Philosophy 103
Linguistics 103
Yet, still, Even further More and more
Introductory Logic:
Critical Thinking
Dr. Robert Barnard
Last Time:
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Deductive Argument Forms
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Modus Ponens
Modus Tollens
Disjunctive and Hypothetical Syllogism
Reductio ad Absurdum
Formal Fallacies
Counter Example Construction
Plan for Today
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Introduction to Categorical Logic
Aristotle’s Categories
Leibniz, Concepts, and Identity
Analytic – Synthetic Distinction
Essence and Accident
Necessary and Sufficient Conditions
Welcome to
The Land of Big Thinkers
The Science we now call
LOGIC started as an attempt
to codify certain well accepted
an idealized patterns of
reasoning. Logic is practiced
by e.g. Plato, but it is first
laid out by Aristotle in his
ORGANON:
The Topics
The Categories
The Prior Analytics ***
The Posterior Analytics
What is Categorical Logic
Categorical Logic is the Logic of Aristotle
(with some further developments)
• Aristotle thought that everything in the
universe was definable using a set of related
categories in Nature.
• The methods of Logic could then be used to
explain or understand the natural world.
What is a “Category”
In Categorical Logic, a CATEGORY is a class or group
of things (or at least of description of such a
class).
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All Dogs
All Dogs with fleas
All Brown Dogs with Fleas
All Brown Dogs with Fleas in Mississippi
Barak Obama (an individual is a class with 1 member)
Categorical Propositions
The basic Unit of Categorical Logic is the
CATEGORICAL PROPOSITION.
• Every Categorical Proposition relates two
terms: Subject Term and Predicate Term
• Both terms denote classes or categories.
Categorical Propositions
Categorical Propositions relate one category (in
whole or part) to another category (either
affirmatively or negatively):
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All houses have roofs
Some buildings are houses
No eggs are shatterproof
Some people are not paying attention
Aristotle’s Categories
1.
2.
3.
4.
5.
Substance
Quantity
Quality
Relation
Place
6. Time
7. Position
8. State
9. Action
10. Affection
Categories Explained I
1. Substance. -- is defined as that which can be said to be
predicated of nothing nor be said to be within anything.
– "this particular man" or "that particular tree" are substances.
– Aristotle calls these particulars "primary substances," to distinguish them
from "secondary substances," which are universals.
– Hence, "Socrates" is a primary Substance, while "man" is a secondary
substance.
2. Quantity. This is the spatial extension, size, dimension of
an object.
– The house is 30 feet wide.
– The man is tall.
Categories Explained II
3. Quality. This is a determination which
characterizes the nature of an object.
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The Tree is wooden.
The apple is red.
4. Relation This is the way in which one object may be related to
another.
– The car is to the left of the tree
– All ducks are smaller than Elephants
Categories Explained III
5. Place Position in relation to the surrounding
environment.
– The Student is on the Grove
– Some fish are in the river.
6. Time Position in relation to the course of events.
– Tom came home today
– Fred opened the door first
7. Position - a condition of rest resulting from an action:
‘Lying’, ‘sitting’.
– All boats on the lake are floating
– All pilots are flying
Categories Explained IV
8. State The examples Aristotle gives indicate that he meant a
condition of rest resulting from an affection:
– Fred is well fed
– The horse is shod
– The Soldier is armed
9. Action The production of change in some other object.
10. Affection The reception of change from some other object.
action is to affection as the active voice is to the passive.
Thus for action Aristotle gave the example, ‘to lance’, ‘to
cauterize’; for affection, ‘to be lanced’, ‘to be cauterized.’
Affection is not a kind of emotion or passion.
Using Aristotle’s Categories
Aristotle thought that:
a) Everything able to be said was said using the
various categories.
b) Everything that is or that happens can be
explained by appealing to the 10 categories.
Explain Rain:
Rain is wet. Rain is water. Rain falls. Rain wets the
ground. …
Essence and Accident
We can use CATEGORY terms to talk about a
thing or substance, but there is a difference
between what a substance IS and how it
seems or appears.
What a thing IS -- is determined by its ESSENCE
• How a thing seems or appears is determined by its
ACCIDENTS
• The same attribute can sometimes be essential or
accidental
Essential Properties
An ESSENTIAL attribute of a thing is that which
the thing MUST have in order to be THAT
THING:
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Color is an essential attribute of something red
Maleness is an essential attribute of a father
Being an egg-layer is an essential attribute of a hen.
