Syntax Semantics Tableau Completeness Free Logic The Adequacy of First-Order Logic. Predicate Logic. Daniel Bonevac.

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1 March 18, 2013

2 Aristotle s theory of syllogisms, the first real logical theory, took predicates expressions that are not true or false in themselves, but true or false of objects as its basic units of analysis. It thus examined a finer level of structure than propositional logic. It looked inside the structure of what propositional logic would call atomic sentences.

3 Conditionals The Stoics and medieval philosophers developed propositional logic, even while Aristotelian logic remained the central logical theory. These two schools of thought stood side-by-side for centuries, with only a few analogical connections between them. In the nineteenth century, however, George Boole and others tried to unify them by seeing them as reflecting a single underlying algebraic structure. In 1879, Gottlob Frege and Charles Sanders Peirce independently combined them to develop what has become known as classical first-order logic.

4 Vocabulary Classical first-order logic, also known as predicate logic or quantification theory, adds to propositional logic several new vocabulary items. We will add a set of constants, a set of variables, a set of n-ary predicate symbols, and the quantifiers.

5 Formulas We define formulas: 1. A constant or variable is a term. 2. If t 1...t n are terms, and P is an n-ary predicate symbol, then P(t 1,..., t n ) is a(n atomic) formula. 3. If A and B are formulas, so are A, B, (A B), (A B), (A B), and (A B). 4. If A is a formula and v a variable, va and va are formulas.

6 Variables A variable v is bound if it occurs in the scope of a quantifier on that variable; otherwise it is free. A formula with no free variables is closed. A v (c) is the formula obtained by substituting c for each free occurrence of v throughout A.

7 Models An interpretation or model is a structure M =< D, v >, where D is a nonempty set known as the domain, v a function from constants into D and from n-ary relation symbols to n-place relations on D. v(c) D v(r n ) D n.

8 Models v( Fred ) = Fred v( human ) = {x : Human(x)} v( loves ) = {< x, y >: Loves(x, y)}

9 Assignments There are various ways of writing truth conditions. A common one uses the concept of an assignment, a function that takes variables into objects of the domain. An assignment a is a v variant of assignment a iff a and a agree on all variables except perhaps the variable v. Then the truth conditions for quantified statements va and va talk about all or some v-variants.

10 Witnesses Priest does it differently. Extend the language L to L(M), so that every object in the domain has a name. Then, for every d D, there is a constant k d such that v(k d ) = d.

11 Truth Conditions Our truth definition uses the usual clauses for the connectives, plus: v(pa 1...a n ) = 1 < v(a 1 ),..., v(a n ) > v(p); otherwise it is 0. v( xa) = 1 for all d D, v(a x (k d )) = 1; otherwise it is 0. v( xa) = 1 for some d D, v(a x (k d )) = 1; otherwise it is 0.

12 Implication If, on model M, v(a) = 1, we can write M = A. Similarly, if on M v(b) = 1 for every B X, we can write M = X. We define implication as before: X = A every model of X is a model of A. That is, X = A for all M (M = X M = A).

13 Facts In every model, v( xa) = v( x A) v( xa) = v( x A) v( x(px A)) = v( x(px A)) v( x(px A)) = v( x(px A)) v( xa) = 1 for all c C, v(a x (c)) = 1 v( xa) = 1 for some c C, v(a x (c)) = 1

14 Substitutional Quantification The last two could have been taken as defining the truth conditions of quantified sentences. That approach is known as truth value semantics or substitutional quantification. When all objects in the domain have names, it agrees with objectual quantification. If there are objects without names, they can diverge. What if there are names without objects? That question leads to free logic.

15 Tableaux We can define tableau rules for quantifiers in the standard way. Where X is finite, X A means that there is a closed tableau with the initial list X, A. Open branches induce countermodels: for each constant b appearing on the branch, add an object δ(b) to D, and let v(p) be the set of n-tuples < δ b1,..., δ bn > such that Pb 1...b n appears on the branch. Note that some tableau may be infinite. (Consider one showing that x ysxy is not valid.)

16 Tableau Rules va A v (c) c new va A v (a) any a

17 Tableau Rules va A v (a) any a va A v (c) c new

18 Soundness Lemma Soundness Lemma: Suppose M makes every formula on the initial list of a tableau true. Applying a rule of inference to the branch b produces at least one extension b of the branch such that there is a model M of b.

19 Soundness We must consider each rule. This is trivial in the propositional and negated quantification cases, so the only interesting cases are those for and. Say we apply the rule that takes us from xa to A x (a) for each constant a on b. Since M = xa, M = A x (k d ) for each d D. Let d be such that v(a) = v(k d ). Then M = A x (a). So, we can take M = M.

20 Existential Instantiation Say we apply the rule taking us from xa to A x (c). This is the trickiest rule, since, considered all by itself, it does not appear to be sound; it allows us to move from xfx to Fc, and that is not a valid inference! But its presence in the system does no harm, because the constant c has to be new to the tableau. It cannot have appeared in any premise or in the conclusion. In the tableau, then, it acts to introduce an arbitrary name for an object assumed to exist.

