Notes on first-order logic
Notes on first-order logic Phil 26403 9 October 2016 We’ve developed a complete analysis of truth-functional logic: The truthtables for the connectives give a theory of meaning, which yields a notion of semantic consequence. On the other hand, the method of tableaux gives a system of proof.1 As we’ll see later on in the course, the theories of meaning and of proof align: the relation of semantic consequence coincides with the relation of provability. However, truth-functional logic is not all of logic! Consider, for example • “all humans are mortal, Confucius is immortal, so Confucius is not human” • “all horses are mammals, so all horses’ tails are mammals’ tails”. These arguments are logically valid. However, their validity cannot be represented truth-functionally. So we’ll now develop a richer logical system which is adequate to analysis of arguments like the above. The validity of those depends crucially on the concepts “all” and “some”. These concepts are known as quantifiers. We’re now going to study first-order logic. Just as truth-functional logic is the logic of truth-functions, so first-order logic is the logic of quantifiers.2
1
The possible worlds of first-order logic
You can think of logic as investigating the way that the formal features of statements constrain what distributions of truth and falsehood are possible for them. An approach to this study is that of truth-functional logic. There, we study truth-functional combinations of statements they contain. E.g., the truthvalue of “either it’s raining or it’s sunny” is a function of the truth-value of “it’s raining” and “it’s sunny”. However, not all logically complex statements are truth-functions of simpler statements they contain. Consider, e.g., “all humans are mortal”. To determine whether this is true or false, you have to consider each thing x in the universe, and check whether or not it is human but immortal. Thus it depends on every statement of the form “if x is human then x is mortal”. And there are way 1 According to that method, a proof of a formula is just a closed tableau beginning with the formula labelled F . 2 So, sometimes it’s called “quantificational logic”.
1
too many of these to write down in a single sentence. Moreover, in knowing how to write them down, we’d have to know what are all the objects that exist; but that goes beyond what can be determined by logic alone. In other words, it sometimes comes about that the truth or falsehood of a statement depends essentially on what objects there are, and on what those objects are like. Now, the truth and falsehood of statements like “Socrates is human” is not a matter for logic to decide. Logic doesn’t in general determine what is true or false, but rather determines the way in which truth and falsehood of statements is interdependent. Such interdependencies can be clarified using the notion of possible world: for example, two statements are incompatible if there is no possible world in which both are true. So it remains to find a suitable notion of structure for the logic of quantification. As we’ve already concluded, the truth or falsehood of quantified formulas depends on what objects there are and on what those objects are like. This suggests that a notion of possible world W which is suitable for the logic of quantification can be given by choosing • which objects exist in W, and • how those objects happen to behave in W, i.e., – which objects have which properties, and – which objects stand to each other in which relations. The set of objects which exist in W is called the domain of W. The set of objects which have a property F in W is called the extension of F ; the set of pairs of objects standing to each other in some relation R is called the extension of R. We will write the domain of W as ∗W , and write the extension of a property F , or of a relation R, as F W or as RW . A possible world for first-order logic is called a structure. Defining a structure W is a bit like playing creator. You simply • choose some objects, say Alice, Bob, Carol to form the domain ∗W , and • decide e.g. that Alice and Bob are happyW , that Bob is smilingW , and that Alice lovesW Bob while Bob lovesW Carol. It is usual to insist that the domain of a structure must not be empty. Therefore in any possible world, at least one thing exists. You may or may not find this weird, but it simplifies many technical details. We’ll present structures in the following format: ∗ : {Alice, Bob, Carol} Happy : {Alice, Bob} W= Smiling : {Bob} Loves : {(Alice, Bob), (Bob, Carol)} A few comments about notation may be in order here. An expression {o1 , o2 , . . . , } represents the set whose elements are the objects o1 , o2 , . . .. The 2