Being strong enough to support a person’s weight is
an essential attribute of a chair
• Being four sided is an essential attribute of a square.
Accidental Properties
An ACCIDENTAL attribute of a thing is that which the thing
does have, but need not have:
• Color is an accidental attribute of something wooden
• Maleness is an accidental attribute of a human
• Being an egg-layer is an accidental attribute of a animal.
• Being strong enough to support a person’s weight is an
accidental attribute of a piece of rope
• Being four sided is an accidental attribute of a table.
Particular and Universal Kinds
To give an account of a General or Universal
kind, one need only give an account of the
essential attributes of that kind .
To identify a particular thing at a particular time
and place requires that we list both essential
and accidental attributes.
ATTRIBUTES OF A BOOK:
Essential
Attributes:
•Has Covers
•Has Pages
Accidental
Attributes:
•Written in
English
•Has Pictures
•Cover is Leather
•Has 200 Pages
Questions?
ANNOUNCEMENT!!!!!
Thursday, September 13, 2007
4:00 PM
Bryant 209
Philosophy Forum Talk –
“Einstein on the Role of History and Philosophy of
Science in Physics”
Dr. Don Howard – University of Notre Dame
Extra Credit: 1 page reaction, due in 2 weeks (9/27)
Categorical Logic!
The MODERN Way…
The German philosopher and
mathematician LEIBNIZ
adapted the Aristotelian
system to give a more
complete account of a
substance and its properties.
Aristotle thought that general
kinds were in nature and that
individuals were special cases
of general kinds.
Leibniz thought that particular
individuals were basic in
nature. Thus each substance
was a particular, and every
property was essential.
Leibniz’s Concepts
A CONCEPT for Leibniz is like the INTENSION
that defines a thing.
A COMPLETE CONCEPT is the set of all
characteristics that a thing has.
Each COMPLETE CONCEPT determines a
particular SUBSTANCE
Necessary Truths
When the predicate term of a Categorical
Proposition is a term that is part of the
complete concept of the Subject Term, then it
is impossible for that Categorical Proposition
to be False. -- Thus it is a NECESSARY TRUTH.
– All Tuesdays are days
– All bluebirds are colored
– All fish have gills.
Necessary Truths II
• A Test: If denying a categorical proposition
yields a contradiction, then that categorical
proposition is a necessary truth.
• When these necessary truths are understood to
follow from our ability to reason alone, they are
sometimes called TRUTHS OF REASON.
• When necessary truths are understood to follow
from the linguistic meaning of the terms
involved then they are often called ANALYTIC
TRUTHS
Leibniz’s Law of Identity
There is a special case of a Leibnizian Necessary
Truth: Leibniz’s Law of Identity:
(a = b)   F (F (a)  F (b) )
This says: If two things called ‘a’ and ‘b’
have all and exactly the same properties,
attributes, or characteristics, then a is
identical to b.
Contingent Truths
When a truth is not necessary, it is said to be
CONTINGENT. (Denying a contingent truth
does not yield a contradiction.)
Contingent Truths are usually truths of
experience or of science:
– Some people prefer fish to chicken
– The tree has red leaves
– All liquids boil
– All falling objects accelerate
Contingent Truths II
A Contingent Truth is sometimes called a
SYNTHETIC TRUTH.
SYNTHESIS is the process of combination. A
SYNTHETIC TRUTH combines or relates two
distinct Concepts.
Questions?
The Logical Payoff!! -- Conditionals
To FULLY understand a CONDITIONAL STATEMENT
such as:
‘If the interest rate drops below 3.2% then we
can expect increased inflation.”
We need to recognize that we can read this as both
a necessary claim and as a contingent claim.
Conditional Statements
A CONDITIONAL PROPOSITION expresses a
logical relation between the ANTECEDENT and
the CONSEQUENT
[C] If (x is y) then (a is b).
(x is y) = the Antecedent Proposition
(a is b) = The Consequent Proposition
Conditional Statements II
[C] If (x is y) then (a is b)
[C] says: (i) if a condition (x is y) is satisfied then
a logical consequence (a is b) MUST also
obtain.
Because (a is b) MUST obtain when (x is y)
obtains we say that (a is b) is a NECESSARY
CONDITION FOR (x is y).
Conditional Statements III
[C] If (x is y) then (a is b)
[C] also says: (ii) if condition (x is y) is satisfied,
then that is SUFFICIENT reason to infer that (a
is b) also obtains.
Thus, we say that (x is y) is a SUFFICIENT
CONDITION for (a is b).
