1 Introduction to Logic Reading: Metalogic Part I, 1-6; Part II,15-19 Contents 1.1 Preliminary Remarks . 3 1.1.1 Preliminary motivating remarks . 3 1.1.2 What is a Function? . 6 1.1.3 Use and Mention/Object Language and Meta Language . 9 1.1.4 Alternative Notations . 10 1.2 The System P: Informal Semantics . 11 1.2.1 Propositional Logic: Introductory Remarks . 11 1.2.2 Well-Formed Formulas . 13 1.2.3 Rules of Resolution . 15 1.3 Historical Remarks . 17 1.1 Preliminary Remarks 1.1.1 Preliminary motivating remarks In characterizing a formal system, we state the Formation rules, i.e., the rules for setting down the sequences of expressions that will count as meaningful. Then we state the Transformation rules, i.e., the rules for generating the the- orems of the system. And finally, we state the Translation rules, i.e., the rules identifying what the meaningful expressions specified by the Formation rules are supposed to mean, so that the theorems of the system will be interpreted as truths. The Formation and Transformation rules of the system are fre- quently called the Syntax of the system; the Translation rules are frequently called the Semantics of the system.1 1 The terms Formation Rules and Transformation Rules are Carnap’s (1937). (Translation Rules is my term); in the days when logicians were suspicious of 4 1 Introduction to Logic One of the paradoxical results many of us are familiar with from elementary logic is that everything follows from a contradiction. An argument to this effect, which does not depend on the well-known peculiarities of the material conditional, goes as follows: 1. p ∧ ¬p Assumption 2. p Simplification, 1 3. p ∨ q Addition, 2 4. ¬p Simplification, 1 5. q Disjunctive Syllogism, 3, 4 So, assuming a contradiction, ‘p ∧ ¬p’, we have been able to derive an arbi- trarily chosen statement, ‘q’. This is an example of a derivation within a system of natural deduction, which we might call S. A derivation is a sequence of steps, each of which is a premise or an assumption or the result of applying one of the Deduction (i.e. Transformation) Rules in the system to earlier steps in the derivation. Since we have ‘q’ on the last line of the derivation and ‘p ∧ ¬p’ our only assumption, we say ‘q’ is derivable from ‘p ∧ ¬p’ in the system S. In mathematical notation, we express this as follows: ‘p ∧ ¬p’ `S ‘q’ (‘`’ is called the turnstile.) Three Deduction Rules have been used in the derivation: A ∧ B Simplification: A A Addition: A ∨ B A ∨ B, ¬A Disjunctive Syllogism : B To apply the rule of Simplification to line 1. in the derivation, we had to be able to identify the formula on line 1. as a conjunction. Our notation must be sufficiently precise so that the identification can be done purely on the basis of the shapes in the notation: a machine, or an individual lacking any understanding of the intended meaning of the shapes, should be able to make this identification. This is not a totally trivial problem. There are two connectives in the formula, and it must be clear that the conjunction- sign is the main connective. Similarly, to apply the rule of Addition, we had anything semantical in logic, to specify a formal system was just to state the Formation Rules and the Transformation Rules. 1.1 Preliminary Remarks 5 to identify the formula on line 3. as a disjunction. So, we require a precise characterization of the notion of a meaningful sequence of expressions and a precise characterization of the notion of the main connective in a formula. Once we are satisfied that our notation is precisely characterized, then, to complete our description of the syntax of our language, we must specify the derivation rules and what is to count as a derivation. If we were to continue the list of rules in the spirit of those already given, we would produce a list of Introduction-and-Elimination Rules, or, as they are fondly known, Intelim Rules, for each of the logical symbols of our language. Addition is an intro- duction rule for ∨ because it enables us to introduce a formula using ∨ into the derivation; Disjunctive Syllogism is an elimination rule for ∨ because it enables us to break up a previously derived disjunction and write one of the disjuncts on a distinct line. Each of the derivation rules in the language is intended to capture some elementary valid