Classical Propositional Logic

Transcription

1 The Language of A Henkin-style Proof for Natural Deduction January 16, 2013

2 The Language of A Henkin-style Proof for Natural Deduction Logic Logic is the science of inference. Given a body of information, we can sensibly ask, Is that information consistent? What follows from it? What additional information would be consistent with it? What information would imply it? We can also ask whether individual sentences are truths of logic, contradictions, or contingent statements that give real information about the world.

3 The Language of A Henkin-style Proof for Natural Deduction Implication A logic is a theory of inference. The most fundamental concept of logic is implication or entailment, symbolized by =.

4 The Language of A Henkin-style Proof for Natural Deduction Notation We will use capital Italic letters (generally from the beginning of the alphabet) to stand for sentences or formulas or a formalized language. I ll generally use capital letters from the end of the alphabet for sets of sentences or formulas. So, we can think of a logic as a theory specifying when X = A. Generally we ll restrict ourselves to finite sets unless otherwise indicated.

5 The Language of A Henkin-style Proof for Natural Deduction Syntax and Semantics Logics generally characterize implication in two ways. They formalize a language that reveals relevant aspects of structure, allowing the identification of valid patterns of reasoning, and specify a semantics, or theory of meaning, for that language.

6 The Language of A Henkin-style Proof for Natural Deduction Semantics The semantics usually, but not always, takes the form of a theory of truth. It characterizes implication, specifying when X = A. But it does not always characterize it in a very useful form. We want to be able to test arguments for validity; we want to be able to construct arguments that are valid. We want, in other words, to be able to identify valid inferences.

7 The Language of A Henkin-style Proof for Natural Deduction Logical Systems Logics thus also specify valid patterns of inference or provide tests of valid inference that depend on form. They might be axiomatic, specifying axioms and rules of inference that allow us to move from axioms to theorems. They might be natural deduction systems, which forgo axioms and consist entirely of rules of inference. They might be tableaux systems, that test for validity using trees. Or they might take other forms. If the inference from X to A passes the test (or is possible given the axioms and rules, etc.), we can write X A.

8 The Language of A Henkin-style Proof for Natural Deduction Implication Implication itself is a semantic notion;, provability, is a syntactic or structural notion. The goal of logic is typically to find a syntactic method that matches the semantics. This is not always possible, as we shall see.

9 The Language of A Henkin-style Proof for Natural Deduction Vocabulary The language of propositional logic is remarkably simple. Its vocabulary consists of propositional parameters (also called sentence letters) p 0, p 1,..., connectives,,,,, and parentheses ( and ).

10 The Language of A Henkin-style Proof for Natural Deduction Syntax The syntax: The set of formulas is the smallest generated by the inductive clauses: Propositional parameters are formulas; If A and B are formulas, so are A, (A B), (A B), (A B), and (A B). The conditional is often written ; the biconditional, as.

11 The Language of A Henkin-style Proof for Natural Deduction Semantics A model, interpretation, or assignment v is a function from propositional parameters to truth values in classical logic, {0, 1}, where 0 represents falsehood and 1 represents truth.

12 The Language of A Henkin-style Proof for Natural Deduction Truth Clauses We can extend v to a function from formulas to truth values by way of an inductive truth definition: v( A) = 1 v(a) = 0 (and = 0 otherwise; hereafter I omit this) v(a B) = 1 v(a) = v(b) = 1 v(a B) = 1 v(a) = 1 or v(b) = 1 v(a B) = 1 v(a) = 0 or v(b) = 1 v(a B) = 1 v(a) = v(b)

13 The Language of A Henkin-style Proof for Natural Deduction Implication A set X of formulas implies a formula A (X = A) iff every assignment making every member of X true also makes A true. A is logically true or valid iff = A.

14 The Language of A Henkin-style Proof for Natural Deduction Adequacy How well does classical propositional logic represent items in natural language? Negation is meant to mirror not ; conjunction, and ; disjunction, or ; and the conditional, if. Do they succeed?

