FOL Practice

1. FOL Practice

2. Models • A model for FOL requires 3 things: • A set of things in the world called the UD • A list of constants • A list of predicates, relations, or functions

3. UD: Ashley, Clarence, Rhoda, Terry, and their marbles a: Ashley c: Clarence r: Rhoda t: Terry B(x): x is blue G(x): x is green R(x): x is red S(x): x is a shooter C(x): x is a cat’s-eye T(x): x is a steely M(x): x is a marble B(x,y): x belongs to y W(x,y): x wins y G(x, y, z): x gives y to z The Marble World

4. All the cat’s-eyes belong to Rhoda. All the marbles but the shooters are cat’s eyes. Some, but not all, of the cat’s-eyes are green. All of the shooters that are steelies belong to Terry. (x)(C(x)  B(x,r)) (x)[(M(x) & ~S(x))  C(x)] (x)(C(x) & G(x)) & ~(y)(C(y)  G(y)) (x)[(S(x) & T(x))  B(x,t)] Some Examples

5. WFFs and Truth in FOL • Before we can decide if a sentence in FOL is true, we need to make sure it is well-formed • Once this is determined, we can test if a sentence is true in the model. • Let’s look at a less complicated model to practice this…

6. UD: positive integers a: 1 b: 2 E(x): x is even O(x): x is odd L(x,y): x is larger than y Eb & (Ob  Lba) WFF? True? Test for truth by looking at the truth of each sub-sentence Eb: 2 is even (TRUE) Ob: 2 is odd (FALSE) Lba: 2 is larger than 1 (TRUE) T & (F  T) => T & T =>T The Positive Integer Model

7. More Truth • When quantifiers are involved, it is more complicated • x means “No matter what you choose for x…” • UD = Giant Bag o’ Stuff • A universal statement has to be true for every item in the UD • x means “There is at least one x such that…” • An existential statement is true as long as it is true for at least one thing in the UD

8. Practice • Every positive integer is either odd or even and no positive integer is both. • Paraphrase: No matter what x you choose, either x is even or x is odd andit is not the case thatthere is a y such that both y is even and y is odd. • Symbolization: x(Ex  Ox) & ~y(Ey & Oy)

9. x(Ex  Ox) & ~y(Ey & Oy) • Is this true in the model? • Remember that &-statements are true if both conjuncts are true. • Is x(Ex  Ox) true? • This statement is false if we can find just one item from the UD that makes it false, which we can’t! It’s TRUE. • Is ~y(Ey & Oy) true? • This statement is true if y(Ey & Oy) is false. Is there something in the UD that is both even and odd? There isn’t, so y(Ey & Oy) is false which makes ~y(Ey & Oy) TRUE. • Therefore, the entire statement is true in the model.

10. Creating Dummy Models • So far, we have looked at whether a sentence is true in a given model. • Models don’t have to be complicated things dealing with real properties of real world things. • We can create a dummy model using a diagram to help predict if a FOL sentence or argument has particular properties.

11. What We Can Do with Models • True/False on a model • We can give a model that makes a sentence true/false • True/False on all models • We can show that a sentence is true/false on all models or if it is indeterminate • Consistency • We can check if a set of sentences is consistent by giving a model where they are all true • Validity • We can check if an argument is invalid by giving a model where the premises are true but the conclusion is false

12. Strategy for Model Creation • Represent constants with a dot • Represent properties with a circle • Anything that falls in the circle has that property; anything not in the circle doesn’t • Represent relations with an arrow • Getting a model from the picture • Put any dots in the UD • List the dots as constants • For each predicate, give a set that lists the dots in the predicate’s circle • For each relation, give a set of ordered pairs

13. a a d p Example 1 • Give a model that makes the sentence Nad -> ~Nda true. • Give a model that makes the sentence Iap -> (Ipa -> Iaa) false.

14. G B a b Example 2 • Show that (Ga & xBx)  yBy is not true for every model • Paraphrase: If both a is in G and there is something in B, then everything is in B. UD: a, b G(x): {a} B(x): {b} a: a b: b

15. Example 3 • Show that xGx & yz(Fyz  Gz) is not true for every model • Paraphrase: There is something in G and everything that is pointed to with F is in G. UD: a, b a: a b: b G(x): {a} F(x,y): {<a,b>, <b,a>} G a b

16. What good are dummy models? • Dummy models are quick way to discover a property of a sentence • Once we know that property, it will still hold no matter what the model is • So, if we can show that a sentence is true on at least one model, we know we can come up with a real world model that it is also true on

17. Back to Example 2 • We know that (Ga & xBx)  yBy is not true for every model. • We can use the dummy model as a template for a real world model UD: a, b UD: Homer, Otis Gx: {a} Gx: x is yellow Bx: {b} Bx: x is black a: a a: Homer b: b b: Otis

18. Arguments • An argument consists of a set of premises and a conclusion • An argument is valid if and only if it is not possible to give a model that makes the premises all true and the conclusion false • So, to show that an argument is invalid, give a model where the premises are true and the conclusion is false • An invalid argument is always invalid! It may be possible to give a model that makes it seem like a good argument (politicians do this all the time!), but you can use the dummy model method to easily figure out if the argument is really any good.

19. Example 4 • Show that the following argument is invalid: x(Fx  Gx) “Everything in F is in G” ~xFx “Nothing is in F” ~xGx “Nothing is in G” UD: a, b UD: Homer, Shai (the very furry bunny) Fx: {} Fx: x is a cat Gx: {b} Gx: x is furry a: a a: Homer b: b b: Shai F G a b

20. That’s it! • As you are working on the argument you give in your paper, you may want to think about how it might look in FOL • Is your argument valid??? • Remember: Paper outlines are due on Friday by the end of the day (e-mail before 11:59 pm is OK) • No reading for this week!