phr103
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Logic is one of the five main branches/sub-disciplines of philosophy; however, there are
areas of logic that are also part of mathematics, artificial intelligence, linguistics, and
computer science. The five main branches of philosophy are:
1. Metaphysics
2. Epistemology
3. Ethics (or Moral philosophy)
4. Aesthetics
5. Logic
All of these branches of philosophy attempt to construct theories (just like in the natural
and social sciences), give explanations of various phenomena, investigate issues, answer
questions, and identify problems in their respective domains. Their domains may be
learned easily by considering first order questions. First order questions are fundamental
questions about the nature of aspects of the universe and everything in it. First order
questions take the following form:
What is the nature or fundamental nature of _____________?
where the blank is to be filled in with a word or words. Here are the first order questions
of the five main branches of philosophy.
Metaphysics: What is the nature of reality?
Epistemology: What is the nature of knowledge?
Ethics: What is the nature of the right and good?
Aesthetics: What is the nature of beauty and art?
Logic: What is the nature of correct reasoning?
Traditionally, logic is divided into two main sub-branches:
1. Informal logic
2. Formal logic
Informal logic addresses reasoning and arguments in natural languages.
Formal logic addresses reasoning and arguments in formal languages.
Natural languages include the roughly 6,000 to 7,000 languages that have more or less
naturally evolved during the history of Homo sapiens. Natural language ability seems to
be innate in members of Homo sapiens. Following are examples of natural languages:
English, Spanish, German, French, Chinese, Korean, Hindi, Arabic, Turkish, Russian,
Polish, and Swahili. Linguists have categorized natural languages into language families
and sub-families and have traced the origins of most natural languages.
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All of you are competent in at least one natural language; some of you are competent in
more than one natural language (e.g., English and Spanish).
Formal languages are artificial, synthetic, or constructed languages that are deliberately
constructed by humans for a purpose. Following are examples of formal languages:
arithmetic, algebra, the calculus, computer programming languages (Fortran, Basic,
Cobol, C, C+, C++, HTML, Java). Each of these formal languages has a particular
purpose. Consider the main functions of these formal languages:
What is the purpose or function of arithmetic, algebra, C++?
Natural languages have a purpose, too. But their purpose is general communication.
All of you are (more or less) competent in at least two formal languages: basic arithmetic
and basic algebra. Some of you may know (say) computer programming languages.
To be a language, whether natural or formal, three things are needed. In other words,
these are the necessary conditions for being a language.
Languages must have a:
1. Symbol set or alphabet
2. Syntax or Grammar
3. Semantics or Interpretation
A symbol set can be sounds, inscriptions in some medium (paper, monitor, sand, dirt,
etc.). The symbols set for English are the twenty-six upper case (capital) and lower case
letters of the Roman alphabet, plus punctuation, and a few other symbols. English also
has sounds that correspond with many of these symbols (sounds of alphabet letters, for
example). The symbol set for basic arithmetic is:
operators = {+, -, x, ÷, =}
natural numbers = {1, 2, 3, 4, 5, …} (this set is infinite)
dividers = { (, ), [, ], {, } } (parentheses, brackets, braces)
A syntax or grammar is a set of rules that determine how to put the symbols together.
Usually a syntax takes the form of, or can be reduced to, a set of rules. Sometimes these
rules are expressed in a meta-language (a language outside of the object language that
refers to the object language and can say things about the object language). Following
are ungrammatical or non-syntactical strings of symbols in the natural language of
English and the formal language of arithmetic.
boy dog cat how
1+2-4x÷3+
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These strings of symbols make no sense (literally, nonsense), because they do not follow
the syntax/grammar rules all of us (who are competent in English and arithmetic) have
internalized.
Semantics is the assignment of meanings to strings of symbols (e.g., words, sentences,
―+,‖). In natural languages, we assign meanings to words and sentences. In formal
languages, we assign meanings to symbols.
In this course, we will study reasoning and arguments in English first. Subsequently, we
will learn a new formal language trivially and boringly called ―SL.‖ ―SL‖ stands for
Symbolic Logic. There are other names for this formal language, but it is not important
what one calls it—it is important to have a name or label for it in order for there to be
something to which one refers. Like most formal languages, there are many variations of
SL, and many complements and alternatives to SL (alternatives are often called ―deviant‖
logics). SL is the formal language of the classical first order logic. Classical here means
that the formal system abides by, and perhaps codifies, the following axioms or
postulates.
1. The Law of Non-contradiction. A sentence or expression in language cannot be both
true and false.
