Predicate Logic combines the distinctive features of syllogistic and propositional logic.
◦ The fundamental component in predicate logic is the predicate, which is always symbolized with upper case letters. ◦ Symbolic predicates may be used to translate three kinds of statements:
Singular Statements: Socrates is dead.
Universal Statements: All S are P or No S are P .
Particular Statements: Some S are P or Some S are not P. Translating singular statements requires lower case letters called individual constants and the logical operators of propositional logic.
◦ “Socrates is mortal” is symbolically translated as “Ms.”
◦ “If Paris is beautiful, then André told the truth,” becomes Bp Ta. Translating universal statements requires universal quantifiers, individual variables, and the horseshoe operator:
◦ “All S are P” is translated as x (Sx Px), meaning, for any x, if x is an S, then x is a P. ◦ “No S are P” is translated as x (Sx ~Px), meaning, for any x, if x is an S, then x is not a P. ◦ The Venn diagrams responding to these statements are: Translating particular statements requires existential quantifiers, individual variables, and the dot operator. ◦ “Some S are P” is translated as (Ǝx) (Sx • Px) or “There exists such an x such that x is an S and x is a P.” ◦ “Some S are not P” is translated as (Ǝx) (Sx • ~Px) or “There exists such an x such that x is an S and x is not a P.” ◦ The Venn diagrams responding to these statements are: For both universal and particular statements, we need to distinguish between free and bound variables.
◦ Free variables are not bound by any quantifier. Bound variables are. Four rules govern the removal and introduction of quantifiers:
◦ Universal Instantiation (UI): This allows universal quantifiers to be removed. It requires using either a constant or a variable.
◦ Universal Generalization (UG): This allows universal quantifiers to be introduced. Universal Generalization cannot be performed if the instantial letter is a constant. ◦ Existential Generalization allows existential quantifiers to be introduced. There are no restrictions on Existential Generalization.
◦ Existential Instantiation allows for existential quantifiers to be removed. Existential instantiation requires that the existential name must be a new name that does not occur on the line that indicates the conclusion to be derived. This name must be new and many not occur earlier in the proof. Common Misapplications:
◦ All four of these rules of inference require that the rules be applied only to whole lines in the proof. As long as negation signs or ~s precede quantifiers, statements cannot be instantiated; thus, conclusions cannot be derived. The Quantifier Negation Rule allows us to remove the negation signs.
The Quantifier Negation Rule (QN) is expressed as: (x) Fx :: ~(Ǝx) ~Fx ~(x) Fx :: (Ǝx) ~Fx (Ǝx) Fx :: ~(x) ~Fx ~(Ǝx) Fx :: (x) ~Fx One type of quantifier can be replaced by the other type if and only if it appears immediately before and after the new quantifier:
1. Tilde operators that were originally present are deleted.
2. Tilde operators that were not originally present are inserted. Many arguments with conclusions that are either difficult or impossible to derive by the conventional method can be handled with ease in using either conditional or indirect proof. 1. (x) (Hx Ix) / (Ǝx)Hx (Ǝx) Ix 2. (Ǝx) Hx ACP 3. Ha 2, EI 4. Ha Ia I, UI 5. Ia 3, 4, MP 6. (Ǝx) Ix 5, EG 7. (Ǝx)Hx (Ǝx) Ix 2-6, CP UG must not be used within the scope of an indented sequence if the instantial variable y occurs free (unbound by any quantifier) in the first line of that sequence. The following defective proof shows why:
1. (x)Rx (x)Sx / (x) (Rx Sx) 2. Rx ACP 3. x(Rx) 2, UG (invalid) 4. x(Sx) 1, 3, MP 5. Ia 4, UI 6. Rx Sx 2-5, CP 7. (x) (Rx Sx) 6, UG If Rx means “x is a rabbit,” and Sx means “x is a snake,” then the premise translates as, “If everything in the universe is a rabbit, everything in the universe is snake.”
