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8/9/2019 Logic Notes 1
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Spring 2015 Logic Lecture Notes #1
Jared Warren
This course (Modern Deductive Logic, Spring 2015) has no assigned textbook.All required material will be covered during course lectures. In addition, each
week I will prepare and send out, via e-mail, lecture notes for purposes of review
and self-study. In this the first set of lecture notes for the course Ill intro-
duce some basic set theoretic notions and terminology that well use throughout
the course, introduce the idea of an argument, and introduce the language of
sentential logic that well be focusing on in the first part of the course.
1 Some Basic Set Theory
Aset is, very roughly, a collection of objects. The objects in a set are called
the elements or members of that set. The reason this characterization is
rough is that we also allow sets to have only a single member or even no
members.
We can denote sets by writing down the names of the members of the
set and enclosing the list in curly brackets, e.g., {0, 1, 2, 3} is a set whose
members are 0, 1,2, and 3. And {a, b} is a set whose members are a and
b. We can also denote a set by writing down a condition that all and onlymembers of the set satisfy, e.g., {x: x is a philosopher}would denote the
set of all philosophers. So Aristotle and Plato are both members of this
set.
We can express that a is a member of the set S in symbols as follows:
a S; we can express that a is not a member of S as follows: a / S.
So a {a, b} but c / {a, b} and Plato {x: x is a philosopher} but
Katy Perry /{x: x is a philosopher} (presumably).
Sets are identical just in case they have all and only the same members.
This means that it doesnt matter if we write {1, 2} or {2, 1} these
sets are the same, they both have 1 and 2 as elements and nothing else.
Similarly, it is redundant to write {1, 1}, since this must be the same setas {1} the set has 1 as an element and nothing else.
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The empty set (sometimes called the null set) is denoted by ; it is
the set with no members. The empty set is unique, i.e., there is only one
empty set, not two, three, or nineteen.
One set is a subset of another if every member of the first set is a member
of the second set, we write this as follows: A B. So, e.g., {1} {1, 7}.
And if setA is not a subset of set B, we write A * B.
Because all of the members of any setA are, trivially, members ofA, every
set is a subset of itself. A subset of a set A that is not identical to A is
called a proper subset ofA.
The empty set, , is a subset of every other set. This is a bit strange
at first glance, but think about it like this: if a set A is not a subset of
a set B, then there must be some member ofA that isnt a member of
B, but ifA is the empty set, there cant be a member ofA that isnt a
member ofB, since the empty set has no members, so A (the empty set)
is automatically a subset of any set B.
An ordered pair is written < a, b > and, unlike sets, the order in whichthe members of a pair are listed matters.
For two ordered pairs< a, b >and < c, d >,< a, b >=< c, d >if and only
ifa= c and b=d. This means that the pair isnt the same pair
as the pair the first pair has 1 as its first element and 2 as its
second, but the second pair has 2 as its first element and 1 as its second.
We can repeat elements in ordered pairs, e.g., < a, a >. Unlike with sets,
this is not redundant.
We can generalize this notion to that of ordered triples, e.g., < 1, 2, 3 >
is a triple with 1 as its first element, 2 as its second, and 3 as its third.
Obviously we could also introduce the notion of an ordered quadruple,which would have four ordered elements. Or an ordered quintuple, which
would have five ordered elements, etc. The widest generalization is the
notion of an orderedn-tuple, for any natural number n. So we could work
with a notion of an ordered 17-tuple, if we needed to or so desired.
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2 Arguments and Validity
Formally an argument is a pair < , > where is a set of sentences
and is a sentence. The members of are the premisesof the argument
and is the conclusion of the argument. When we write out an argument
we typically list the premises and then indicate the conclusion by sayingsomething like therefore.
Although Ive used our newly introduced notion of an ordered pair to give
a formal definition, really all that is important is that arguments contain
both premises and conclusions. Something we really havent talked about
yet in class that well later discuss: in an argument, the premise set can
be empty.
A clarification: many types of natural language sentences (commands,
questions, etc.) arent truth-apt, i.e., they arent the kinds of things that
can be sensibly said to be either true or false. Truth-apt sentences, some-
times called declarative sentences, will be our focus the premises and
conclusion of an argument must consist of declarative sentences.