inference. The rule of Simplification only says that, given a line containing a conjunction, we can add on a new line to the derivation containing one of the conjuncts. (So it is an ∧-elimination rule.) Intuitively, this is supposed to capture the fact that if a conjunction is true, each conjunct is true. Addition holds because a disjunction is true if one of the disjuncts is true. We want the steps in our derivation to be such that if we start out with true premises, then each step the rules enable us to add on to the derivation will also be true. If we start out with truths and stay on the path of righteous- ness (in this context, that means obeying the laws of logic), then we will never stray into falsehood. We want the syntactical manipulations admissible in the setting up of a derivation to mimic the intuitive semantic notion of “follows from” or “is a logical consequence of.” We say that ‘q’ is a logical consequence of ‘p ∧ ¬p’ in the structure S. which is expressed in mathematical notation as: ‘p ∧ ¬p’ |=S ‘q’ (‘|=’ is called the double turnstile.) if any interpretation which makes ‘p ∧ ¬p’ true in a structure also makes ‘q’ true. We need, then, to set up the semantics of our language. We have to make precise the notion of an interpretation, which is supposed to set out what the elements of the notation mean. For the case of truth-functional logic, the intended structure will consist of the set of truth values T,F and the boolean functions on this set. An interpretation links up elements of the notation with elements of the structure. ‘∧’, for example, gets interpreted as the boolean function f∧ : {T,F } × {T,F } → {T,F } such that T if x = T and y = T f (x, y) = ∧ F otherwise 6 1 Introduction to Logic That is, a conjunction of two truths is true; a conjunction of any other com- bination of truths and falsehoods is false. There are two key theorems we want to prove. The first assures us that our rules are truth-preserving, i.e., that whenever we can derive a statement from a set of statements that the argument is intuitively a valid one. Let Γ be a set of statements and let γ be an individual statement. Then we can state this theorem as follows: Soundness If Γ ` γ, then Γ |= γ The second assures us that our rules capture all of those inferences we in- tuitively believe to be valid under the intended interpretation of the logical symbols: Completeness If Γ |= γ, then Γ ` γ 1.1.2 What is a Function? We associate with each open sentence ‘Px’, that we intuitively take to express a property, a set P = {x | P x}.2 If we were to take ‘Px’ to be ‘x is a parent’, then P is the set of parents. If ‘Pa’ is true, we say a ∈ P, i.e., a is an element or member of P. Let C be the set of children. Since every parent is a child, we have the following relation holding between the two sets: If x ∈ P then x ∈ C. We abbreviate this P ⊆ C, i.e., P is a subset of C. Since there are children who are not parents, i.e., since there exists at least one x such that x ∈ C but x 6∈ P, P is said to be a proper subset of C, denoted P ⊂ C. And since there exists at least one x such that x ∈ C but x 6∈ P, the two sets are not identical, for sets are identical iff their members are: P = C ↔ (V x)(P x ≡ Cx). We distinguish the element x from the singleton {x}: x ∈ {x} but x 6= {x}. We distinguish the set {x, y} from the ordered pair < x, y > which is frequently represented as {{x}, {x, y}}. The null or empty set, designated Φ is a set that has no elements so that for every set S, Φ ⊆ S. Here are some important operations on sets: • The intersection of P and C P ∩ C = {x | P x ∧ Cx}. P ∩ C is the set of all things that are in both sets, P and C. Following through with our example, it is the set of all things that are both parents and children.3 • The union of P and C P ∪ C = {x | P x ∨ Cx} P ∪ C is the set of all things that are in one or the other of the two sets P and C. It is the set of all things that are either parents or children. 2 Of course, such an unrestricted Comprehension Axiom of Set Theory would allow the paradoxes to be generated; precise statements guard against them. 3 Since in this particular case P ⊆ C, P ∩ C = P 1.1 Preliminary Remarks 7 • The Power Set of P 2P = {S | S ⊆ P}. 2P is a set, each of whose elements is a set. It consists of all possible groupings of parents.