Today’s Topics

1. Today’s Topics Universes of Discourse Truth Functional Expansions of Quantified Formulae Expansions, Consistency and Validity

2. 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.

3. Truth Functional Expansions • A quantified formula makes a claim about the elements in its universe of discourse. • Replacing a variable with an element in the universe, we get an instance of the formula. • Consider the formula ‘(x)(Fx  Gx)’ and the 2 element universe {a, b}. • ‘Fa  Ga’ is one instance, ‘Fb  Gb’ is another. • If every element is substituted for the bound variables, we get a truth functional expansion.

4. 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.

5. 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.

6. 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

7. 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

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

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

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

11. 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

12. 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.

13. 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.

14. 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.

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