y [Class(y) & Skipped(x,y)]}.

Compositionality: the meaning of the whole sentence is built entirely from the meanings of its parts and the logical way they re combined. For example, if you know the meaning of the words every, student, skipped, and class , and you also know the logical form that combines them, then you can predict the sentence s meaning. There is no extra meaning hiding in the sentence.

h. Principle of Substitutivity (Salva Veritate, Leibniz s Law)

Expressions may be interchanged without changing truth-value in extensional contexts (e.g. Mark Twain wrote Adventures of Huckleberry Finn and Samuel Clemens wrote Adventures of Huckleberry Finn are both true).

However, this fails in intensional contexts (e.g. Alice believes that the morning star is bright might be true, while Alice believes that the evening star is bright might be false).

i. Presupposition vs. Entailment

Entailment: whenever A is true, B must also be true (A ⟶ B) (e.g,. John is a bachelor entails John is unmarried ).

Presupposition: background assumptions that must hold for a sentence to make sense (e.g. Jane stopped smoking presupposes she once smoked).

j. Implicature vs. Convention

Conventional meaning: the literal content encoded by words or grammar (e.g., some bachelors own pets means there is at least one bachelor who owns a pet).

Conversational implicature: extra meaning a speaker conveys by drawn from context and cooperative principles (e.g. by saying some, the speaker implies not all bachelors own pets. ).

k. Deixis & Indexicality

Deixis: how certain expressions pick out different referents depending on context (e.g., I refers to whoever is speaking, Here refers to the speaker s current location, Tomorrow refers to the day after the utterance day).

Indexical resolution: the hearer uses context (who, where, when) to slot in the correct referent for each indexical

l. Propositional Attitudes; De Dicto vs. De Re

Propositional attitudes: mental states toward propositions (e.g., to believe P, to hope P, to fear P).

De dicto: attitude toward the proposition itself (e.g., Alice fears that the murderer will escape ).

De re: attitude toward an object under some description (e.g., Alice fears the man with the scar ).

m. Rigid vs. Non-Rigid (Floppy) Designators (Kripke)

Rigid: denote the same object in every possible world where it exists (e.g., proper names like "Abraham Lincoln", natural-kind terms like water ).

Non-Rigid: can pick out different objects in different worlds (e.g., the forty-seventh president of the U.S. , which in some possible worlds would be Kamala Harris).

n. Speech-Act Theory (Austin, Searle)

Illocutionary act: the action performed in saying something, such as warning, promising, ordering (e.g., uttering I promise to call you performs a promise).

Performatives: first-person, present-tense verbs that name the act performed by the utterance itself (e.g., I apologize, I declare, I name this ship Queen Elizabeth ).

Felicity conditions: requirements for a speech act to succeed, such as the speaker s authority, sincerity, appropriate context (e.g., I now pronounce you married only works if the speaker is an authorized officiant and the legal prerequisites are met).

Perlocutionary force: the actual effect that an utterance has on the listener, such as to persuade, scare, amuse (e.g., saying Watch out! may startle someone).

o. Semantic Externalism

Meaning depends partly on factors outside the speaker s mind, such as one s environment, social practices, natural kinds. (e.g., Putman's Twin-Earth example, where water refers to H2O in this Earth, but on Twin Earth it refers to XYZ, even though speakers thoughts are identical, the external substance differs).

p. Classical Definitions vs. Prototype Theory

Classical definitions: categories defined by necessary and sufficient features (e.g., triangle = a plane figure with exactly three-sided figure).

Prototype theory: categories organized around typical examples and family resemblances (e.g., Bird centers on robin-like features such as feathers, beak, ability to fly, so penguins feel less birdlike than sparrows).

q. Ambiguity vs. Vagueness

Ambiguity: a single expression has two or more distinct meanings (e.g., bank = a financial institution or a riverbank).

Vagueness: an expression lacks precise boundaries without clear cutoff (e.g. with tall , there is no clear cutoff height where someone stops being tall. ).