most fundamental fact about sets is this: that no two sets have precisely the same elements. Therefore, for example, {2, 1 + 1, 1} = {2, 2, 1} = {2, 1} = {1, 2}. It is also sometimes useful to keep in mind that a set may have no elements. From what I’ve said already, it follows that there is only one such empty set. Second, an expression (o, p) represents the ordered pair whose first element is o and whose second element is p. Ordered pairs are identical just in case they have the same first and second elements. So, (1, 2) = (1, 1+1) but (1, 2) 6= (2, 1). The first and second elements of an ordered pair may be the same object, as in for example (1, 1). A set can contain any objects whatsoever. This includes other sets, or also ordered pairs. For example, if you decide to build a structure in which Alice loves Bob, then you are putting the pair (Alice, Bob) into the set of all pairs whose first element loves the second.3
2
Using quantifiers
The logical purpose of a notion of possible world is to generate a suitable notion of truth. So after introducing the notion of structure, our main task becomes to spell out the conditions under which a quantificational formula is true or false. But let’s not plunge into that right away. Rather, let’s begin by trying to describe the structure W we just built, using English sentences which would be true if the actual world were like W. The applicability of the intuitive notion of truth to these English sentences will motivate the formal account of truth for quantificational formulas. To begin with, suppose you have built the above world W and somebody asks you what W is like. That is, they are asking you to make some statements which are true in W. In response, some things you might say are pretty much baked into W. You might say • Alice is happy, • Bob loves Carol,. . . . Getting a little more creative, these baked-in truths can be combined using the resources of truth-functional logic. E.g., • Carol is not happy • Alice is happy or Carol loves Alice if and only if neither Carol nor Alice loves Alice. Of course, these can be formalized as in • ¬Happy(Carol) 3 In fact, it is possible to define the concept of ordered pair using the concept of set. This is left as an exercise.
3
• Happy(Alice) ∨ Loves(Carol, Alice) ↔ ¬Loves(Carol) ∧ Loves(Alice, Alice). So far there’s nothing really new here. But then you might also notice the obtaining or nonobtaining of patterns and want to express this. For example, W makes true all of the following: • Smiling(Alice) → Happy(Alice) • Smiling(Bob) → Happy(Bob) • Smiling(Carol) → Happy(Carol) These statements all exemplify a certain pattern, namely • Smiling(x) → Happy(x) But moreover, Alice, Bob, Carol are all of the objects that exist in W. Hence, the pattern holds universally in W. That is to say, the following is true in W: • every object x is such that: Smiling(x) → Happy(x) This observation can be fully formalized with the help of a new symbol ∀x, a so-called universal quantification of x: • ∀x(Smiling(x) → Happy(x)) It could have been expressed idiomatically: • everything smiling is happy In a similar vein, you might notice that although in W not everything happy is smiling, nonetheless something is. That is to say, • there is an x such that x is happy and x is smiling. This can be rephrased using the new symbol ∃x, the so-called existential quantification of x: • ∃x(Happy(x) ∧ Smiling(x)) Or idiomatically, the same thought is • something happy is smiling. Let’s continue developing new forms of observation about W. In particular, note that patterns can themselves involve objects. Consider, for example, the pattern • Alice loves x. It is a fact about W that this pattern is satisfied by at least one thing (which?). This fact-in-W can itself be stated, like this: • ∃x Loves(Alice, x)
4
Just as we found a pattern involving Alice, there is a similar pattern involving Bob instead: • Bob loves x And indeed, this pattern too is satisfied by at least one thing according to W. Hence in W the following statements both hold: • ∃x Loves(Alice, x) • ∃x Loves(Bob, x) Now it’s clear that Alice and Bob themselves have a commonality: you get a truth-in-W by taking either of them to be the value of y in • ∃x Loves(y, x) This commonality between Alice and Bob might suggest new generalizations. Unfortunately, the commonality is not universal: it is not enjoyed by the third object Carol. But, let’s try to make some generalization in the vicinity. First, note that Alice and Bob are distinguished from Carol by being happy. Thus, the following are all true: • Happy(Alice) → ∃x Loves(y, x) • Happy(Bob) → ∃x Loves(y, x) • Happy(Carol) → ∃x Loves(y, x) What we’ve observed, then, is that every object in the domain satisfies the following condition: • Happy(x) → ∃x Loves(y, x) Thus, the following formula is true in W: • ∀y(Happy(y) → ∃x Loves(x, y)) How would you express this fact in English?
