Ch 1 Sets, Relations And Arguments Flashcards
What is a set?
A collection of objects
What are the objects in a collection called?
Elements of that set
When are sets identical?
Iff they have the same elements
What is the set that contains no elements called?
The empty set
Definition of a binary relation?
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
A binary relation is reflexive on a set S iff
For all elements d of S the pair is an element of R
A binary relation R is symmetric on a set S iff
For all elements d, e of S: if € R then € R
A binary relation R is asymmetric on a set S iff
For no elements d,e of S: €R and €R
A binary relation R is antisymmetric on a set S iff
for no two distinct (that is, different) elements d,e of S:
€R and €R
A binary relation R is transitive on a set S iff
For all elements d,e,f of S:
if €R and €R, then also €R
A binary relation R is …
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
What properties does the empty relation have?
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
What is an equivalence relation?
A binary relation R is an equivalence relation on S iff R is reflexive on S, symmetric on S and transitive on S
A binary relation is a function iff
for all d,e,f: if €R and €R then e=f
The domain of a function R is the set
{ d: there is an e such that €R }
The range of a function R is the set
{e: there is a d such that €R }
R is a function into the set S iff
All elements of the range of the function are in S
If d is in the domain of a function R one writes
R(d) for the unique object w such that is in R
What is a ternary relation?
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
What are declarative sentences
Sentences that are true or false
What is an argument?
An argument consists of a set of declarative sentences (the premises) and a declarative sentence (the conclusion) marked as the concluded sentence
How many conclusions in an argument?
Always exactly one
What is logical validity?
An argument is logically valid iff there is no interpretation under which the premises are all true and the conclusion is false
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
‘Hagen is a bachelor. Therefore Hagen is not married.’
This is inductively valid but not logically valid
Does a valid argument need to have a true conclusion?
No
What is propositional validity?
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
What is consistency?
A set of sentences is logically consistent iff there is at least one interpretation under which all sentences of the set are true
How can you define validity in terms of consistency?
An argument is valid iff the set obtained by adding the negation of the conclusion to the premises is inconsistent
Can arguments with no premises still be logically valid?
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.
What is a logical truth?
A sentence is logically true iff it is true under any interpretation
What is a contradiction
A sentence is a contradiction iff it is false under all interpretations
What is logical equivalence?
Sentences are logically equivalent iff they are true under exactly the same interpretations
