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

Last Time : Introduction to Categorical Logic Categorical Propositions – Parts and Characteristics – Conditional and Conjunctive Equivalents – Existential Import

Plan for Today Venn Diagrams for Propositions Existential Import in Diagramming Traditional Square of Opposition

REVIEW: THE 4 TYPES of CATEGORICAL PROPOSITION UNIVERSALPARTICULAR AFFIRMATIVE ALL S is PSOME S is P NEGATIVE NO S is PSOME S is not P

REVIEW: A, E, I, and O TERM Proposition FormQuantityQuality AALL S IS PUNIVERSALAFFIRMATIVE ENO S IS PUNIVERSALNEGATIVE ISOME S IS PPARTICULARAFFIRMATIVE OSOME S IS NOT PPARTICULARNEGATIVE

Diagramming Propositions… Diagramming is a tool that can be used to make explicit information that is both descriptive and relational. Geometric Diagrams Blueprints Road Maps Flow Charts

…is FUN!!! We can also diagram CATEGORICAL PROPOSITIONS. They describe a relationship between the subject term (class) and the predicate term (class).

Focus on Standard Diagrams Since there are 4 basic standard form categorical propositions, this means that there are exactly 4 standard diagrams for Categorical Propositions. BUT – there are two flavors of diagrams we might use!

Euler Diagrams (not Standard) AALL S is P ENO S is P ISOME S is P OSome S is not P X X S S P P

Pro and Cons: Pro: Euler Diagrams are very intuitive Con: Euler Diagrams can represent single propositions but are difficult to combine and apply to syllogisms. Con: Euler Diagrams Cannot capture Existential Import in both the Aristotelian AND Modern modes. (more later)

Alternative: Venn Diagrams Venn Diagrams are less intuitive to some people than Euler Diagrams Venn Diagrams Can easily be combined and used in Syllogisms. Venn Diagrams CAN represent alternative modes of Existential Import.

The Basic VENN Diagram SUBJECT CIRCLE PREDICATE CIRCLE S P LABEL RULE 1: SHADING = EMPTY RULE 2: X in a Circle = at least one thing here! X

Questions?

THE UNIVERSAL AFFIRMATIVE TYPE A : ALL S is P Conceptual Claim

THE UNIVERSAL NEGATIVE TYPE E : No S is P Conceptual Claim

THE PARTICULAR AFFIRMATIVE TYPE I: Some S is P At least one thing X is Both S and P Existential Claim

THE PARTICULAR NEGATIVE TYPE O: Some S is not P At least one thing X is S and not P Existential Claim

EXISTENTIAL IMPORT ONLY a proposition with EXISTENTIAL IMPORT requires that there be an instance of the SUBJECT TERM in reality for the proposition to be true. Diagrams with an X indicate EXISTENTIAL IMPORT.

PROPOSITIONS ABOUT INDIVIDUALS In CATEGORICAL LOGIC a proper name denotes a class with one member. Fred Rodgers is Beloved by Millions Fred Beloved

How are the 4 standard CPs related? The Traditional Square of Opposition

Contraries The A Proposition is related to the E proposition as a CONTRARY X is CONTRARY to Y = X and Y cannot both be true at the same time. Thus if A is true: If E is True: If A is False: E is False A is False E is UNDETERMINED

Contraries: Not Both True If both are TRUE then S is all EMPTY and there is no UNIVERSAL Proposition asserted!!!! A E

The Traditional Square of Opposition

Sub-Contraries The SUBCONTRARY RELATION holds between the I- Proposition and the O-Proposition. Sub-Contrary = Not both False at the same time If I is False then O is true If O is False then I is true If O (or I) is True, then I (or O) is undetermined

Sub-Contrary: Not Both False IF both are FALSE, then there is no PARTYICULAR Proposition asserted!!! I O

The Traditional Square of Opposition

Contradictories Contradictory Propositions ALWAYS take opposite TRUTH VALUES A and O are Contradictories E and I are Contradictories

A – O Contradiction If BOTH are True then the Non-P region of S is BOTH empty and contains an object! A O

E – I Contradictories If Both are TRUE, then the overlap Region is EMPTY and contains an object. I E

The Traditional Square of Opposition

Subalternation What is the relation between the UNIVERSAL and the PARTICULAR? If All S is P, what about Some S is P? If No S is P, what about Some S is not P? Subalternation claims that if the Universal is true, then the corresponding Particular is true.

Some Subalternations: If All dogs are Brown, then Some dogs are brown. If All Fish have Gills, then Some Fish have Gills. If All Greeks are Brave, then Some Greeks are Brave

The TRADITIONAL Interpretation The TRADITIONAL or ARISTOTELIAN interpretation allows SUBALTERNATION Because FOR ARISTOTLE all category terms denote REAL objects. -- Every name picks out something in the world.

TRADITIONAL A and E When we want to clearly indicate a TRADITIONAL - ARISTOTELIAN interpretation we need to adapt the A and E Diagrams! XX A E

The Traditional Square of Opposition

Questions?