2. The Law of Excluded Middle or Bivalence. A statement is either true or false; there is
no third or fourth option.
3. Law of Identity. An object or event or sign/symbol is identical to itself. (a=a, 1=1,
2=2)
These axioms are so basic that many logicians and philosophers have called them ―laws
of thought.‖ However, not all logicians accept these axioms as true or useful or good.
Non-classical or deviant logics deny or do not abide by one or more of these three laws.
Examples of non-classical logics are: Intuitionist, Many-valued logics, Quantum logic,
Paraconsistent, and many others. We will not study these logics in this course.
So, what is logic?
There are many things one can say in defining ―logic.‖ Our textbook defines logic
simply as the ―science that evaluates arguments‖ (Hurley, p. 1). This is not a
comprehensive definition of logic, but we will use it for now. ―Argument‖ is a technical
term in logic and philosophy. Be sure you understand what ―science‖ and ―evaluate‖
mean; otherwise, you will not understand the definition just given.
To evaluate (noun = evaluation) something is to determine its value, such as being good
or bad. Often, we have a richer, more robust vocabulary for ―good‖ and ―bad.‖ For
example, when evaluating people, we are concerned with specific aspects of persons such
as their moral life, physical attributes, psychic/mental attributes, et cetera. We say:
beautiful, ugly, attractive, unattractive, sexy, hot, desirable, smart, intelligent, savvy,
cool, nerdy, quick, friendly, and many virtue and vice terms are used.
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A science consists of two major components. We will always use the physical or natural
sciences as are paradigm examples of sciences: physics, chemistry, and biology. (There
may be other sciences such as social sciences (anthropology, sociology, economics,
linguistics, et cetera).) A science is a body of knowledge or a set of facts. The second
component expresses how those facts are obtained: the 1 scientific method. There are
many features of the scientific method, many of which are controversial. Some are less
controversial. The goal of science is the truth, not money, profit, or making people feel
good. Science concerns testable statements/propositions (testability). This means that
there is a way/method/procedure in principle to verify (noun = verification) or refute
(noun = refutation) the claim. ―Verify‖ means to demonstrate that a claim is true, and
―refute‖ means to demonstrate that a claim is false. One must be able to do a study or an
experiment to determine whether the claim is true or false. This brings us to another
important feature of the scientific method: falsification. Any claim (except for
mathematical statements, tools, et cetera) used in a scientific theory or explanation, or as
an hypothesis must have the property of being able to be false. It must be possible for the
claim to be false. Statements that always true, i.e., it is impossible for them to be false,
are called tautologies (singular = tautology, adjective = tautologous). We will learn about
them in Chapter 6.3. The scientific method also involves methods, tools, procedures,
experiments, studies that are replicable. This means that the experiments or studies may
be done many, different times by different people and achieve the same results.
So, what is an argument?
First, in logic, ―argument‖ does not mean a debate, disagreement, shouting match, to row
(not long ―o‖), or quarrel. An argument is:
a piece of discourse
at least two statements or propositions
at least one premise (or ―premiss‖)
at least one conclusion
at least one inference
Discourse is any form of communication in any medium (verbal/oral, written,
gesture, Braille (tactile)). Following are examples of discourse: newspaper, a STOP
sign, a traffic light, telephone conversation, semaphore flag movements, Morse code
sounds and written code, displaying the middle finger to someone, singing, smoke
signals, magazines, internet world wide web pages with text, sounds and images, CD
recordings, MP3 recordings, waving to someone, etc.
What is a statement?
This is a very important term in logic and in our course.
1 Some historians and philosophers of science would not use the definite article here (―the‖); they would
use the indefinite article (―a‖).
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[We will take a brief detour from our definition of ―argument‖ in order to learn about
statements and learn our first skill (Skill# 1).]
A statement is:
a kind of sentence that has a particular grammatical feature and logical feature. The
grammatical feature is that a statement must in English be a complete sentence (not a
fragment) with a NOUN PHRASE (NP) + VERB PHRASE (VP). The logical feature is
that a statement must have a truth-value.
We will use the descriptive definition of a sentence (not a normative definition). A
sentence is the sequence of words and punctuation up to a full-stop. A full-stop is
typically a period (.) but may also be a question mark (?) or exclamation mark (!).