F. LINGUISTIC CONCEPTS IN OLD LOGIC BOOKS (bonus material, not included on test)

The above issues concerning language and logic are of comparatively recent interest to logicians. However, for much of logic's 2,000-plus year history, Aristotle's system dominated, and older discussions of language and logic drew on this. During the nineteenth-century, a new set of linguistic concepts was introduced into logic, most notably in John Stuart Mill's A System of Logic (1843), and these began appearing in introductory logic books of the time along with the Aristotelean concepts. Both Aristotle's and Mill's discussions are obsolete from the perspective of recent logic, but they remain important in the history of philosophy.

1. Aristotelean concepts

a. Laws of Thought (from Aristotle's Metaphysics)

The Law of Identity: A is A (e.g., Bob is Bob).

The Law of Contradiction: A is not not-A (e.g., it is not the case that Bob is not-Bob).

The Law of Excluded Middle: Everything is either A or not-A (e.g., everything is either Bob or not-Bob).

Later additions:

The Law Causation (or sufficient reason): every event must have a cause.

The Law of the Uniformity of Nature: the same antecedents are invariably followed by the same consequents.

b. Predicables (from Aristotle's Categories, modified in middle ages)

Predicable: ways that predicates stand in relation to subjects (that which can be predicated).

Five predicables

Genus: a term which refers to two or more subordinate classes, e.g., "man"

Species: an attribute that represents one of the subordinate classes in a genus, e.g., English man, French man

Differentia: an attribute that distinguishes one species from another,

Property: an attribute that helps make a term what it is

Accident: an attribute that does not help make a term what it is

Examples

This trigon (species) is a polygon (genus) of three sides (differentia) and three angles (property) the sum of the angles being equal to two right angles (accident).

A moonbeam (species) is a ray of light (genus) from the moon (differentia).

A dwelling (species) is a building (genus) where people live (differentia).

c. Meaning of terms (from Aristotle's Categories)

Univocal words: have only one meaning, often technical terms, e.g., molecule, angle, vertebrate.

Equivocal words: have more than one meaning, e.g., "bit" means morsel or part of a horse s harness.

Analogical: meanings are partly, but not wholly the same, e.g., "love", where king loves subjects as parent loves children.

2. Kinds of terms (from John Stuart Mill's System of Logic)

a. Abstract vs. Concrete

Abstract terms: name qualities or relations apart from any particular thing. (e.g., sweetness denotes the quality you taste in sugar.)

Concrete terms: name individual objects or collections of them. (e.g., apple names a fruit; honey names a substance you can touch.)

b. Singular vs. General vs. Collective

Singular terms apply to one specific thing. (e.g., the large dictionary in this room").

General terms apply to any number of things sharing a feature. (e.g., dog, holiday, member of Congress").

Collective terms treat a group as a single entity, ignoring individual members. (e.g., family, tribe, army").

c. Absolute vs. Relative

Absolute terms: describe something in itself, without reference to anything else. (e.g., stone, building").

Relative terms: describe something only in relation to something else. (e.g., shepherd one who tends sheep; patient one who is treated).

d. Positive vs. Negative vs. Privative

Positive terms imply the presence of a quality. (e.g., heavy, efficient").

Negative terms imply the absence of a quality. (e.g., light , not heavy; inefficient , not efficient).

Privative terms imply the absence of a quality normally expected. (e.g., maimed , lacking a limb; blind , lacking sight; orphaned , lacking parents).

e. Connotative vs. Non-connotative

Connotative terms both denote a subject and imply an attribute. (e.g., man denotes an adult human male and implies rationality.)

Non-connotative terms denote a subject without implying anything beyond its reference. (e.g., blueness names the quality of being blue, nothing more.)

f. Extension vs. Intension

Extension of a term is the set of all things it applies to. (e.g., The extension of triangle is every individual triangle (isosceles, scalene, equilateral, etc.).

Intension is the set of criteria or features that something must have to fall under the term. (e.g., The intension of triangle is a closed figure with three straight sides and three angles").