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Lecture

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Philosophy

Philosophy 2250

Lorne Falkenstein

Fall

Description

Introductory Lecture Kayla Ann Marie
This is a course in basic formal logic
Formal logic is the study of an argument’s form.
When we reason we argue from a collection of known things: the
premise of our argument.
To conclude to another thing that is not so well known.
The forms arguments take can be good or bad.
EX: good
o I have five toes on my left foot. The left and right sides of
animals are generally approximate reflections of one another.
I have five toes on my right foot.
EX: bad
o I have five toes on my left foot. Stephen Harper is the Prime
Minister of Canada.
Different arguments can follow a common form or pattern:
All A’s are B’s. All B’s are C’s. All A’s are C’s.
Some A’s are B’s. Some B’s are C’s. Some A’s are C’s.
a, b, and c are A’s. d is like a, b, and b. d is an A
If P then Q. But not-P, Not-q. If P then Q. P, Q.
a is a B. Most B’s are C’s. a is a C.
What separates good and bad arguments?
Good argument forms are of two kinds:
o Deductive
o Inductive
We are concerned with deductively good arguments in this course.
o Inductively good argument forms establish the likelihood of
the truth of the conclusion.
When an argument form is deductively good, the form itself makes
it impossible to have all true premises and a false conclusion. So,
when an argument form is deductively good, it is impossible to
think of a substitution instance (an argument with that form) that
does have either: at least one false premises OR a true conclusion.
EX:
o All A’s are B’s. All B’s are C’s. All A’s are C’s.
o All lawyers are greedy. All greedy people are untrustworthy.
All lawyers are untrustworthy o All cats are felines. All felines are mammals. All cats are
mammals.
Method for proving an argument form is bad: a counter example,
think up a single argument that shares that form that has all true
premises and a false conclusion.
EX:
o Some cats weigh over 12 pounds. Some animals weighing
over 12 pounds are dogs. Therefore, some cats are dogs.
o How do you prove that a deductive argument form is a good
one?
First step: logicians construct a formal language that
represents just those features of natural language that
are relevant to the forms of arguments.
Most basic and general formal elements have to do with
the way entire sentences are compounded in arguments
using logically relevant “connective” expressions such
as “and,” “or,” and “not.”
This course begins with the study of that logic, sentence
logic.
Why the study of formal deductive logic is important:
Constructing formal language that represents logical features of
natural language tells us something about how natural languages
work. It also tells us how to construct machines that will compute
the right results.
Tells us how to understand how the mind processes information.
nd
It was once thought that mathematics could be reduced to 2 order
predicate logic, the study of formal logic contributes to
understanding the foundations of mathematics.
Formal logic gives us an example of an especially rigorous and
precise way of approaching any topic. The way you learn to pove
results in formal logic is highly instructive for effectively proving
results in any other field.
F.L has been the model for discourse in contemporary analytic
philosophy and a basic understanding of it is required to understand
papers on a wide range of philosophical topics, from philosophy of
language to mathetics. When we reason we make claims that are generally well known to be true,
and then draw conclusions from those claims that are less well known to be
true.
To make a claim that is generally regarded as trust is to say or
write a certain kind of sentences: a “declarative” sentence.
Sentences of this sort are the most basic thing that logic is
concerned with.
Exercise 1.3#1: Declarative Setences
When we reason we make claims that are generally well known to be true,
and then draw conclusions from those claims that are less well known to be
true.
Arguments are therefore used to persuade and to make discoveries.
They are to be distinguished from collections of sentences that
merely serve to report a number of facts without attempting to
justify any of them.
Also to be distinguished from collections of sentences that make
claims that are generally not well known to be true and appeal to
those claims to explain why something that is much better known
came to be.
In this courses, arguments will be put in “standard form.” This means
numbering and listing premises first and placing the conclusion last under a
bar.
Deductive Validity:
An argument form is deductively valid if and only if each
substitution instance of that form either has at least one false
premise or has a true conclusion.
An arugment form is deductively invalid as long as there is even on
substitution instance of that form that has all true premises and a
false conclusion.
An argument is deductively valid if and nly if it has a deductively
valid form.
An argument is deductively invalid if and only if it does not have a
deductively valid form.
Deductively Valid Arguments can have: o True premises and true conclusions.
o False Premises and true conclusions.
o False premises and false conclusions.
o False premises and true conclusion.
Deductively Invalid Arguments can have:
o True premises and true conclusions.
o True premises and false conclusions.
o False premises and true conclusions.
o False premises and false conclusions.
Deductive Soundness:
An argument is deductively sound if and only if it both has a
deductively valid form and it has all true premises.
An argument is not deductively sound if and only if it either has at
least one false premise or has deductively invalid form.
Deductively sound arguments must have true conclusions.
This is not necessarily the case for deductively valid arguments
Objects of Classical Formal Logic:
Sentences
Sets of sentences
Arguments
Sentence of Classical Logic: a set of sentences of classical logic
must have one or the other, but not both, of exactly two truth
values, true (T) and false (F). Other sentences do not exist as far as
classical logic is concerned.
Set of sentences of classical logic: a set of sentences of classical
logic is a list of from none to infinitely many sentences of classical
logic placed between braces ({,}). A set with no members , is the
empty or null set. A set with exactly one member is a unit set.
Argument: an argument is a collection of two or more sentences,
one of which is identified or intended as a conclusion.
The Properties and Relations of these Objects:
Logical truth, falsity, and indeterminacy
Logical equivalence
Logical entailment
Deductive Validity and invalidity Logical truth, falsity, and indeterminacy are properties of
sentences.
o It is improper to describe arguments or sets as true, false, or
indeterminate.
Logical equivalence is a relation between certain pairs of sentences.
Logical consistency and inconsistency are properties of sets of
sentences.
o It is improper to describe sentences or arguments as
consistent or inconsistent.
Logical entailment is a relation that holds between certain sets and
certain sentences.
Deductive validity and invalidity are properties of arguments.
o It is improper to describe sentences or sets as valid or invalid.
These properties and relations can be defined in terms of:
Form (just done for deductive validity)
Non-contradiction (handout definitions)
Possibility (textbook definitions)
Contradiction: a contradiction is either a sentence of the form “P
but not-P” or two sentences, one of the form “P,” and the other of
the form, “not-P.”
o You are caught in a contradiction when you end up saying
something that reduces to a contradiction.
E.g., denying that all fish are fish reduces to saying that
there something that is a fish but that thing is not a
fish.
Definitions in terms of contradiction:
A sentence of classical logic is logically true if and only if you
would get caught in a contradiction were you to deny it. E.g., “All
fish are fish.”
A sentence of classical logic is logically false if and only if you
would get caught in a contradiction were you to affirm it. E.g.,
“Some fish are not fish.”
A sentence of classical logic is logically indeterminate if and only
if there is no contradiction in either affirming or denying it. E.g.,
“There are more than 80 people in this room.” A pair of sentences of classical logic is logically equivalent if and
only if there would be a contradiction in supposing that they have
different truth values. E.g., “Some hawks are osprey.” “Some
osprey are hawks.”
A pair of sentences of classical logic is not logically equivalent if
and only if there would be a contradiction in supposing that they
have different truth values. E.g., “All whales are mammals.” “All
mammals are whales.”
A set of sentences of classical logic is logically consistent if and
only if there is no contradiction in suppos

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