A NP contains the subject of the sentence and anything that modifies the subject of the
sentence (such as an adjective or adjectival phrase or relative clause construction). A VP
contains the verb, auxiliary verb (if any), the predicate or object (direct or indirect
object), and anything that modifies the verb (adverbs) or the object. The general idea is
that statement must express a complete thought. Complete thought are usually expressed
through the connection between a subject (e.g., noun, proper noun, gerund) and a
predicate. Statements cannot be fragments.
Possessing a truth-value means to have the capacity to be true (T) or false (F). Not all
sentences have the capacity to be true or false. One does not need to know whether the
sentence is in fact true or in fact false; one just needs to know that it is or can be true or
false. One does not need to know whether the sentence is actually true or false. For
example, suppose you do not know the capitol of California. The statement ―Los
Angeles is the capitol of California‖ still has a truth-value even though you may not
know what it is. Determining that a sentence has a truth-value does not depend upon the
state of anyone‘s knowledge or lack thereof.
Following are examples of statements:
John runs.
John runs home.
Jerky John runs home.
Jerky John runs home quickly.
John, who is a jerk, runs home quickly.
John ran.
Jill swims.
Jill swims home.
George Washington was the second president of the United States of America.
George Washington was the first president of the United States of America.
God exists.
God does not exist.
Abortion is immoral.
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Abortion is morally permissible.
Democracy is the best political system.
Fascism is dead.
Napoleon was Italian.
Turin is in Spain.
Turin is in Italy.
Following sentences are non-statements:
John.
Runs.
John, who is a jerk
Wow!
Ouch!
Ouch.
Whoaa.
Slow down, cowboy.
Shut the door.
Close the window.
Sammy, play it again.
Play it again, Sam.
What is your name?
Where does Sam live?
Because arguments must contain statements, and arguments are our central concern in
this course, one needs to be able to classify sentences as a statement or non-statement. If
one is confronted with non-statements, then one will know that one is not dealing with an
argument.
Skill # 1: Classify English sentences into the categories of statement or non-statement.
How can one determine whether a sentence is a statement or non-statement?
There are two methods: the question/flow chart method, and the contrast method. Most
of the time, one should try to use both methods. The use of both methods yields two
results: confirmation of one‘s answer, and the determination of what kind of non-
statement sentence (Directive, Expressive, Interrogative, Speech Act).
Question Method
Following is the question/flow chart method. This method follows directly from the
definition of a statement. Statements have two necessary conditions: NP + VP and truth-
value. When confronted with a sentence, ask first whether the sentence has the proper
grammatical structure (NP + VP). If not, then the sentence is a non-statement. If yes,
then continue by asking whether the sentence can be true or false (possess a truth-value).
If yes, then the sentence is a statement. If not, then the sentence is a non-statement.
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English sentence = X
X is an input
English sentence X
Does X have a NP + VP?
YES NO STOP Non-Statement
Does X have a truth-value?
YES NO STOP Non-Statement
STOP
Statement
Contrast Method
The contrast method requires that one learn and know the five main types or kinds of
sentences that are uttered in natural languages. Most natural languages have all five
types, but some natural languages lack one or more of these types. This is similar to the
fact that not all natural languages possess all of the tenses (or the same tenses), aspects,
moods, parts of speech, et cetera. The five types of sentences have different names
depending on what discipline is discussing them. These labels are roughly synonymous.
Five Kinds/Types of Sentences in Natural Languages
# Linguistics/Function Grammar Logic/Other 1. Informative Declarative Statement
2. Directive Imperative Command
3. Expressive Exclamatory
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4. Interrogative
5. Performative Illocutionary Speech Act
Only type # 1 are statements and have truth-values. # 2 – 5 do not have truth-values
when they are functioning as non-statements. However, # 2 – 5 may have NP + VP.
Directive sentences are commands. Typically, in English, directive sentences begin with
a verb. Examples: ―Shut the door.‖ ―Close the window.‖ These sentences do not have a
truth-value. In English, Directive sentences may even be in NP + VP form.
The primary function of expressive sentences is to express or arouse emotions, feelings,
and attitudes. Their purpose is not to inform. Expressive sentences do not have truth-
values.
The primary function of interrogatives is to inquire, ask questions. This includes all of
the ―wh‖ questions (where, what, when, why, (w)how) and others. Obviously, questions
do not have truth-values. In English, we use questions marks (?) as the full-stops to
interrogative sentences. Other natural languages use different symbols and grammatical
constructions to indicate questions.
