Today’s Topics

1. Today’s Topics • Limits to the Usefulness of Venn’s Diagrams • Predicates (Properties and Relations) • Variables (free, bound, individual, constants)

2. Limits to Venn’s diagrams • We cannot construct a proper diagram (no region left out, none duplicated) for an argument involving 6 or more terms. • We CAN construct 16 and 32 region “Venn diagrams” for both 4 and 5 term arguments, but they are really hard to use. • This limit is linked to what mathematicians call the 4 color problem. Any may can be colored with 4 colors and have no adjoining regions the same color.

3. A 16 region Venn-type diagram for a 4 term argument

4. An Imposter Venn-type diagram for a 4 term argument

5. A 32 region Venn-type diagram for a 5 term argument

6. The Joker “Venn Diagram”

7. Propositional and syllogistic logic have serious limitations. • Some arguments that are clearly valid cannot be shown valid in our system (remember DeMorgan’s valid sentence). • Propositional logic misses the internal structure of sentences. • Syllogistic logic cannot deal with more than 4 terms, nor can it deal with relations. • We need a new, more powerful, tool: Predicate Logic.

8. The two central concepts in predicate logic are: • The Predicate (Property Constant) • The Variable

9. A PREDICATE is either a property of an object or a relation between a group of objects. • A predicate is represented symbolically by a capital letter followed by one or more lower case variable letters. • Hx = ‘x is happy’ • Txy = ‘x is taller than y’

10. Monadic and Polyadic (Relational) Predicates • A predicate that expresses a property of an object is monadic, it applies to only one thing. • A predicate that expresses a relation between objects is polyadic or relational, it applies to an ordered set of objects.

11. A VARIABLE is a place-holder in a formulaic expression. • An individual variable is a true place holder, expressed symbolically with lower case letters taken from the end of the alphabet, t, u, v,...z. • An individual constant stands for a specific individual, represented by a lower case letter taken from the beginning of the alphabet).

12. Predicates plus variables allow us to describe individual and relations more fully • Let Hx = ‘x is happy’ and Txy = ‘x is taller than y’ and ‘a’ be the individual constant for Alice and ‘b’ the individual constant for Bob. • Hb says that Bob is happy. • Tba says that Bob is taller than Alice

13. Sentences containing free individual variables are called open sentences and have no truth value. • Hx says only that x, whoever that is, is happy. Since the value of x is open, we can’t assign a truth value to Hx. • Replacing free variables with individual constants turns the open sentence into a closed sentence with a truth value.

14. Another way to close an open sentence is to BIND the free individual variables with QUANTIFIERS. • There are only 2 quantifiers in English: All and Some.

15. Symbolizing quantifiers • The universal quantifier, all, is represented with an upside down A-- -- followed by a variable letter ( x says ‘for any x’) • Some systems of logic (the text, LogicWorks) use a variable in parens—(x)--as the symbol for the Universal Quantifier. We shall simply be symbolically polyglot. • The existential quantifier, some, is represented with a backwards E----followed by a variable letter ( x says ‘there exists an x such that’)

16. Any variable that falls within the scope of a quantifier is bound by that quantifier (see pp. 362-372). • The parentheses following a quantifier mark its scope. • xHx says “everybody is happy.” • x(Txb  Hx) says “someone is taller than Bob and that person is happy.”

17. If all the variables in a sentence are either bound or individual constants, the sentence is closed. • So, while ‘Hx’ is an open sentence, ‘ xHx,’ in which the second ‘x’ is a bound variable, is closed and means ‘someone is happy.’ • However, in ‘xHx  Txy’ both the third x and the y are free, outside the scope of any quanitfier, and thus the sentence is open.

18. In which, if any, of the following WFF’s are there free variables? • x(Fx  (Ga ● Hx)) • x(Fx  (Ga ● Hy)) • xFx y(Fy ● Rxy) • x(Fx y(Fy ● Rxy)) • xFx y(Fy ● Gy) • x(Tx  (Se▼Bxe)) • x(Tx y(Sy▼Bzy))

19. Answers • 1. No free variables • 2. The ‘y’ is free • 3. The second ‘x’ is free • 4. No free variables • 5. No free variables • 6. No free variables • 7. The ‘z’ is free

20. Symbolizing with Quantifiers: • The material inside the parenthesis following a quantifier is called the matrix of the formula. • The dominant operator in the matrix of a universally quantified proposition will almost always be the conditional. • The word “are” indicates the dominant operator • Relative clauses (All ’s who are ’s are ’s) indicate a compound antecedent. • The dominant operator in the matrix of an existentially quantified proposition will almost always be conjunction.

21. Common Errors in Symbolizing with Quantifiers • Sentences beginning with “A” do not follow strict rules: • ‘A barking dog never bites’ is a universal claim, but • ‘A barking dog is in the road’ is an existential claim.

22. Common Errors in Symbolizing with Quantifiers • Sentences beginning with “A” do not follow strict rules: • ‘He who’ sentences are universal claims

23. Common Errors in Symbolizing with Quantifiers • Sentences beginning with “A” do not follow strict rules: • ‘He who’ sentences are universal claims • ‘He who lives by the sword dies by the sword’ is a universal claim

24. ‘Common Errors in Symbolizing with Quantifiers • Sentences beginning with “A” do not follow strict rules: • ‘He who’ sentences are universal claims • ‘Any’ and ‘every’ are not synonymous when following negations • ‘Hamner is not taller than any NBA player’ is false, but • ‘Hamner is not taller than every NBA player’ is true.

25. Common Errors in Symbolizing with Quantifiers • Sentences beginning with “A” do not follow strict rules: • ‘He who’ sentences are universal claims • ‘Any’ and ‘every’ are not synonymous when following negations • The problem of ‘only’

26. The problem of ‘only’ • In English sentences beginning with ‘only,’ the grammatical subject is the logical predicate. ‘Only freshmen are eligible’ means ‘All who are eligible are freshmen.’

27. Troubling occurrences of ‘and.’ • Sometimes ‘and’ does not signal conjunction. • ‘Hamner and Peggy are married’ indicates a relational predicate, not a conjunction • ‘Women and children are exempt’ says that whoever is either a woman or a child is exempt, NOT that whoever is a woman/child is exempt. • ‘Some dogs and cats do not make good pets’ does not, the cartoon notwithstanding, indicate that there are cat-dogs who do not make good pets

28. Try some on your own. • Download the Predicate Study Guide from the Handouts section and review it. • Download the Handout entitled Predicate Translation Exercises and try some. Post and discuss your answers. Be careful to use quantifiers when necessary, but remember that we do not quantify across individual constants—names have a fixed value.