3
Formal syntax
Let’s now make the previous discussion systematic. The first task is to specify a formal language, call it FOL, which serves to formalize the varieties of expression just introduced. Specifying the language FOL basically requires determining what are its formulas; but it will be useful to define some other notions too along the way. A sentence of FOL is supposed to say how things are. This has two components: mentioning objects, and saying something about them. The way to mention objects is to use a sign which refers to them. A sign which refers to objects is a term. On the other hand, a predicate is a sign which, when combined with terms for some objects, thereby says that the objects are in a certain way. So, to build up the language of FOL, we need to introduce terms and predicates. These combine to yield formulas of the simplest kind, atomic formulas. Further formulas will result from formulas by applying truth-functional connectives and quantifiers. 5
Atomic formulas. The language of first-order logic contains two kinds of term: constants and variables. A constant is like a proper name, in that as part of its meaning, there is some fixed object it denotes. A variable, on the other hand, is like a pronoun; its meaning is rather just to denote whatever was denoted by some previous occurrences of terms. More formally, • a constant of FOL is any of the signs a, b, . . . , w, a0 , b0 , . . . , a1 , . . .. • a variable of FOL is any of the signs x, y, z, x0 , y0 , . . .. • the terms of FOL are the constants and variables of FOL. The system of predicates is just a tiny bit fussier. The simplest kind of predicate just applies to single things, saying of a thing e.g. that it is happy or smiling; such predicates are said to be one-place, or monadic. But some predicates are to be applied not to one thing but to more than one; such predicates are said to be polyadic. Specifically, a dyadic predicates applies to an ordered pair of things, saying of the pair e.g. that the first element loves the second element. Similarly there are triadic, tetradic, and in general n-place predicates. We will also allow zero-place “predicates”; these were the atomic formulas of FOL. Many textbooks begin by introducing predicate logic first with monadic predicates only. Here, I’ll present the grammar of the whole system. Typographically, monadic predicates should be distinguished from dyadic, triadic, etc. So, we’ll use accents to indicate the number of argument places. The predicates of FOL are now defined as follows: • • • •
a zero-place predicate is any of the signs A, B, . . . , Z, A0 , B0 , . . . , A1 . . .; a one-place predicate is any of the signs A0 , B 0 , . . . , Z 0 , A00 , B00 . . . , . . . A01 . . .; a two-place predicate is any of the signs A00 , B 00 , . . . , Z 00 , A000 , B000 , . . . , A001 , . . ., and more generally an uppercase letter with n primes is an n-place predicate.
In practice, it is annoying to write 0 all the time. But if we agree not to use predicates like A0 and A00 together in the same context, then no ambiguity will arise.4 As promised, the terms and the predicates now can be said to combine to yield atomic formulas: • an atomic formula of FOL is an n-place predicate followed by n occurrences of terms. Officially, then, atomic formulas include things like D, B 0 a, A0 x, E 00 ax, F cxbxy, etc. But in practice we will often simply write D, Ba, Ax, Eax, F cxby, etc., taking the superscript of the predicate to correspond to the length of the subsequent string of terms. 00000
4 It is sometimes considered good manners to use letters like F, G, H . . . as monadic predicates, and R, S, T . . . for polyadic predicates.
