Today’s Topics

1. Today’s Topics • Symbolizing with Quantifiers • Truth functional expansions

2. 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.

3. 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.

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

5. 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

6. ‘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.

7. 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’

8. 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.’

9. 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

10. Truth Functional Expansions • Every quantified formula ranges over a universe of discourse. • The elements in the universe are the things that have the properties or stand in the relations described by the formula, they are the values of the bound variables. • Quantified formulas make claims about the entire universe of discourse.

11. Specifying a universe of discourse • We can specify a universe of discourse. • Consider the 3 element universe {a, b, c} • ‘(x)(Fx  Gx)’ says of these elements that ‘[(Fa  Ga)  (Fb  Gb)]  (Fc  Gc)’ • ‘(x)(Fx  Gx)’ says of these elements that ‘[(Fa  Ga) v (Fb  Gb)] v (Fc  Gc)’ • Now, if we know the properties of ‘a,’ ‘b,’ and ‘c,’ we can determine if either of the formulas is true in the universe.

12. The truth-functional expansion of a universally quantified proposition is a conjunction of the instances, the truth-functional expansion of an existentially quantified proposition is a disjunction of the instances.

13. Interpretations of the Extension of Predicates • Knowing the properties of the elements in a universe means knowing the extension of the predicates being used—the set or subset of objects having the property or standing in the relation • A description of the extension of the predicates in a universe is called an interpretation of that universe

14. Consider the following chart: FG a + - b + + c - - This chart presents an interpretation of the 3 element universe {a, b, c} and the predicates F and G. It says that ‘a’ is an F but not a G, ‘b’ is both F and G, and ‘c’ is neither F nor G. (x)(Fx  Gx) is FALSE in this universe, but (x)(Fx  Gx) is TRUE

15. The truth-functional expansion of a universally quantified proposition is a conjunction of the instances, the truth-functional expansion of an existentially quantified proposition is a disjunction of the instances.

16. Expansions can be used to show that a statement, or set of statements, is consistent or that an argument is non-valid.

17. To show that a statement, or set of statements, is consistent, show that there is some interpretation in which all the statements are true

18. Let Px = x is a philosopher • Mx = x is male • Fx = x is female • (x)(Px  Mx) and (y)(Py  Fy) are consistent. • Consider the 2 element universe {a, b} where a is Al Hayward and b is Bambi Robinson. • In that universe, both claims are true, so the pair is consistent

19. To show that an argument is non-valid, first generate a truth functional expansion for the premises and the conclusion, then use the abbreviated truth table method to show non-validity, I.e., that the premises can be true and the conclusion false.

20. Consider the argument: • (x)(Px  Mx), (x)(Qx  Mx) (x)(Px  Qx) • Expand this argument across the 2 element universe {a, b} to get: • {[(Pa  Ma) (Pb  Mb)]  (Qa  Ma)}  (Qb  Mb) • (Pa  Qa)  (Pb  Qb) • If Pa, Pb, Ma and Mb are true, while either Qa or Qb is false, the non-validity of the argument is established.

21. Alternatively, you can simply create an interpretation of the predicates under which the argument is shown to be non-valid • In the previous example, let Px = x is greater than 6, Mx = x is greater than 4, Qx = x is greater than 10.

22. Try some on your own. • Download the Expansions Study guide and review it. • Download the Expansions Exercises and create some expansions of your own and then determine whether the quantified formulas are true or false in a specified uinverse.