15 The Language of A Henkin-style Proof for Natural Deduction Bivalence The principle of bivalence states that each sentence is either true or false. Classical logic also assumes that no sentence is both true and false. Many-valued and paraconsistent logics reject those assumptions. For now, assume that every sentence is either true or false, and not both.

16 The Language of A Henkin-style Proof for Natural Deduction Adequacy Negation, conjunction, and disjunction are reasonably accurate representations of the meanings of natural language connectives. Some people think that there is a tensed aspect of conjunction; compare John fell down and got up to John got up and fell down, or Joan got married and got pregnant to Joan got pregnant and got married. But this effect occurs with or without and ; it seems to be a function of narrative discourse, not of conjunction.

17 The Language of A Henkin-style Proof for Natural Deduction Adequacy Some also think that disjunction is ambiguous between inclusive (p or q or both) and exclusive (p or q but not both) senses. But that s a hard case to make. Many putative examples of exclusive disjunction involve permission, which is notoriously tricky to analyze. Others rely on background information. ( The car was either a Dodge or a Plymouth. Given the background information that no car is both, inclusive and exclusive senses are equivalent.)

18 The Language of A Henkin-style Proof for Natural Deduction The Conditional The chief problem with classical propositional logic, as a theory of natural language meaning, lies with the conditional. The material conditional is not a very accurate representation of the meaning of if in English. Much of this course, in fact, will be devoted to finding better representations. The search for a logic of the conditional adequate to natural language conditionals is a, and perhaps the, main motivation for developing nonclassical logics.

19 The Language of A Henkin-style Proof for Natural Deduction History Tableau or tree-based systems for evaluating arguments stem from Gerhard Gentzen s 1936 dissertation (Investigations into Logical Deduction). There are many differences, however, between Gentzen s method and the increasingly popular tableaux technique. First, he thought of the method as syntactic. Second, and more importantly, he made explicit use of the deduction symbol; the basic unit in a Gentzen sequent calculus is a sequent (or, in Belnap s Abelard-inspired terminology, consecution) of the form A 1,..., A n B.

20 The Language of A Henkin-style Proof for Natural Deduction History In 1958, E. W. Beth and K. J. J. Hintikka independently realized that a simplified version of Gentzen s method reflected semantic truth clauses very closely and yielded an intuitively and computationally appealing way to evaluate arguments for validity. Tableaux make metatheoretic proofs easy, because the rules for each connective so closely match its semantics.

21 The Language of A Henkin-style Proof for Natural Deduction Strategy The general strategy underlying tableaux rests on the fact that X = A iff X, A (which abbreviates X { A}) is contradictory. If X = A, every model of X is a model of A. But then no models of X are models of A, so, X, A is unsatisfiable, that is, contradictory. Tableaux in effect formalize W. V. O. Quine s Main Method (Methods of Logic, 1950): to find out whether X implies A, assume X and A, and try to deduce a contradiction. To assess whether X = A, we construct a tableau with X, A as its initial list.

22 The Language of A Henkin-style Proof for Natural Deduction Tableau Rules Define the rules as in Priest:

23 The Language of A Henkin-style Proof for Natural Deduction Negation A A

24 The Language of A Henkin-style Proof for Natural Deduction Conjunction A B A B

25 The Language of A Henkin-style Proof for Natural Deduction Negated Conjunction (A B) A B

26 The Language of A Henkin-style Proof for Natural Deduction Disjunction A B A B

27 The Language of A Henkin-style Proof for Natural Deduction Negated Disjunction (A B) A B

28 The Language of A Henkin-style Proof for Natural Deduction Conditional A B A B

29 The Language of A Henkin-style Proof for Natural Deduction Negated Conditional (A B) A B

30 The Language of A Henkin-style Proof for Natural Deduction Biconditional A B A B A B

31 The Language of A Henkin-style Proof for Natural Deduction Negated Biconditional (A B) A B B A

32 The Language of A Henkin-style Proof for Natural Deduction Closure A tableau branch is closed iff some formula and its negation both appear on it. A tableau is closed iff all its branches are. A tableau is completed iff every rule that can be applied on it has been applied. Throughout, assume that X is finite. Say that X A iff there is a completed, closed tableau with X, A as initial list.