6
Formulas. We now need to explain the general notion of formula. Rather than just mentioning particular formulas, we are now going to generalize about them. Recall that in arithmetic when you mention particular integers, you do so with expressions like ‘0’, ‘1’, ‘2394’, etc. But when you want to generalize, then instead you use variables x, y, z . . . which range over all integers. And similarly to generalize about formulas, we need variables which range over all formulas, or all predicates, terms or whatever. I will use boldface letters for this purpose. In the new notation we can restate the definition of atomic formula like this: • An atomic formula is an expression of the form Fa1 · · · ak , where F is a k-place predicate and a1 , . . . , ak is a series of k terms. Now the first part of the definition of formula is easy. • Every atomic formula of FOL is a formula of FOL. How about more complicated formulas? As before, you can construct some using truth-functional connectives. So provided that P and Q are formulas of FOL, then also • ¬P, (P ∧ Q), (P ∨ Q), (P → Q), (P ↔ Q) are formulas of FOL. But the key new ingredient is construction of formulas with quantifiers. Let’s say that x is a variable of FOL, so that it is one of the letters x, y, z, x0 , . . .. Now provided that P is a formula of FOL, then also • ∀xP and ∃xP are formulas of FOL. So, formulas of FOL include for example F a, ¬Rab, ∀xF x → ¬∃yRxy, etc. Order of operations, a review. Recall that for truth-functional logic we adopted a conventional order of operations. Those conventions remain in force for first-order logic. Moreover, we now stipulate that • The scope of a quantifier ∀x or ∃x is the smallest complete formula immediately to its right. So the precedence order for connectives of first-order logic is ∃x, ∀x, ¬, ∧, ∨, →, ↔ . For example, in ∀y(F y → Gd), the ∀y governs F y → Gd, while in ∀yF y → Gd it governs just F y. You should now be able to draw the formation tree of any formula of first-order logic. Draw for youself the tree for ∀x(¬∀yF y → Rxy) → ∃zRxz. For later developments we’ll need a few more syntactical ideas, so let’s develop them now. A variable can occur in a formula in one of two mutually exclusive ways: either as free, or as bound by a quantifier. Every occurrence 7
of a variable in an atomic formula is free. A variable occurs free in a truthfunctional combo ¬P, P ∧ Q, etc., just where it occurs free in P or in Q. The free occurrences of variables other than x in ∀xP or in ∀xP are precisely the free occurrences of variables other than x in P. But wherever x occurs free in P, then it is bound by the indicated occurrences of ∀x and ∃x in ∀xP or ∃xP. For example, in the formula ∃xF x → (F x ∧ F y) the variable y has one occurrence and that occurrence is free. The quantifier ∀x, on the other hand, binds the first subsequent occurrence of x but not the last one, so that the last, and only the last, occurrence of x is free. You should think of a variable as something like a pronoun. Thus, for example, if the predicate F expresses the property of being square, then the formula F x says something like “it is square”. The proper response to that formula is “which thing?” In contrast, a quantification with respect to x, say ∃x, completes the thought: thus ∃xF x amounts to “there is some thing or other such that: it is square.” In that case the meaning of the pronoun is fixed by the antecedent noun phrase “some thing or other”. Thus, you may think of a formula as “expressing a complete thought” when no variable occurs free in it. A formula in which no variable occurs free is said to be closed. Only a closed formula, then, represents a complete thought. The concept of truth for predicate logic will apply to closed formulas only. Finally, we’ll need one more syntactical notion, which is a special kind of substitution on expressions. Roughly speaking, substitution is a sort of “copyand-paste” operation which replaces occurrences of one expression with another. So, for example, the result of substituting “t” for “b” in “babble” is “tattle”. There are also notions of substitution which are more specific. For example, the result of substituting “r” for all occurrences of “b” which are not adjacent to another “b” in “babble” is “rabble”. One such more specific notion we’ll use repeatedly. • If P is a formula, t is a term, and x is a variable, then P[x/t] is the result of substituting t for all free occurrences x in P. Suppose, for example, that P is the formula ∃xF x → (F x ∧ F y). Since only the last occurrence of x in P is free, therefore P[x/a] is ∃xF x → (F a ∧ F y). Underlying this special concept of substitution derives from the fact that quantifiers express generalizations. A formula P[x/t] is said to be an instance of a universal or existential generalization ∀xP or ∃xP. Thus, we will be able to say that a universal generalization implies each of its instances, while an existential generalization is implied by each of its instances.