There is an exception to questions: rhetorical questions. Rhetorical questions have the
surface grammar (sequence of words/letters and punctuation) of a question, but no
answer to the question is expected or desired because the purpose of the sentence
(question) is to make a statement. There are several reasons for the rhetorical question
phenomenon. Two main reasons are politeness (or impoliteness), and persuasion. When
people are too assertive they can be offensive, and therefore ruse or obnoxious. So,
people may phrase a statement as a question in order not to be offensive (―We are going
to the movies, right?‖). When rhetorical questions are used in arguments, or
presentations, or speeches, they are used to draw in an audience to the speaker‘s/writer‘s
point of view or conclusion. Again, when someone comes off as too assertive, people
may be turned off, and the speaker loses the confidence of the audience even if the
audience members agree with what the speaker is saying. (―Abortion is immoral.‖ versus
―Ladies and gentlemen, isn‘t abortion immoral?‖). Rhetorical questions should rewritten
as declarative sentences and classified as statements.
Speech acts are sentences that accomplish an action besides uttering. Some actions can
only be accomplished by uttering a sentence. ―Saying so makes it so.‖ Some actions can
only be accomplished by the proper authority, and thus that authority must utter the
speech act sentence (baptism, pronouncing people married, knighting). Examples of
actions that can only be accomplished by uttering: apologizing, promising, hiring, firing,
declaring war, welcoming, inviting, sports phenomena by umpires, referees, and other
officials (―Out!‖ ―Safe!‖ ―Out of bounds!‖ ―Ball!‖ ―Strike!‖ ―Touchdown!‖ ―Foul!‖).
Since speech acts are actions, and actions themselves cannot have truth-values (they are
not the proper objects of truth and falsehood, only bits of language can be true or false),
speech acts do not have truth-values. Often, speech acts have NP + VP. Following are
examples of speech act sentences.
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I apologize for my behavior.
I am sorry that I hurt your feelings.
You are hereby hired.
You are fired.
Your employment with this firm is hereby terminated.
The United States of America (hereby) declares war on Japan.
I baptize you ―John.‖
I promise to pay you back.
Please come to our party.
You are invited!
You and a guest are invited.
Welcome to our home!
Welcome to Fantasy Island.
Contrast Method: if one clearly identifies a sentence as directive, expressive, an
interrogative, or a speech act, then classify that sentence as non-statement.
Tricky Cases
There are several difficult or tricky cases in classifying sentences. Many English
sentences may belong to more than one category or kind depending on the context in
which it is used or uttered.
Consider: ―I love you.‖ ―Go to hell!‖
(There will be a separate treatment of tricky/difficult cases.)
Oratio obliqua is indirect speech. Indirect speech is a sentence that reports what
someone said or reports a psychological state (propositional attitude report/that-clauses).
direct discourse
indirect discourse
Oratio directa is direct speech. When someone asserts something, like ―it is raining,‖ she
is using direct speech or oratio directa.
Examples:
―It is raining.‖ Oratio directa
She said, ―it is raining.‖ Oratio obliqua-direct discourse
She said that it is raining. Oratio obliqua-indirect discourse.
Oratio directa (when the sentence in quotation marks or spoken is in fact a statement)
and oratio obliqua sentences are statements. One must distinguish between the nested
sentence and the reporting sentence. This is important because of truth conditions. In
other words, what makes ―it is raining‖ true is different from what makes ―she said, ‗it is
raining‘‖ true. If it really is raining outside, then the first sentences is true but the second
one is not. The truth conditions of the direct discourse sentence depend upon whether the
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referent of ―she‖ (whomever that may be) said the sequence of words indicated within the
quotation marks.
She said that it is raining.
She said, ―it is raining.‖
She said get out of here.
She said, ―get out of here.‖
She yelled, ―Shut the door.‖
Reports of what someone said are statements even though the content of what the person
said may not be a statement. When someone is uttering an oratio obliqua sentence, that
person is not asserting or saying the nested sentence (whatever it may be). In oratio
obliqua sentences, the nested sentence is being mentioned not used. The use/mention
distinction is an important one in philosophy, logic, linguistics, and the law. Following
example should make the difference between using words (or other symbols) and
mentioning words (or other symbols). Consider two sentences.
1. The cat is on the mat.
2. ―Cat‖ has three letters.
In sentence 1 the word ―cat‖ is being used. In sentence 1, ―cat‖ refers to an actual, real
cat (the animal). In sentence 2, ―cat‖ is being mentioned, and ―cat‖ refers to the sequence
of letters <c, a, t> or the word ―cat.‖ In sentence 2, ―cat‖ refers to the word not to the
animal.
Now back to the definition of ―argument.‖
Lecture # 2 is next.