33 The Language of A Henkin-style Proof for Natural Deduction and We can now ask whether our rules are sound, that is, whether X A X = A, and also whether we have enough rules, that is, whether X = A X A. These are relatively new questions. Throughout the history of logic, soundness was an intuitive notion, and asked rule-by-rule; the assumption seems to have been that a logical system is sound if and only if all its rules are sound. As we will see, this is usually, but not always, true; Quine s system for predicate logic is sound, but not rule-by-rule.

34 The Language of A Henkin-style Proof for Natural Deduction Truth Preservation The intuitive idea behind the soundness theorem is truth-preservation. A valid inference is truth-preserving: If the premises are true, the conclusion is true. If our logical system consists in a set of rules, we might show that each rule is truth-preserving, and then use induction to show that any proof using those rules yields valid conclusions.

35 The Language of A Henkin-style Proof for Natural Deduction Fidelity A tableau system requires a variation on this strategy. We need to show that closed tableaux establish validity; if something can be proved from a set of premises by way of trees, it follows from them. We define the fidelity of an assignment to a branch, and then show that the rules are fidelity-preserving.

36 The Language of A Henkin-style Proof for Natural Deduction Fidelity Say that an assignment v is faithful to b iff v(a) = 1 for every formula A on b. An assignment is faithful to a branch if it makes every formula on the branch true.

37 The Language of A Henkin-style Proof for Natural Deduction Theorem The Theorem: X A X = A.

38 The Language of A Henkin-style Proof for Natural Deduction : Proof Proof: We ll prove the contrapositive: If X =/ A, then X A. Assume that X =/ A. Then there is an assignment v such that v(b) = 1 for all B X but v(a) = 0, so v( A) = 1. We show that X A. Consider a completed tableau with the initial list X, A. Notice that v is faithful to that initial list. It makes everything in X true and it also makes A true.

39 The Language of A Henkin-style Proof for Natural Deduction Lemma Lemma: Rules preserve fidelity. If v is faithful to b, and a tableau rule is applied to b, v is faithful to at least one resulting branch.

40 The Language of A Henkin-style Proof for Natural Deduction Proof of Lemma Proof: Assume that v is faithful to b, to which a tableau rule is applied. There are as many cases as rules. Here are two representative ones.

41 The Language of A Henkin-style Proof for Natural Deduction Conjunction Case Case 1: The rule for A B is applied. Then A and B both appear on the branch. Since v(a B) = 1, by the truth definition v(a) = v(b) = 1. So, v is faithful to the resulting branch.

42 The Language of A Henkin-style Proof for Natural Deduction Negated Conjunction Case Case 2: The rule for (A B) is applied. Then two branches result, one with A, and one with B. By the truth definition, since v( (A B)) = 1, v( A) = 1 or v( B) = 1. Thus, v is faithful to at least one of the resulting branches.

43 The Language of A Henkin-style Proof for Natural Deduction Proof by Induction With lemma in hand, we can now proceed to show that v is faithful to an open branch on the completed tableau by induction on the number of rules applied to complete the tableau.

44 The Language of A Henkin-style Proof for Natural Deduction Base Case Base: No rules need to be applied. Then the initial list is itself a completed tableau, and v is faithful to it.

45 The Language of A Henkin-style Proof for Natural Deduction Inductive Hypothesis Inductive step: Assume that v is faithful to an open branch up to the nth rule application. By the lemma, v is faithful to at least one resulting branch.

46 The Language of A Henkin-style Proof for Natural Deduction Inductive Step So, by induction, v is faithful to a branch on the completed tableau. If it were closed, no assignment could be faithful to it, since no assignment makes both a formula and its negation true. So, the branch is open, and thus X A.

