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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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