21 Existential Instantiation The rule encodes the common mathematical practice of introducing a name in such contexts, e.g., in Brouwer s fixed-point theorem: Every continuous function f from a closed disk to itself has at least one fixed point. So, if f from closed disk d to d is continuous, there is a point call it x 0 at which f(x 0 ) = x 0. That does no harm as long as we make no other assumptions about x 0 as such.

22 Soundness Since M = xa, there is a d D such that M = A x (k d ). Let M = M except that v (c) = d. Since c does not occur in A x (k d ) (since it was new to the branch), M = A x (c). Since v (c) = d = v (k d ), M = A x (c). Also, since c was new, no other formula on the branch is affected by the switch from M to M ; they all still come out true.

23 Soundness Theorem Soundness Theorem: For finite X, X A X = A.

24 Soundness Theorem Suppose X =/ A. Then there is an M such that M = X and M =/ A. Consider a completed tableau with the initial list X, A. Whenever we apply a rule to an open branch, there is a model M that makes every formula on the extension true, including the initial list. So, we can find a model M for the entire branch. It follows that the branch must be open; if it were closed, there would be a formula B such that both B and B were on the branch, and no model could make both true. So, the tableau is open; X A.

25 Completeness It is quite easy to extend our proof of the completeness of our propositional system to cover full predicate logic.

26 Completeness Lemma Completeness Lemma: Induced interpretations are faithful. Let b be a completed open branch of a tableau and let M be induced by b. Then for any assignment v, if A is on b, v(a) = 1, and, if A is on b, v(a) = 0.

27 Completeness Proof. We proceed by induction on the complexity of A. The atomic case is obvious from the definition of being induced by a branch; the propositional cases are exactly as for the propositional lemma. So, the only interesting cases are quantificational.

28 Completeness Case 1: xa is on b. Since b is complete, the rule for universal formulas has been applied in every possible case, so, for every constant c on b, A x (c) is on b. By inductive hypothesis, then, v(a x (c)) = 1 for every constant c on b. Since D is just the set of objects named by those constants, however, v( xa) = 1.

29 Completeness Case 2: xa is on b. Since b is complete, the rule for existential formulas has been applied, so there is a constant c on b such that A x (c) is on b. By inductive hypothesis, then, v(a x (c)) = 1 for some constant c on b. Since D is just the set of objects named by the constants on the branch, however, v( xa) = 1.

30 Completeness Theorem Completeness Theorem: For finite X, X = A X A.

31 Completeness Proof: Say X A. Then the tableau with initial list X, A is open. There is then an interpretation induced by an open branch making X true and A false. So, X =/ A.

32 Completeness We can extend these theorems to arbitrary sets X by going through the premises of an infinite set in order.

33 Compactness Theorem 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. Proof: Set up a tableau with initial list X, A. Since X = A, it closes. Each branch closes after a finite number of steps; the whole tableau, therefore, closes after a finite number of steps. (This relies on König s Lemma: a tree with an infinite number of nodes has an infinite branch.)

34 Löwenheim-Skolem Theorem Löwenheim-Skolem Theorem: Every set of formulas with a model has a countable model. Proof: Set up a tableau for the set; it remains open. Choose an open branch; it induces a model. Construct the domain by adding any an object for any constant in the first formula, the second, and so on. Even if the branch is infinite, there can be at most countably many objects that are in the domain.

35 Löwenheim-Skolem Theorem If we have identity in the language, the proof of completeness is slightly more complicated; we build equivalence classes of constants, and correlate objects not with constants but with equivalence classes of constants. This affects the Lowenheim-Skolem theorem, for countably many variables might constitute only finitely many equivalences classes. Consider the set of formulas consisting solely of x y x = y. In a language with identity as a logical symbol, therefore, we must say only that any set of formulas with an infinite model has a countable model.

36 Skolem s Paradox This leads to Skolem s paradox: First-order theories of uncountable domains have countable models. A theory of the real numbers, for example, might allow us to prove Cantor s theorem that there are uncountably many reals. In ZF, we can prove the existence of an uncountable set. Nevertheless, these theories have countable models. There are uncountably many sets can be true even in a countable domain!

37 First-order logic commits us to the existence of something, for x x = x and x(fx Fx) are valid. This marks it as an exclusive logic; it excludes the null domain. It also commits us to the existence of everything: x x = c and Fc xfx are valid for every c. So, in classical first-oder logic, every singular term denotes something. Think of the way we constructed a model from an open branch; we let every constant (or, with identity, equivalence class of constants) stand for a distinct object in the domain.

38 Free logic is logic free of existential assumptions for its singular terms. That is, free logic allows for the possibility of nondenoting singular terms. This includes descriptions such as The present King of France as well as names such as Pegasus or function terms such as My BMW.