47 The Language of A Henkin-style Proof for Natural Deduction A logical system is (strongly) complete if and only if every valid inference can be proved valid within the system. That is, a system is complete iff, whenever X = A, X A. (It is weakly complete if and only if every logical truth is provable within it; if, that is, = A only if A.) Surprisingly, no one seems to have asked the question of completeness until David Hilbert and Wilhelm Ackermann raised it in their Principles of Theoretical Logic (1928) and Kurt Gödel, in his 1930 dissertation (The of the Axioms of the Functional Calculus of Logic), answered it.

48 The Language of A Henkin-style Proof for Natural Deduction Strategy The central strategy of the completeness theorem is to show that valid inferences can always be proved valid by showing the contrapositive. Assume that an inference cannot be shown to be valid. In a tableau system, this means there is at least one open branch of any tableau constructed to evaluate the inference. Derive, from this open branch, a set of formulas here, the set of literals, that is, atomic formulas and their negations that appear on the branch.

49 The Language of A Henkin-style Proof for Natural Deduction Strategy Construct, from this set of formulas, a semantic interpretation here, the interpretation assigning p truth if it appears on the branch and falsehood if p appears on the branch. Show that the interpretation is a countermodel of the original inference, thus concluding that it was not valid.

50 The Language of A Henkin-style Proof for Natural Deduction Strategy Putting this in a diagram: Open branch Set of literals Induced assignment Countermodel

51 The Language of A Henkin-style Proof for Natural Deduction Open Branches We can use open branches to build models and countermodels. An open branch of a tableau not only tells us that there is an assignment making the initial list true, but also specifies it. Say that an assignment v is induced by an open branch b iff, for every propositional parameter p on b, v(p) = 1, and for every p on b, v(p) = 0. Induced assignments thus respect the values implicitly assigned to propositional parameters on the branch.

52 The Language of A Henkin-style Proof for Natural Deduction Induced Assignments Induced assignments, that is, make all propositional parameters and their negations on a branch true; faithful assignments make every formula on it true. All faithful assignments are induced, by definition, but whether all induced assignments are faithful depends on the tableau rules. Indeed, showing that all induced assignments are faithful is the key lemma in the completeness theorem. It accomplishes steps (3) and (4), showing that the assignment induced by the branch is a countermodel of the original inference.

53 The Language of A Henkin-style Proof for Natural Deduction Theorem The Theorem: X = A X A.

54 The Language of A Henkin-style Proof for Natural Deduction Proof of Proof: We prove the contrapositive. Suppose that X A. Then there is no completed closed tableau with X, A as initial list. So, every completed tableau with X, A as initial list has at least one open branch b.

55 The Language of A Henkin-style Proof for Natural Deduction Induce an Assignment Any completed open branch b, moreover, induces an assignment. For any propositional parameter p, let v(p) = 1 if p is on b and v(p) = 0 if p is on b. Assign any other parameters any value you like.

56 The Language of A Henkin-style Proof for Natural Deduction Lemma Lemma: All induced assignments are faithful. If v is induced by b, v is faithful to b. That is, if b, a completed open branch of a tableau, induces v, then: if A is on b, v(a) = 1; if A is on b, v(a) = 0.

57 The Language of A Henkin-style Proof for Natural Deduction Proof by Induction Let b, a completed open branch, induce v. We show that for every A on b, v(a) = 1, and for every A on b, v(a) = 0, by induction on the complexity of A, that is, the number of connectives in A.

58 The Language of A Henkin-style Proof for Natural Deduction Base Case Base: Say A is a propositional parameter on b. Then v(a) = 1, since b induces v. Say A on b is p for some propositional parameter p. Then v(a) = 0, since b induces v.

