Ch 1 Sets, Relations And Arguments

Question 1

What is a set?

Answer 1

A collection of objects

Question 2

What are the objects in a collection called?

Answer 2

Elements of that set

Question 3

When are sets identical?

Answer 3

Iff they have the same elements

Question 4

What is the set that contains no elements called?

Answer 4

The empty set

Question 5

Definition of a binary relation?

Answer 5

A set is a binary relation if and only if it contains only ordered pairs

The empty set is a binary relation as it does not contain anything that is not an ordered pair

Question 6

A binary relation is reflexive on a set S iff

Answer 6

For all elements d of S the pair is an element of R

Question 7

A binary relation R is symmetric on a set S iff

Answer 7

For all elements d, e of S: if € R then € R

Question 8

A binary relation R is asymmetric on a set S iff

Answer 8

For no elements d,e of S: €R and €R

Question 9

A binary relation R is antisymmetric on a set S iff

Answer 9

for no two distinct (that is, different) elements d,e of S:

€R and €R

Question 10

A binary relation R is transitive on a set S iff

Answer 10

For all elements d,e,f of S:

if €R and €R, then also €R

Question 11

A binary relation R is …

Answer 11

Symmetric iff it is symmetric on all sets

Asymmetric iff it is asymmetric on all sets

Antisymmetric iff it is antisymmetric on all sets

Transitive if it is transitive on all sets

Question 12

What properties does the empty relation have?

Answer 12

It’s diagram is empty

It is reflexive on the empty set but on no other set

It is symmetric as there is no arrow for which there is not any arrow in the opposite direction.

It is also asymmetric and antisymmetric because there is no arrow for which there is an arrow in the opposite direction.

It is also transitive

Question 13

What is an equivalence relation?

Answer 13

A binary relation R is an equivalence relation on S iff R is reflexive on S, symmetric on S and transitive on S

Question 14

A binary relation is a function iff

Answer 14

for all d,e,f: if €R and €R then e=f

Question 15

The domain of a function R is the set

Answer 15

{ d: there is an e such that €R }

Question 16

The range of a function R is the set

Answer 16

{e: there is a d such that €R }

Question 17

R is a function into the set S iff

Answer 17

All elements of the range of the function are in S

Question 18

If d is in the domain of a function R one writes

Answer 18

R(d) for the unique object w such that is in R

Question 19

What is a ternary relation?

Answer 19

One containing only triples

An n-place relation is a set containing only n-tues. An n-place relation is called a relation of arity n

1-tuples are a special case, a 1-place or unary relation is just some set

Question 20

What are declarative sentences

Answer 20

Sentences that are true or false

Question 21

What is an argument?

Answer 21

An argument consists of a set of declarative sentences (the premises) and a declarative sentence (the conclusion) marked as the concluded sentence

Question 22

How many conclusions in an argument?

Answer 22

Always exactly one

Question 23

What is logical validity?

Answer 23

An argument is logically valid iff there is no interpretation under which the premises are all true and the conclusion is false

Question 24

Give an example of a argument in which the conclusion is bound to be true if the premiss is true but is not logically or formally valid, that is, valid in virtue of its form

Answer 24

‘Hagen is a bachelor. Therefore Hagen is not married.’

This is inductively valid but not logically valid

Question 25

Does a valid argument need to have a true conclusion?

Answer 25

No

Question 26

What is propositional validity?

Answer 26

An argument is propositionally valid iff there is no (re-)interpretation of the sentences in the argument such that all the premises are true and yet the conclusion is false

Question 27

What is consistency?

Answer 27

A set of sentences is logically consistent iff there is at least one interpretation under which all sentences of the set are true

Question 28

How can you define validity in terms of consistency?

Answer 28

An argument is valid iff the set obtained by adding the negation of the conclusion to the premises is inconsistent

Question 29

Can arguments with no premises still be logically valid?

Answer 29

Yes

All metaphysicians are metaphysicians

A conclusion that is always true. Therefore no interpretation under which all the premises are true (there is none) and the conclusion is false. Therefore, the argument is logically valid.

Question 30

What is a logical truth?

Answer 30

A sentence is logically true iff it is true under any interpretation

Question 31

What is a contradiction

Answer 31

A sentence is a contradiction iff it is false under all interpretations

Question 32

What is logical equivalence?

Answer 32

Sentences are logically equivalent iff they are true under exactly the same interpretations