◦ The statement is true because the antecedent is false: not everything in the universe is a rabbit. However, the conclusion is false because it asserts that all rabbits are snakes. Thus, the argument is invalid. The Counterexample Method consists of finding a substitution instance of a given invalid argument for or a given invalid symbolized argument that has true premises and a false conclusion:
◦ Some animals are not mammals. ◦ All cats are mammals. ◦ Therefore some cats are not mammals.
(Ǝx) (Ax • ~Bx) (x) (Cx Bx) / (Ǝx) (Cx • ~Ax) The Finite Universe Method can be used to establish the validity of an invalid argument expressed in terms of a single variable. ◦ Conditional Logical Equivalence: We shrink the universe from billions of things to one thing, named Abigail. “Everything in the universe is perfect” is equivalent to “Abigail is perfect.” “Something in the universe is perfect” is also equivalent to “Abigail is perfect” because “Abigail” is “something” as well as “everything.” (x)Px :: Pa (Ǝx)Px :: Pa The method for proving an argument invalid consists in translating the premises and conclusion into singular statements, then testing the result with an indirect truth table. ◦ A universe of one is tried. If it is possible for the premises to be true and the conclusion false in this universe, the argument is immediately identified as invalid. ◦ If a contradiction results from this assumption, a universe of two is tried. If, in this second universe, it is possible for the premises to be true and the conclusion false, the argument is invalid. ◦ If not, a universe of three is tried, and so on. Monadic (one-place) predicates assign attributes to individual things while Relational (N-place) Predicates establish connections between or among individuals.
Translating Relational Statements: ◦ The Willis Tower is taller than the Empire State Building = Twe. ◦ However, as in monadic predicates, Tew = The Empire State Building is taller than the Willis Tower (which is not true). Overlapping Quantifiers:
◦ When two quantifiers of the same sort appear adjacent to each other, the order does not make a difference.
◦ When different quantifiers appear adjacent to one another the order makes a difference, even when the statement function is nonrelational. Using the Rules of Inference: The quantifier negation rule is applied in basically the same way as with single quantifiers. As the tilde operator is moved past a quantifier, the quantifier in question is switched to its correlative. 1. (Ǝx)(Ǝy)Pxy 2. (Ǝy) Pay 1, EI 3. Pab 2, EI 4. (Ǝx)Pxb 3, EG 5. (Ǝy)(Ǝx)Pxy 4, EG ◦ The CQ rule is used in the same way.
◦ UI, EG and EI are all used in the same way.
◦ UG must not be used if the instantial variable y is free in any preceding line obtained by EI. Translating and Negating Identity Relations:
◦ “The only friend I have is Elizabeth. Elizabeth is not Nancy. Nancy is a Canadian. Therefore, there is a Canadian who is my friend.” This relationship involves the following signs: “=” and “”
◦ Simple Identity Statements: Whoopi Goldberg is Caryn Johnson: w = c Whoopi Goldberg is not Roman Polanski: w r. ◦ “Only,” “The Only,” and “No…Except”: Only Nolan Ryan has struck out 5,000 batters: Sn • (x) (Sx x = n) The only opera written by Beethoven is Fidelio. Of • Bf • (x) [Ox • Bx) x = f] No nation except Australia is a continent: Na • Ca • (x) [(Nx • Cx)] x = a]
◦ “All Except” All painters except Jackson Pollack make sense: Pj • ~Mj • (x) [(Px • x j) Mx] ◦ Superlatives: The largest planet is Jupiter: Pj • (x)[(Px • x j) Ljx]
◦ Numerical Statements: There is at most one god: (x) (y) [Gx • Gy) x = y]
◦ Definite Descriptions: The author of Middlemarch was a Victorian freethinker: (Ǝx) [Wxm • (y)(Wym y = x) • Vx • Fx] ◦ Using the Rules of Inference: ◦ Special Rules, known collectively as Id govern the identity relation: Rule 1: a=a Rule 2: a=b is logically equivalent to b=a Rule 3: If a=b and b=c then a=c