Notes on Propositional Calculus

Home , Argument, Atomic sentence, Propositional calculus, Transposition (logic)

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.

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 {p1, p2, ..., pN ` T } 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 {A1, A2, A3 ` q ⇔ ¬r} 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. Now, the first one is ok but the second is false. Fix this by trying FFT. Bingo. We only had to test three sets of values. If you continue in this manner and fail to find a set that works, the axioms are likely to be inconsistent. However, it is possible that you didn’t test all possibilities unless you keep track carefully. If you did test all, you’ve done all the work required for the truth table. If you didn’t, you can either finish the table or try to derive a contradiction. 4. Problem: Using the same propositions A1, A2 and A3 as in problem 3, show that {A1, A2, A3 ` ¬p ∨ ¬r)} is a valid argument. Solution: [[This could be shown using a truth table but I’ll show it using Rules of Inference.]]

Theorem Rule of Inference & number Proposition Theorems used Axioms used 1. p ⇒ ¬q Ax 1 2. r ⇒ ¬q Ax 2 3. (p ∧ r) ⇒ q Ax 3 4. p ∧ r HAx 4 5. q MP 3,4 3,4 6. p S 4 4 7. ¬q MP 1,6 1,4 8. q ∧ ¬q C 5,7 1,3,4 9. ¬(p ∧ r) RC 4,8 1,3 10. ¬p ∨ ¬r DMT 9 1,3

4 Table 2: Truth table for all possible binary operations with interpretations below. From Wikipedia’s Truth table page.

p q B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 B13 B14 B15 B16 TT FFFFFFFFTTTTTTTT TF FFFFTTTTFFFFTTTT FT FFTTFFTTFFTTFFTT FF FTFTFTFTFTFTFTFT

Binary Formal name Explained in words. Notation (not unique) operation B#(p, q) is true when... (using only ∧, ∨, ¬, ⇒ ) B1 Contradiction ... (never true). p ∧ ¬p B2 NOR ... neither p nor q are true. ¬(p ∨ q) B3 Converse non-implication ... q is true but p is not. ¬p ∧ q, also ¬(q ⇒ p) B4 Negation on ﬁrst operand ... p is not true. ¬p B5 Material non-implication ... p is true but q is not. p ∧ ¬q, also ¬(p ⇒ q) B6 Negation on second operand ... q is not true. ¬q B7 Exclusive disjunction ... either p or q but not both. (p ∧ ¬q) ∨ (¬p ∧ q) B8 Non-conjunction ... not both p and q are true. ¬(p ∧ q) B9 Conjunction ... both p and q are true. p ∧ q B10 Biconditional ... both true or both false. (p ⇒ q) ∧ (q ⇒ p) B11 Projection (second operand) ... when q is true. q B12 Implication ... q is true whenever p is true. p ⇒ q B13 Projection (ﬁrst operand) ... when p is true. p B14 Converse implication ... p is true whenever q is true. q ⇒ p B15 Disjunction ... either p is true or q is true. p ∨ q B16 Tautology ... (always true). p ∨ ¬p

Note that B12 is sometimes expressed in words as “p implies q” or as “if p then q” or as “q only if p”. The same is true for B14, just reverse the p and q in each of these. As a result, B10 is often expressed as “p if and only if q”. Be sure you understand the diﬀerence between “if p then q” and “p only if q” as these are not the same thing. The easiest way to sort them out is to notice that in one of them, the conditional clause is “if p” and in the other it is “if q”.

5 Table 3: Common operators of PC and their LATEX notation. *The last entry in the table (“ensure that”) is not an operator in that operators take propositions as input and return a proposition as output. “`” is more like an entry in a PC instruction manual in that it tells you that if the input propositions (those to the left) are theorems then the output proposition (the one to the right) is also a theorem. A statement of the form {p ⇒ q, p ` q} is not a proposition with a truth value. It is simply a concise way of stating a rule for getting a new theorem from old theorems. If you mistake it for an operator, it will seem indistinguishable from “⇒” which it is not.

Operator Symbol LATEX implies ⇒ \Rightarrow if and only if ⇔ \iff not ¬ \lnot and ∧ \land or ∨ \lor *ensure that ` \vdash

Table 4: Rules of Inference. The notation A ` B means from statement A we can infer statement B as a valid argument. The validity of all these can (and should) be veriﬁed using truth tables. Lifted partially from Wikipedia’s Propositional calculus page.

Rule name Formalism Modus Ponens (MP) ((p ⇒ q) ∧ p) ` q Modus Tollens (MT) ((p ⇒ q) ∧ ¬q) ` ¬p Disjunctive Syllogism (DS) ((p ∨ q) ∧ ¬p) ` q Projection (P) (p ∧ q) ` p Conjunction (C) p, q ` (p ∧ q) Addition (A) p ` (p ∨ q) De Morgan’s Theorem (1) (DMT) ¬(p ∧ q) ` (¬p ∨ ¬q) De Morgan’s Theorem (2) (DMT) ¬(p ∨ q) ` (¬p ∧ ¬q) Distribution (1) (D) (p ∧ (q ∨ r)) ` ((p ∧ q) ∨ (p ∧ r)) Distribution (2) (D) (p ∨ (q ∧ r)) ` ((p ∨ q) ∧ (p ∨ r)) Double Negation (DN) p ` ¬¬p Transposition (T) (p ⇒ q) ` (¬q ⇒ ¬p) Material Implication (MI) (p ⇒ q) ` (¬p ∨ q) Material Equivalence (ME) (p ⇔ q) ` ((p ⇒ q) ∧ (q ⇒ p)) Rule of Deduction (RD) (A1,A2,A3, ··· AN , p ` q) ` (A1,A2,A3, ··· AN ` p ⇒ q) Rule of Contradiction (RC) (A1,A2,A3, ··· AN , p ` q ∧ ¬q) ` (A1,A2,A3, ··· AN ` ¬p) (For this last rule, A1,A2,A3, ··· AN must be consistent.) The following rules can be “skipped steps”: Commutation (1) (p ∨ q) ` (q ∨ p) Commutation (2) (p ∧ q) ` (q ∧ p) Commutation (3) (p ⇔ q) ` (q ⇔ p) Association (1) (p ∨ (q ∨ r)) ` ((p ∨ q) ∨ r) Association (2) (p ∧ (q ∧ r)) ` ((p ∧ q) ∧ r)