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

Last Time: Deductive Argument Forms Formal Fallacies 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 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). 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 Substance Quantity Quality Relation Place 6. Time 7. Position 8. State 9. Action 10. Affection

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.

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

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

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: 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. …

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: 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: 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)]

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 44

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