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Notes on Propositional Calculus

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Notes on Propositional Calculus Notes on Propositional Calculus Learning goals 1. Distinguish between inductive and deductive inference. Provides examples to illustrate each one. 2. Provide definitions for Propositional Calculus (PC) terminology. See list below. 3. Translate propositions from English into PC. 4. Check consistency of axioms. 5. Check validity of arguments using truth tables (for simple cases). 6. Check validity of arguments using rules of inference listed on Table 4. Supplementary readings There are several files posted on the course web site that are worth reading. Wikipedia also has several useful pages that address various aspects of propositional calculus. These include page entitled Argument, Atomic sentence, Axiomatic system, Boolean logic, Consistency, List of logic symbols, Logic, Truth table, Deductive reasoning, Rule of inference. Definitions for common PC terminology Proposition A proposition is a well-formed grammatical statement that can be either true or false. It can be either an atomic proposition or a set of atomic propositions combined using unary and binary operators. Atomic proposition An atomic proposition is a proposition that cannot be written in terms of the output of operators acting on more basic propositions. These are represented by single letters. Operator An operator is a kind of function that take a proposition(s) as input and returns a proposition as output. Unary operator A unary operator is an operator that takes one proposition and returns a new proposition (e.g., :). [[In general, an operator can be written in function notation (e.g., U(p) where U is the operator, p is a proposition) but, in practice, they are usually written in a way that reflects the manner in which they are spoken. For example, negation can be written as N(p) (returning the proposition \p is false" which is true precisely when p is false) but it is usually written as :p which is read as \not p".]] Binary operator A binary operator is an operator that takes two proposition and returns a new composite proposition (e.g., )).[[As with a unary operator, a binary operator can be represented in function notation as B(p; q). In practice, we write them as, for example, p _ q. Table 2 gives a truth table and explanations for all binary operators of PC.]] Tautology A tautology is a proposition that is true no matter what truth values are given to its constituent parts. (e.g., p _:p). Contradiction A contradiction is a proposition that is false no matter what truth values are given to its constituent parts (e.g., p ^ :p). 1 Truth table A truth table provides a method for presenting all possible instances of truth values for a set of propositions. [[For example, suppose we are dealing with two atomic proposition, p and q, and two composite proposition, p ) q and p _ q. Each atomic proposition has two possible truth values, so a set of n atomic propositions have 2n possible distinct sets of truth values. In this case, n = 2 so the table has 4 rows.]] p q p ) q p _ q TTTT TFFT FTTT FFTF Contingent proposition A contingent proposition is a proposition that is neither a tautology nor a contradiction (e.g., p ) q). That is, its truth value is not certain until the truth values of the atomic propositions are ascertained. Axiom An axiom is proposition that we assume to be true and from which we intend to derive conclusions using rules of inference. Theorem A theorem must be defined in the context of a particular set of axioms. It is a proposition that is true whenever the atomic propositions involved are given truth values that make all the axioms true. [[For example, an axiom is always a theorem. Any proposition that follows from the axioms by a valid argument is also a theorem.]] Argument An argument consists of a set of axioms and an alleged theorem. Valid argument A valid argument is an argument in which the alleged theorem is actually a theorem. Theory A theory is a set of axioms along with all the theorems that follow from those axioms. Consistency of axioms A set of axioms is consistent provided there is at least on set of truth values for the underlying atomic propositions that render all axioms true. If the axioms are not true for any set of atomic truth values, the set of axioms is said to be inconsistent. Consistent theory A consistent theory is one which is based on a consistent set of axioms. [[A contradiction cannot be a theorem in a consistent theory. Interest- ingly, a contradiction can be (and always is) a theorem in an inconsistent theory. The fact that human thought is capable of housing contradic- tions (usually inadvertently) has interesting implications for the nature of human thought if interpreted as in terms of formal theories.]] Rule of inference A rule of inference is a tool for the valid derivation of new theorems from previously established theorems and axioms. [[It is expressed in the form fp1; p2; :::; pN ` T g which we read as \If p1; p2; ::: and pN have all been established as theorems then T is also a theorem. A rule of inference can be verified by constructing a truth table and checking that whenever p1; p2; :::; pN are all true, T is also true.]] Examples 1. Problem: Show that the propositions (A1) p ) (q _ r), (A2) r ):q, and (A3) p form a consistent set of axioms. 2 Solution: To show consistency, we must find truth values for p, q and r for which all three axioms are simultaneously true. To do this, we present the truth table: p q r q _ r p ) (q _ r) :q r ):q p TTT T T F F T TTF T T F T T TFT T T T T T TFF F F T T T FTT T T F F F FTF T T F T F FFT T T T T F FFF F T T T F [[Note: Columns 1, 2 and 3 give truth values for the atomic propositions. Eight rows allow for all possible combinations. Columns 4 and 6 allow explicit calculation of parts of the axioms and columns 5, 7 and 8 show the resulting truth values for the axioms themselves.]] Notice that in row 2 and 3, all three axioms are true. This means these axioms are consistent. 2. Problem: Using the same propositions A1, A2 and A3 as in problem 1, show that fA1, A2, A3 ` q ,:rg is a valid argument. Solution: [[There are two ways to show that an argument is valid, either construct a truth table and check that the conclusion is always true whenever the axioms are all true or use the Rules of Inference to infer the conclusion from the axioms. I'll show both here.]] Using a truth table. From the previous problem, we know that the axioms are all true in only two circumstances, either p is true, q is true and r is false or p is true, q is false and r is true. So we need only construct a truth table with these two rows. p q r :r q ):r :r ) q q ,:r TTF T T T T TFT F T T T This demonstrates that the conclusion is true whenever all the axioms are true and so the argument is valid. Using Rules of Inference. Theorem Rule of Inference & number Proposition Theorems used Axioms used 1. p ) (q _ r) Ax 1 2. r ):q Ax 2 3. p Ax 3 4. q _ r MP 1,3 1,3 5. :q ) r MI 4 1,3 6. :r ) q T 5 1,3 7. q ):r T 2 2 8. q ,:r ME 5,7 1,2,3 3. Problem: Determine whether the following set of axioms are consistent or not. (A1) p ):q, (A2) r ):q, (A3) (p ^ r) ) q. 3 Solution: [[Not knowing whether the axioms are consistent or not, we can either build the truth table, \guess" at a set of truth value for p; q and r that make the axioms all true or try to derive a contradiction. The truth table approach would determine consistency immediately but can get tedious when large numbers of atomic propositions are involved (the size of a truth table is 2n where n is the number of atomic propositions). If you suspect the axioms are inconsistent, it is often more efficient to derive a contradiction using Rules of Inference. If you suspect consistency, try choosing combinations of truth values for the atomic propositions, changing one at a time until you find a combination that make all axioms true. Looking at the axioms, I suspect they are consistent so I'll show the truth table and explain how you might arrive at a set of truth values without using a table.]] Using a truth table. p q r p ):q r ):q p ^ r (p ^ r) ) q TTT F F T T TTF F T F T TFT T T T F TFF T T F T FTT T F F T FTF T T F T FFT T T F T FFF T T F T The 4th, 6th, 7th and 8th rows demonstrate truth values for the atomic propositions that make all the axioms true. Thus the axioms are consistent. [[To do this problem without making a truth table, I would start with all atomic propositions true (TTT) and test the axioms. The first one comes out false but that can be fixed by making p false so I test FTT.
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