39 Free logic, first developed by Leonard in 1956, adds to classical first-order logic an existence predicate, usually written E!. An appealing approach, though far from the only one, is to employ a semantics of inner and outer domains, pioneered by Leblanc and Thomason in Let a model be a triple < D, E, v >, which differs from a classical model only in considering D its outer domain, which we might think of as subsisting objects, and thinking of E D as its inner domain, the domain of existing objects. In any model, v(e!) = E.

40 Truth Conditions Free logic alters the truth clauses for the quantifiers: v( xa) = 1 for all d E, v(a x (k d )) = 1; otherwise it is 0. v( xa) = 1 for some d E, v(a x (k d )) = 1; otherwise it is 0.

41 Truth Conditions Correspondingly, we have v( xa) = 1 for all c C such that v(e!c) = 1, v(a x (c)) = 1 v( xa) = 1 for some c C such that v(e!c) = 1, v(a x (c)) = 1

42 Tableaux Tableau rules for free logic split the tableau for universal quantification, allowing for the possibility that E!a, and yield E!c as well as A x (c) for the existential quantifier. Now x x = c and Fc xfx are no longer valid. Neither are x x = x or x(fx Fx). Free logic is thus inclusive. E may be empty.

43 Tableau Rules va va A v (c) E!c c new E!a A v (a) E!a any a

44 Tableau Rules va va E!a A v (a) E!a any a A v (c) E!c c new

45 Countermodels We can read countermodels off tableau just as for classical logic, with the addition that v(e!) = E. It is easy to show that these tableau rules are sound and complete with respect to the semantics for inner and outer domains.

46 Inner and Outer Domains There are at least two reasons why people have felt unsatisfied by the semantics of inner and outer domains. First, it seems to invoke an inner domain of subsisting objects that are somewhat mysterious. What are these objects? What are their identity conditions? How many possible fat men are in that doorway, to ask Quine s famous question? The enterprise seems Meinongian, and that has often been thought to be a bad thing.

47 Quin es Dictum Note that the semantics of inner and outer domains observes Quine s dictum that to be is to be a value of a variable. That has been a major methodological guidepost in the development of free logic. The quantifiers are tied to E, not to D; E appears in the truth clauses for the quantifiers, and we could even define E!a as x x = a. Most free logicians have wanted to observe Quine s dictum; they ve sought to preserve the intuition that the answer to What is there? is Everything.

48 Other That s not the only way to go. We could add quantifiers that range over all objects, existing and nonexisting, So, second, people (such as Routley) have sometimes felt that this semantics is actually too restrictive. They have wanted outer quantifiers ranging over outer objects as well as inner quantifiers ranging over existing objects.

49 Two Sets of Quantifiers? In such a logic, it is possible to introduce two sets of quantifiers, one ranging over D, the other over E. Actually, however, this becomes otiose. The inner pair are definable in terms of the outer pair: E xa x(a E!x), and E xa x(e!x A).

50 Existence as a Predicate But now the quantifiers seem to obey the classical rules; Fc xfx no longer seems to commit us to the existence of c. Neither does x x = c. So, presumably, we can use classical rules; we simply have a special predicate E! that represents existence.

51 Free Modal Logic Free logic remains something of an acquired taste, except in the context of quantified modal logic, where it seems essential (pun intended). For any c, it emerges there not only that = x x = c but also that = x x = c. That is, not only does everything exist: it necessarily exists! That goes too far. But if we have to adopt free logic once we combine quantifiers and modality, what s the argument for resisting it with quantifiers alone?

52 Adequacy How does classical or free first-order logic do as a theory of quantification? Once again, it depends on what we want it to accomplish. As an approach to reasoning in mathematics, it largely succeeds, though there is a serious argument to be made that mathematical reasoning is essentially higher-order. First-order arithmetic has nonstandard models; second-order arithmetic is categorical (meaning all its models are isomorphic).

53 Lindström s Theorem Lindström s theorem tells us that first-order logic is the strongest logic that is both compact and has the Löwenheim-Skolem property. It is also the strongest logic that is complete and has the Löwenheim-Skolem property. Both of these properties are strengths, but also arguably limitations.

54 Natural Language As a theory of natural language quantification, first-order logic has some serious drawbacks. It can define some quantifiers at least n, at most n, exactly n, etc. but there are many that lie beyond its grasp. Some could be added, but some most, for example could not even be added. Also, there are problems with formalization ranging from donkey sentences (Geach, Kamp, Heim) to intensional contexts (Montague) to Bach-Peters sentences to branching quantifiers (Hintikka, Barwise) to sentences requiring sets, pluralities, or higher-order representation (Boolos).

55 Generalized Quantifiers Most Finitely many Infinitely many Countably many Uncountably many More Fs than Gs

56 Donkey Sentences A farmer owns a donkey. A Farmer owns a donkey. He loves it. If a farmer owns a donkey, he loves it.

57 Intensional Contexts I think of a rose. I owe you a horse. I want a sloop. John is looking for a unicorn.

58 Bach-Peters sentences The pilot who aimed at it hit the MIG that chased him.

59 Branching Quantifiers A captain from every regiment and a chief from every tribe met at the conference table.

60 Pluralities Some critics admire only one another.

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