Philosophy 103Linguistics 103

Yet, still, Even further More and more

Introductory Logic: Critical Thinking

Dr. Robert Barnard

Last Time:• Deductive Argument Forms– Modus Ponens– Modus Tollens– Disjunctive and Hypothetical Syllogism– Reductio ad Absurdum

• Formal Fallacies• Counter Example Construction

Plan for Today• 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 TopicsThe CategoriesThe Prior Analytics ***The Posterior Analytics

What is Categorical LogicCategorical 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).

• 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):

• All houses have roofs• Some buildings are houses• No eggs are shatterproof• Some people are not paying attention

Aristotle’s Categories

1. Substance2. Quantity 3. Quality4. Relation5. Place

6. Time7. Position8. State9. Action 10. Affection

Categories Explained I1. 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 II3. 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

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 IV8. 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 CategoriesAristotle 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 PropertiesAn 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.

Accidental PropertiesAn 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 PMBryant 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 ConceptsA 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 TruthsWhen 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 IIA 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 StatementsA 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)

SummaryThus 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 103Linguistics 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

• Traditional Square of Opposition• Modern Square of Opposition• Existential Fallacy• Venn Diagrams for Propositions

Week -

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

Week -

• Categorical Propositions• Conditional and Conjunctive equivalents• Existential Import• Traditional Square of Opposition• Modern Square of Opposition• Existential Fallacy• Venn Diagrams for Propositions

Week-

• Immediate Inferences• Conversion• Contraposition• Obversion

Week-

• 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

Week -

• Logic of Propositions• Decision Problem for Propositional Logic• Symbolization and Definition• Translation Basics

Week -

• Truth Tables for Propositions• Tautology• Contingency• Self-Contradiction

Week -

• 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

Week -

• Logical Truths• Necessity • Possibility• Impossibility