Necessary and Sufficient Conditions
[C] If (x is y) then (a is b)
[C] says:
(i) if a condition (x is y) is satisfied the a logical
consequence (a is b) MUST also NECESSARILY
obtain. [(a is b) is NECESSARY given (x is y)]
(ii) if condition (x is y) is satisfied, then that is
SUFFICIENT reason to infer that (a is b) also
obtains. [(a is b) obtains CONTINGENTLY on
(x is y)]
Necessary and Sufficient Conditions (2)
Term
Definition in terms of ‘IF A THEN B’
NECESSARY
CONDITION
A condition B is said to be necessary for a
condition A, if (and only if) the falsity
(/nonexistence /non-occurrence) [as the case
may be] of B guarantees (or brings about) the
falsity (/nonexistence /non-occurrence) of A.
SUFFICIENT
CONDITION
A condition A is said to be sufficient for a
condition B, if (and only if) the truth
(/existence /occurrence) [as the case may be]
of A guarantees (or brings about) the truth
(/existence /occurrence) of B.
Conditionals expressing Necessary Conditions
• If I live in Mississippi then I live in America
• If Bush is the POTUS, then Bush is at least 35
years old.
• If I roll (5,4) then I roll (9)
• If Mel likes perch then Mel likes fish.
• If Al wears pants then Al wears clothes
• If Jim is a geologist, then Jim studies the Earth.
Conditionals expressing Sufficient Conditions
• If I roll (6,6) then I roll (12)
• If Bush wins the Electoral College, then Bush is
POTUS
• If Mort lives in Memphis, then Mort lives in
America.
• If Lou eats pizza then Lou eats Italian food.
• If I am given two $5 bills, then I am given $10.
Definition by means of Necessary and
Sufficient Conditions
In some cases the set of Necessary conditions
and the set of Sufficient conditions will be the
same. When this is so, we say that, e.g. A is
necessary and sufficient for B. (Abbrev. A iff B)
When A iff B: A is a definition for B and B is a
definition for A
Example: If I roll (1,1) then I roll (2)
Summary
Thus the logic of Aristotle’s Categories, and of
Leibniz’s Complete Concepts can both be
understood as ways of understanding the
necessary (i.e. deductive) and contingent (i.e.
inductive) relationships between two ideas.
Aristotle and Leibniz also provide us with
powerful intellectual tools for thinking about
what an object or idea IS.
Notes:
An excellent resource on Necessary and
Sufficient Conditions is provided by Professor
Norman Swartz at Simon Fraser University:
http://www.sfu.ca/philosophy/swartz/conditions1.htm
Philosophy 103
Linguistics 103
Yet, still, Even further More and yet
more
Introductory Logic:
Critical Thinking
Dr. Robert Barnard
Last Time:
•
•
•
•
•
•
Introduction to Categorical Logic
Aristotle’s Categories
Leibniz, Concepts, and Identity
Analytic – Synthetic Distinction
Essence and Accident
Necessary and Sufficient Conditions
Plan for Today
• Categorical Propositions
– Conditional and Conjunctive Equivalents
– Existential Import
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Traditional Square of Opposition
Modern Square of Opposition
Existential Fallacy
Venn Diagrams for Propositions
Week •
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Categorical Logic
Introduction
Aristotle’s Categories
Leibnizian Concepts
Essence and Accident
Extension and Intension
Realism and Nominalism about Concepts
Necessary and Sufficient Conditions
Week •
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Categorical Propositions
Conditional and Conjunctive equivalents
Existential Import
Traditional Square of Opposition
Modern Square of Opposition
Existential Fallacy
Venn Diagrams for Propositions
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Immediate Inferences
Conversion
Contraposition
Obversion
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Syllogistic Logic
Form- Mood- Figure
Medieval Logic
Venn Diagrams for Syllogisms (Modern)
Week • Venn Diagrams for Syllogisms (traditional)
• Limits of Syllogistic Logic
• Review of Counter-Example Method
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Logic of Propositions
Decision Problem for Propositional Logic
Symbolization and Definition
Translation Basics
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Truth Tables for Propositions
Tautology
Contingency
Self-Contradiction
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Truth Tables for Propositions II
Consistency
Inconsistency
Equivalence
Week • Truth Table for Arguments
• Validity / Invalidity
• Soundness
Week • Indirect Truth Tables
• Formal Construction of Counter-Examples
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Logical Truths
Necessity
Possibility
Impossibility