There are many relationships that can hold between the premises of an
argument and its conclusion, e.g., the truth of the premises can make
the conclusion more likely to be true. Perhaps the most important such
relationship, and the one that we will focus on is validity. Informally, an
argument is valid if and only if it is not possible for all of its premises to
be true and its conclusion false. In other words, in a valid argument, there
is no possible case or situation in which all of the premises are true and
the conclusion false.
We say that the conclusion of a valid argument follows from its premises
or that the premises entailthe conclusion.
An argument is sound if and only if it is both valid and every one of
its premises is true. Soundness is obviously a great feature for an argu-
ment to have, but determining whether an argument is sound will involve
determining whether its premises are true. And so if the premises con-
cern baseball, evaluating the truth or falsity of the premises will involve
knowledge of baseball. As such, logic wont typically be useful in help-
ing you determine whether or not the premises of a given argument are
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true. However, logic will provide you with tools for assessing the validity
of arguments, no matter the subject matter involved.
As mentioned above, there are other goodmaking features that arguments
can have aside from validity and soundness. One is that the premises can
be relevant to the conclusion. Another is that the premises can support
the truth of the conclusion in a way weaker than validity. This brings
us to the distinction between deductive and inductive arguments, which
you may be familiar with. Sometimes this distinction is spelled out by
saying that deductive arguments move from the general to the specific and
that inductive arguments move from the specific to the general, however
this isnt quite accurate. Its more accurate to say that deductively valid
arguments employ a much stronger notion of support than do inductively
good arguments.
3 Introducing Sentential Logic
The system of logic that well study is known as sentential logicor propo-
sitional logic. I prefer to call it sentential logic; Ill often just refer to it
as SL.
SLis a simple formal system that allows us to study the logic of Englishs
sentential connectives (terms like and, or, not, etc.)
We use simple formal models to study reasoning for much the same rea-
son that scientists use simplified models of natural phenomena, viz., the
models are more tractable. If we do our work well, well be able to learn
something about natural language by studying simple formal languages
but we should always be careful not to be too hasty in drawing conclu-
sions about natural languages from consideration of formal languages.
Natural languages like English have an ever-changing and expanding vo-
cabulary, but a formal language like that of SL has a fixed vocabulary
that we will specify in advance.
The vocabulary ofSL comes in three categories:
1. sentence letters: p, q, r, p1, q1, r1, p2, q2, r2,. . . and generally, forany natural number n: pn, qn, rn
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2. punctuation: ( and ) (brackets or parentheses)
3. Connectives: (negation/not). (conjunction/and), (disjunc-
tion/or), (conditional/if...then), (biconditional/if and only if)
An expression of SL is a sequence of items from SLs vocabulary. If we
wanted to define this formally, we could use a generalization of the notionof an n-tuple developed above (strictly for the curious: a function from a
subset of the set of natural numbers to items ofSLs vocabulary).
Natural languages like English also include grammatical rules that de-
termine which expressions drawn from their own vocabulary count as
(grammatical) sentences. Similarly the formal language ofSLwill include
grammatical rules that tell use which expressions ofSLare sentences. As
before though, the rules of our formal language will be incredibly simple
compared to the complex grammatical rules of a natural language.
The grammar ofSL has three clauses:
1. All sentence letters are sentences ofSL
2. Ifand are sentences ofSL, then so too are, ( ),( ),
( ), and ( )
3. Nothing else is a sentence ofSL
Clause (2) ofSLs grammar uses greek letters as variables for sentences
ofSL. These are sometimes called meta-variables or metalinguistic vari-ables. It is important to note that and are not themselves part of
the language ofSL. These need to be used here so that our grammar can
be recursively applied, i.e., any sentences that result as output from appli-
cations of(1) and (2) can then be used as inputs for further applications
of(2).
UsingSLs grammar we can prove that certain expressions are sentences of
SL, e.g., to see that (rp) is a sentence ofSL, note that by(1), sentence
letters r and p are sentences ofSL, so by(2), r is a sentence ofS L,
and so by (2) and what weve already established, (rp) is a sentence
ofSL.
Clause (3) is needed in order to prove that some expressions are not sen-tences ofSL, e.g., the expression p is not an sentence ofSL since no
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application of clauses(1)and(2)give rise to this expression and by clause
(3) all sentences ofSL can be generated using (1) and (2).
So far we have said nothing about the meanings of sentences of SL
well start discussing this next in class next week.
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