FOL Practice Models n A model for FOL

FOL Practice

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

The Marble World 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

Some Examples n n 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. n n ( 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)]

WFFs and Truth in FOL n n n 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…

The Positive Integer Model 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 subsentence Eb: 2 is even (TRUE) Ob: 2 is odd (FALSE) Lba: 2 is larger than 1 (TRUE) T & (F T) => T & T =>T

More Truth n When quantifiers are involved, it is more complicated n x means “No matter what you choose for x…” n n n 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…” n An existential statement is true as long as it is true for at least one thing in the UD

Practice n Every positive integer is either odd or even and no positive integer is both. n n Paraphrase: No matter what x you choose, either x is even or x is odd and it is not the case that there is a y such that both y is even and y is odd. Symbolization: x(Ex Ox) & ~ y(Ey & Oy)

x(Ex Ox) & ~ y(Ey & Oy) n Is this true in the model? n n Is x(Ex Ox) true? n n 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? n n Remember that &-statements are true if both conjuncts are 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.

Creating Dummy Models n n 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. n We can create a dummy model using a diagram to help predict if a FOL sentence or argument has particular properties.

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

Strategy for Model Creation n n Represent constants with a dot Represent properties with a circle n n n 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 n n 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

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

Example 2 n Show that (Ga & x. Bx) y. By is not true for every model n Paraphrase: If both a is in G and there is something in B, then everything is in B. UD: a, b G(x): {a} G B B(x): {b} a b a: a b: b

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

What good are dummy models? n Dummy models are quick way to discover a property of a sentence n n 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

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

Arguments n An argument consists of a set of premises and a conclusion n n 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 n 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.

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

That’s it! n As you are working on the argument you give in your paper, you may want to think about how it might look in FOL n n n 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!