59 The Language of A Henkin-style Proof for Natural Deduction Inductive Step Inductive step: Assume that v(d) = 1 for every D on b with fewer than n connectives and that v(d) = 0 for all D on b with fewer than n connectives. Assume that A has n connectives. There are nine cases to consider:

60 The Language of A Henkin-style Proof for Natural Deduction Negation Case 1: A = B is on b. Since the result holds for B, v(b) = 0, so, by the truth definition, v(a) = 1.

61 The Language of A Henkin-style Proof for Natural Deduction Conjunction Case 2: A = (B C) is on b. By the tableau rules, if A, that is, B C is on b, then so are B and C. But they contain fewer connectives than A, so, by inductive hypothesis, v(b) = v(c) = 1. By the truth definition, therefore, v(a) = 1.

62 The Language of A Henkin-style Proof for Natural Deduction Negated Conjunction Case 3: A = (B C) is on b. By the tableau rules, if A, that is, (B C) is on b, then either B or C is on b. Both B and C have fewer connectives than A, so, by the inductive hypothesis, v(b) = 0 or v(c) = 0. By the truth definition, then, v(b C) = 0, and v (B C) = 1.

63 The Language of A Henkin-style Proof for Natural Deduction Disjunction, Conditional, Biconditional The other six cases are similar.

64 The Language of A Henkin-style Proof for Natural Deduction Completing the Proof We have shown that, if X A, every tableaux with the initial list X, A has an open branch; that such a branch induces an assignment v; and that every induced assignment is faithful to the branch. Since open branch b has the initial list X, A, v, being faithful to it, makes every member of X true, but A false. So, X does not imply A.

65 The Language of A Henkin-style Proof for Natural Deduction Natural Deduction Systems Leon Henkin developed his method in 1948 ( The of the First-Order Functional Calculus ) to prove completeness without immersion in the apparatus of recursion theory. It will be convenient to limit our language to negation and conjunction. We will not begin with a specific logical system in mind, but construct what we need for a complete system from the proof.

66 The Language of A Henkin-style Proof for Natural Deduction Proofs A proof system consists of a (possibly empty) set of axioms and a nonempty set of rules of inference. A proof is a finite sequence of formulas each of which is an assumption, an axiom, or follows from previous lines by a rule of inference. There is a proof of A from X (X A) iff there is a proof including as assumptions formulas in X and terminating in A.

67 The Language of A Henkin-style Proof for Natural Deduction Strategy The general idea of the completeness proof is similar to tableau completeness proofs. The completeness theorem says that X = A X A. So, assume that X A, and construct a model of X, A.

68 The Language of A Henkin-style Proof for Natural Deduction Strategy Let s assume, then, that X A. It follows that X, A are consistent; there is no B such that X, A = B, B. Our strategy is to construct a model of X, A by first extending X, A to a maximally consistent set.

69 The Language of A Henkin-style Proof for Natural Deduction Lindenbaum s Lemma Lindenbaum s Lemma: Every consistent set of formulas extends to a maximal consistent set.

70 The Language of A Henkin-style Proof for Natural Deduction Proof of Lindenbaum s Lemma Proof. Let Y be a consistent set of formulas. Our language is countable, so we can enumerate the formulas of the language: A 1,..., A n,... We construct a series of sets: Y 0 = Y Y n+1 = Y n {A n }, iff that is consistent; if not, Y n+1 = Y n { A n } Y ω = Y n

71 The Language of A Henkin-style Proof for Natural Deduction Maximality Obviously, Y ω Y. Y ω is also maximal: for every formula A, A Y ω or A Y ω. The argument: A appears at some stage in our enumeration of the formulas of the language. Say that A is formula n. Either A is consistent with Y n or not. If so, A Y n+1 Y ω. If not, A Y n+1 Y ω.

