Lecture 10 Compactness/rules for the missing connectives
Todays lecture is the last in propositional logic. What it will contain: Some final remarks on the completeness theorem and its proof A proof of compactness A discussion of how the natural deduction
Homework 2 PHIL 151 Due Wed. Jan 19th in class at 10 am 1. (2 points) Problem 1b on page 20. 2. (8 points)Prove or nd a counter-example to (a) If |= then |= and |= (b) If |= then |= or |= where is a set of propositional formulas (or in other words a subse
Homework 3 PHIL 151 Due Wed. Jan 26th in class at 10 am 1. (10 points) Do exercises 1 (d), 2c and 4 on page 39. In problem 1d), the expression abbreviates the conjunction of two implications. So to prove it you need to prove each direction separately. 2.
Mid-term on propositional logic
You should be prepared to: Provide a recursive definition of a function on an inductively defined set.
Examples: problem 4, hw 1; problem 3, hw3.
Prove by induction that a claim holds of an inductively defined set.
Examples
Midterm PHIL 151 Feb. 5, 2010 Apart from the natural deduction proofs you are asked to provide in problem 6, you may in your proofs use any lemma or theorem from the van Dalen text.
Problem 1. Suppose we have a set of symbols that includes: a propositiona
Lecture 1: Introduction to PHIL 151
Phil 151 is an introduction to the concepts and techniques used in the mathematical analysis of deductive reasoning.
The main pre-mathematical notion being analyzed: The idea that a conclusion follows logically from a s
Lecture 2: the sentences of propositional logic
Four general steps in analysis of the notion of logical consequence
Give a semantics for logic: specify how abstract sentences come to be true and false. Fix a syntactic notion of proof: specify rules by whi
Lecture 3 Inductively Defined Sets and Definition by Recursion
Review Inductively defined sets are generated from operations on base elements. Examples: Natural numbers-generated from 0 by operation of _ + 1 n : : 1 0 PROP- generated from propositional sy
Lecture 4 Semantics for Propositional Logic
Four general steps in analysis of the notion of logical consequence Specify logical form by formulating an abstract notion of sentence. Give a semantics for logic: specify how abstract sentences come to be true
Lecture 5 Algebra of propositional formulas/ Interdefinability of connectives
Definition (by recursion) of substitution function on PROP (def. 1.2.5) Given a propositional formula , the function replaces a propositional formula pi in it with . It is denot
Lecture 6 Proof in Propositional Logic
Four general steps in analysis of the notion of logical consequence Specify logical form by formulating an abstract notion of sentence. Give a semantics for logic: specify how abstract sentences come to be true and f
Lecture 7 Completeness of Propositional LogicPart I
Four general steps in analysis of the notion of logical consequence Specify logical form by formulating an abstract notion of sentence. Give a semantics for logic: specify how abstract sentences come to
Lecture 8 Completeness of Propositional LogicPart II
The completeness theorem
|-
if and only if
|=
Last class we proved the direction: if |- then |= .
In other words, we proved that the formal system of natural deduction with the rules I, E, I, E, , a
Lecture 9 Completeness of Propositional LogicPart III
We are half way through the proof of the hard direction of the completeness theorem. The hard direction, recall, is: if |= then |- The proof proceeds by showing the contrapositive: If not |- then not |
Homework 1 PHIL 151 Due Wed. Jan 12th, in class at 10am
1. (5 points) Prove by induction that n=0 2i = 2n+1 1 i 2. (5 points) Write down explicit denitions of the functions f and g on the natural numbers, where (a) f is dened recursively by f (0) = 1, f (