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

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.

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.

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.

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.

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

Consider the following chart: F G 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

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.

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

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

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

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.

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.

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.

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.