72 The Language of A Henkin-style Proof for Natural Deduction Consistency Moreover, Y ω is consistent. The argument: Suppose not. Then Y ω B, B for some B. Since proofs are finite, and Y is consistent, there must be some first finite stage n + 1 at which Y n+1 B, B. Y n+1 = Y n {A n }, iff that is consistent; since it is inconsistent, Y n+1 = Y n { A n }. So, Y n { A n } B, B. But then Y n A n, provided that our deduction system has rules of indirect proof and double negation. Since Y n A n B, B, however, Y n A, so Y n is inconsistent, contradicting the assumption that Y n+1 is the first finite stage at which an inconsistency appears.

73 The Language of A Henkin-style Proof for Natural Deduction Closure Also, Y ω is closed under proof; Y ω A A Y ω. Let Y ω A but A Y ω. By maximality, A Y ω, so Y ω A, A; Y ω is inconsistent. But we have already demonstrated its consistency.

74 The Language of A Henkin-style Proof for Natural Deduction Countermodel Since, by the Lemma, every consistent set extends to a maximal consistent set, X, A does so. Call that set X ω. We construct a model v from this set as follows: v(p) = 1 p X ω.

75 The Language of A Henkin-style Proof for Natural Deduction Countermodel Claim: v = A A X ω.

76 The Language of A Henkin-style Proof for Natural Deduction Base Case Proof: By induction on the complexity of A. For propositional parameters, it is trivial. Assume that the claim holds for all formulas less complex than A.

77 The Language of A Henkin-style Proof for Natural Deduction Negation Case 1: A = B. v = A v = B v B. By inductive hypothesis, that is true iff B X ω. Since that set is maximal, that is true iff B X ω.

78 The Language of A Henkin-style Proof for Natural Deduction Conjunction Case 2: A = (B C). v = B C v = B and v = C. By inductive hypothesis, that is true iff B, C X ω. So, that is true iff B C X ω, provided that our deduction system has rules corresponding to conjunction introduction and exploitation.

79 The Language of A Henkin-style Proof for Natural Deduction We have now shown that we can construct a model making all of X true but A false, provided that we have rules of indirect proof, double negation, conjunction introduction, and conjunction exploitation. So, X A.

80 The Language of A Henkin-style Proof for Natural Deduction Compactness As a quick corollary to the completeness theorem, we obtain: Compactness: X = A there is a finite X 0 X such that X 0 = A. In an alternate form, a set is satisfiable iff every finite subset of it is satisfiable.

81 The Language of A Henkin-style Proof for Natural Deduction There is a sense in which completeness is a limiting result. Syntactic methods, such as tableaux and proof systems, have limited complexity. Semantic characterizations of entailment can easily have greater complexity. A completeness result thus shows that the syntactic system is strong enough a positive result but also, correspondingly, that the semantic system is weak enough to be captured by a set of rules.

82 The Language of A Henkin-style Proof for Natural Deduction Effective Procedures Say that a method for answering a question determining whether an object has a property, or whether an object belongs to a set is effective iff it is mechanical and infallible. It is mechanical in the sense that it is entirely rule-governed; it is algorithmic. No creativity is required. It is infallible in the sense that it always gives the correct yes or no answer after a finite time.

83 The Language of A Henkin-style Proof for Natural Deduction Decidability A property (or set) is decidable iff there is an effective method (called a decision procedure) for determining whether an object has that property (or belongs to that set). Examples: validity in propositional logic; validity in first-order logic with only monadic predicates; quantifier-free arithmetic.

84 The Language of A Henkin-style Proof for Natural Deduction Decidability Tableaux constitute a decision procedure for classical propositional logic. A tree always has finitely many branches, and each branch is finite. The tableau terminates after finitely many steps. And then it either closes or yields a model of the formulas on the trunk. If the tree is testing the validity of a formula or argument form, that model is a countermodel of the formula or argument form. We have thus shown that classical propositional logic is decidable.

85 The Language of A Henkin-style Proof for Natural Deduction Semi-decidability Sometimes, a method is only half infallible it always gives a correct positive answer after a finite time, say, but may run on infinitely if the answer is negative. Such a method is positively semi-decidable. A method that always gives a correct negative answer after a finite time, but may run on infinitely if the answer is positive, is negatively semi-decidable.

