Phil 201: Introductory Logic, Precept 8. Alejandro Naranjo Sandoval, [email protected] 1879 Hall, Room 225 OH: Tuesday, 10:00am - 10:55am; 12:00pm - 1:00pm; Thursday 10:00am - 10:55am.

1. Where are we heading now?

• Think about to the way we learned propositional logic. On the one side, we talked about what you could prove in the system. Namely, there were some allowed rules of inference (MP, CP, ∧I, ∧E, ∨I, etc.) which we could use to derive, given certain assumptions, further theorems. Syntax is the area of logic that studies what can be proven in a logical system. Hence these rules of inference belonged to the syntax of propositional logic. The crucial relation in syntax, namely, provability, is denoted by the single turnstile: `.

• On the other hand, we studied truth-tables. They allowed us to talk about the truth status of sentences, i.e., whether a sentence is a contingency, a tautology, an impossi- bility, etc. Semantics is the area of logic that studies the truth-status of sentences. Hence truth-tables belong to the semantics of propositional logic. The crucial rela- tion of semantics, namely, semantic entailment, is denoted by the double turnstile: . Recall that Γ  φ iff whenever all the sentences in Γ are true, so is φ. • So far, we’ve studied the syntax of predicate logic. That is, we have outlined the rules of inference that are allowed in this system. The are the rules from propositional logic (CP, MP, ∧I, etc.) plus four other rules: UI, UE, EI, and EE. So, we still need to cover the semantics of predicate logic!! This is what we’ll do for a significant chunk of the rest of the semester.

2. Does that mean that we’ll be doing truth-tables again?

• No! Truth-tables are not the right tool to describe the truth-status of sentences in predicate logic. For recall: truth-tables were tools that allowed us to compute the truth or falsity of a sentence φ based on the sentence letters contained in it. E.g., if φ is P ∧Q, then the truth-table of φ would state that it is true iff both P and Q are true. Now, suppose you want to use truth-table to compute the truth-value of ∀xP x. How would you do this? We can check P a. But clearly this is not enough! Our sentence states that everything is P , not just that a is. Hence to check the truth or falsity of ∀xP x you’d have to check not only P a, but also P b, P c, P d, and any whether, for any other individual, it is P or not. If there are many individuals under consideration, this becomes impractical really quickly. Moreover, if there are infinitely many individuals, this method becomes impossible to use.

• Instead of using truth-tables, then, we need an additional piece of mathematics to introduce the semantics of predicate logic. This piece of mathematics is called...

1 3. Set Theory.

• The central idea of set theory: for any set of elements, we can call collect these elements into a single collection or class of objects. We can call this new object a set. If you have a banana, a cat, and a table, you can form the set containing those elements. Suppose you call them, respectively, b, c, and t. Then we would use the following notation to denote the set containing these three elements in this way: {b, c, t}. Also, to denote the membership of these, we use the symbol ∈. So, we can say b ∈ {b, c, t}, c ∈ {b, c, t}, etc.

• How can you tell two sets apart? That is, what is it that defines different sets or makes them distinct from one another? By looking at their respective elements. That is, sets are individuated by their elements. If two sets have the same elements, they are the same set. If two sets have different elements, then they are different sets. This is what Prof. Halvorson called the Axiom of Extensionality.

• We also have that at least one set exists, i.e., the empty set, which we denote by the symbol ∅. This set is defined as the set such that, for any set x, x∈ / ∅. (Mental check: can you prove that there is only one empty set?)

• Given what we have, we can define y ⊆ x, which is read “y is a subset of x”, as saying that any element of y is also an element of x. So, {b} ⊆ {b, c, t} and, in turn, {b, c, t} ⊆ {b, c, t, s}.

• We also have, for any set x, a set P(x) of its subsets, which we call its power set; and, for any two sets x and y, we have the intersection x ∩ y, defined as the set of the elements which are both in x and in y; and the union x ∪ y, defined as the set of the elements which are either in x or in y.

4. Exercises. Prove the following statements in an “informal but rigorous” way:

i ∩ is commutative, i.e., x ∩ y = y ∩ x.

ii Two sets x and y are identical iff both that x ⊆ y and y ⊆ x.

iii If A ⊆ B, then P(A) ⊆ P(B). iv ∩ distributes over ∪, i.e., x ∩ (y ∪ z) = (x ∩ y) ∪ (x ∩ z).

v If x ⊆ y, then we have both that x ∪ z ⊆ y ∪ z and x ∩ z ⊆ y ∩ z, for any set z.

vi For any sets A and B, P(A) ∩ P(B) = P(A ∩ B).

vii Challenge problem: For any set x, it is never the case that x = P(x).

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