Todays Topics,Limits to the Usefulness of Venns Diagrams Predicates (Properties and Relations) Variables (free, bound, individual, constants),Limits to Venns 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.,A 16 region Venn-type diagram for a 4 term argument,An Imposter Venn-type diagram for a 4 term argument,A 32 region Venn-type diagram for a 5 term argument,Propositional and syllogistic logic have serious limitations.,Some arguments that are clearly valid cannot be shown valid in our system (remember DeMorgans 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.,The two central concepts in predicate logic are:,The Predicate (Property Constant) The Variable,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,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.,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).,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,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 cant assign a truth value to Hx. Replacing free variables with individual constants turns the open sentence into a closed sentence with a truth value.,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.,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),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.”,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.,In which, if any, of the following WFFs 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 (SeBxe) x(Tx y(SyBzy),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,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.,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.,Common Errors in Symbolizing with Quantifiers,Sentences beginning with “A” do not follow strict rules: He who sentences are universal claims,