2. Philosophy 103 Linguistics 103 Yet, still, Even further More and more Introductory Logic: Critical Thinking
Dr. Robert Barnard 2

3. Last Time: Deductive Argument Forms Formal Fallacies
Modus Ponens
Modus Tollens
Disjunctive and Hypothetical Syllogism
Reductio ad Absurdum
Formal Fallacies
Counter Example Construction

4. Plan for Today Introduction to Categorical Logic
Aristotle’s Categories
Leibniz, Concepts, and Identity
Analytic – Synthetic Distinction
Essence and Accident
Necessary and Sufficient Conditions

5. 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

6. 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.

7. 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).
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)

8. 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.

9. Categorical Propositions
Categorical Propositions relate one category (in whole or part) to another category (either affirmatively or negatively):
All houses have roofs
Some buildings are houses
No eggs are shatterproof
Some people are not paying attention

10. Aristotle’s Categories
Substance
Quantity
Quality
Relation
Place
6. Time 7. Position 8. State 9. Action 10. Affection

11. Categories Explained I
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.
Quantity. This is the spatial extension, size, dimension of an object.
The house is 30 feet wide.
The man is tall.

12. Categories Explained II
Quality. This is a determination which characterizes the nature of an object.
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

13. Categories Explained III
5. Place Position in relation to the surrounding environment.
The Student is on the Grove
Some fish are in the river.
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

14. 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.

15. Using Aristotle’s Categories
Aristotle thought that:
Everything able to be said was said using the various categories.
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. …

16. 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

17. Essential Properties
An ESSENTIAL attribute of a thing is that which the thing MUST have in order to be THAT THING:
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.

18. 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.

19. 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.

20. ATTRIBUTES OF A BOOK: Essential Attributes: Has Covers Has Pages
Accidental Attributes:
Written in English
Has Pictures
Cover is Leather
Has 200 Pages

21. Questions?

22. 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)

23. 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.

24. 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

25. 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.

26. 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

27. 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.

28. 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

29. 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.

30. Questions?

31. 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.

32. 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

33. 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).

34. 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).

35. Necessary and Sufficient Conditions
[C] If (x is y) then (a is b)
[C] says:
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)]
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)]

36. 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.

37. 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.

38. 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.

39. 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)

40. 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.

41. Notes:
An excellent resource on Necessary and Sufficient Conditions is provided by Professor Norman Swartz at Simon Fraser University:

44. Philosophy 103 Linguistics 103 Yet, still, Even further More and yet more Introductory Logic: Critical Thinking
Dr. Robert Barnard 44

45. Last Time: Introduction to Categorical Logic Aristotle’s Categories
Leibniz, Concepts, and Identity
Analytic – Synthetic Distinction
Essence and Accident
Necessary and Sufficient Conditions

46. Plan for Today Categorical Propositions
Conditional and Conjunctive Equivalents
Existential Import
Traditional Square of Opposition
Modern Square of Opposition
Existential Fallacy
Venn Diagrams for Propositions

53. Week - Categorical Logic Introduction Aristotle’s Categories
Leibnizian Concepts
Essence and Accident
Extension and Intension
Realism and Nominalism about Concepts
Necessary and Sufficient Conditions

54. Week - Categorical Propositions
Conditional and Conjunctive equivalents
Existential Import
Traditional Square of Opposition
Modern Square of Opposition
Existential Fallacy
Venn Diagrams for Propositions

55. Week-
Immediate Inferences
Conversion
Contraposition
Obversion

56. Week- Syllogistic Logic Form- Mood- Figure Medieval Logic
Venn Diagrams for Syllogisms (Modern)

57. Week - Venn Diagrams for Syllogisms (traditional)
Limits of Syllogistic Logic
Review of Counter-Example Method

58. Week - Logic of Propositions Decision Problem for Propositional Logic
Symbolization and Definition
Translation Basics

59. Week - Truth Tables for Propositions Tautology Contingency
Self-Contradiction

60. Week - Truth Tables for Propositions II Consistency Inconsistency
Equivalence

61. Week -
Truth Table for Arguments
Validity / Invalidity
Soundness

62. Week -
Indirect Truth Tables
Formal Construction of Counter-Examples

63. Week -
Logical Truths
Necessity
Possibility
Impossibility