86 The Language of A Henkin-style Proof for Natural Deduction Enumerability A property is (effectively) enumerable iff there is an effective method of generating or listing all and only its instances. An enumerable set is the range (or the domain) of a decidable relation. Any finite set is enumerable. But there are also infinite enumerable sets. Any decidable set, for example, is enumerable. Examples of enumerable properties that are not decidable: validity in full first-order logic. (Church proved first-order logic to be undecidable in 1936.)

87 The Language of A Henkin-style Proof for Natural Deduction Enumerability and Semi-decidability An enumerable property is positively semi-decidable: take the method to be waiting for the object to emerge on the list. If it is an instance, it will be on the list at some finite stage. Conversely, any positively semi-decidable property is enumerable: test objects one by one, and add to the list those for which the method yields a yes answer.

88 The Language of A Henkin-style Proof for Natural Deduction Craig s Little Theorem Craig s Little Theorem: If P and P are both enumerable, then both are decidable. Start listing instances of P and of P. Any given object must appear on one list or the other (assuming bivalence) after a finite time.

89 The Language of A Henkin-style Proof for Natural Deduction Axiomatizability A theory is a set of sentences closed under logical consequence. Let Cn(X) = {A : X = A}. For any theory T, T = Cn(T). T is finitely axiomatizable iff there is a finite X such that T = Cn(X), and axiomatizable iff there is a decidable X such that T = Cn(X).

90 The Language of A Henkin-style Proof for Natural Deduction Axiomatizability A first-order theory is axiomatizable iff it is enumerable. We want sets of axioms to be decidable; we want to be able to determine effectively whether or not something is an axiom. Similarly, we want proofs to be decidable; we want an effective method for determining whether something is a proof. If a theory is axiomatizable, then the set of its theorems is enumerable, for it is the range of a decidable relation (that of proof) on a decidable set (of axioms).

91 The Language of A Henkin-style Proof for Natural Deduction Partial Enumerability Say that a property is partially enumerable iff it has an infinite enumerable subset. Gödel s incompleteness theorem (1931) states that truth in arithmetic is only partially enumerable; arithmetic is not axiomatizable. But it is partially axiomatizable; that is, there are decidable sets of axioms that are true in arithmetic and whose consequences are all true in arithmetic. But there are truths of arithmetic that are not consequences of them.

92 The Language of A Henkin-style Proof for Natural Deduction Partial Enumerability The general idea of Gödel s proof is to assign symbols, formulas, sequences of formulas, and thus proofs code numbers in a systematic way, and then show that there are arithmetical formulas that represent a sequence being a proof of a formula, a formula s bring provable, etc. It is possible to show that there is an arithmetical predicate Pr such that Pr[A] if and only if A is provable in Peano arithmetic (where [A] is the code number of A). That is, PA Pr[A] if and only if PA A.

93 The Language of A Henkin-style Proof for Natural Deduction Partial Enumerability But then it is possible to construct a sentence G such that PA G Pr[G] that says, in other words, that it is not provable. If G, I am not provable, is true, it is not provable, and there is a true sentence of arithmetic not provable from the axioms. If it is false, then it is provable, and the axioms and rules are unsound, allowing us to prove something false.

94 The Language of A Henkin-style Proof for Natural Deduction Immunity A property is immune iff it is infinite but has no infinite enumerable subsets. Such a property is not even partially enumerable. Any axiom or axiom set would have exceptions; it would have consequences that are not instances of the property. Example: the set of ethical truths, from an intuitionist point of view; the set of truths about God, in Aquinas.

95 The Language of A Henkin-style Proof for Natural Deduction Cardinality Consider the set of natural numbers. It is countable, but its power set the set of all subsets of the set of natural numbers is uncountable. Countably many of those subsets are decidable; countably many others are enumerable but not decidable. Uncountably many are partially enumerable. And uncountably many are immune.