4
Semantics of FOL
Having specified the syntax of a language, it’s time to explain what makes its formulas true or false. In the case of first-order logic, this is a rather interesting problem which takes a couple of steps. I’ll start by defining an appropriate notion of world, and then turn to the concept of truth. 8
4.1
The worlds of FOL, again
To begin with, we have to define a notion of world which is suitable for FOL. This just makes precise the idea informally sketched in §1. Whereas a possible world of truth-functional logic is a row of a truth-tables, a world of FOL is called a structure. A structure W consists two things • a nonempty set ∗W , called the domain of W • an interpretation of the constants and predicates of FOL. In what follows, we’ll mainly consider domains whose elements are among the natural numbers 0, 1, 2, . . .. This is just convenient, since everybody knows what those objects are.5 It remains, then, to stipulate how the constants and predicates of FOL should be interpreted by a structure. A constant is supposed to be like a name, in simply standing for an object. Thus • the W-interpretation aW of a constant a is an element of ∗W . Let’s now consider the case of predicates. First of all, a monadic predicate is combined with a term for an object, in order to say the object has a certain property. The collection of objects which have the property is called its extension. So, a monadic predicate can be interpreted by specifying its extension. • the W-interpretation FW of a monadic predicate F is an set of elements of ∗W . Similarly, a dyadic predicate is combined with a pair of terms for objects, in order to say that the two terms, in that order, stand in a certain relation. The extension of a relation is the set of ordered pairs of objects which, in that order, bear the relation. • the W-interpretation RW of a dyadic predicate R is an set of ordered pairs of elements of ∗W . More generally, • the W-interpretation of an n-place predicate is an set of ordered n-tuples of elements of ∗W . At last, and as in truth-functional logic, • the W-interpretation of zero-adic predicate is a truth-value.
4.2
Truth
Having defined for FOL the concepts of formula and of possible world, it remains to define the concept of truth. Since FOL contains a fair number of moving parts, it will be easiest to proceed in stages. 5 You could instead use, for example, marbles, but it’s hard to talk about particular marbles because few of them have names. Also, the number of numbers is importantly greater than the number of marbles.
9
4.2.1
Closed atomic formulas.
In FOL, an atomic formula consists of an n-place predicate followed by n terms. And a formula is closed if no variable occurs free in it. But any occurrence of a variable in an atomic formula is free in that formula. So, a closed atomic formula must consist of n-place predicate followed by n constants. Now, consider a structure W of FOL. And consider, for example, a closed atomic formula F a. What determines whether F a is true or false in W? Recall that informally, a formula F a represents a statement that a is F . This statement is true just in case the object denoted by a falls in the extension of the predicate F . Fixing denotations of constants and extensions of predicates is the job of a structure. For example, consider ∗ : {0, 1, 2} a : 0 . W= b:1 F : {0, 2} R : {(0, 1), (1, 2)} Then aW = 0, while F W = {0, 2}. So according to W, the object denoted by a does fall into the extension of the predicate F . Therefore, according to W, the formula F a ought to be true. On the other hand, the formula F b ought to be false (why?). Similarly, the ordered pair (0, 1) falls into the W-extension of R. This means that according to W, 0 bears R to 1. Again because a, b refer to 0, 1 therefore the formula Rab ought to be true. On the other hand, the ordered pair (1, 0) does not belong to RW , so the formula Rba ought to be false. More generally, suppose that F is any monadic predicate, and that c is any constant. Then • the formula Fc is true in W iff the object cW belongs to the set PW . Now a bit of notation. Write W |= A to mean that A is true in W. Also, write W 6|= A if it is not the case that W |= A . In the new notation, the clause for handling monadic predicates can be restated like this: • W |= Fc iff cW belongs to FW . So, given the above choice of W, we then have W |= F a but W 6|= F b. Let’s now turn to the case of dyadic predicates. In this case, a predicate is combined with two names, and the result should be true iff the extension of the predicate contains the pair of denotations of those names. That is, suppose R is a dyadic predicate and c, d are constants. Then • W |= Rcd iff (cW , dW ) belongs to RW . Then for example, given the same W from 10, we get W |= Rab but W |6 = Rba. The treatment of general polyadic predicates is similar. If P is an n-place predicate, then 10
W W • W |= Pc1 , . . . , cn iff (cW 1 , . . . , cn ) belongs to P .
Zeroadic predicates are handled just as in truth-functional logic: • W |= P iff PW = T. 4.2.2
Truth-functional combinations.
The logic FOL includes the connectives of truth-functional logic. And they’re evaluated just as usual. That is, we exploit truth-functionality to determine the truth or falsehood of complicated formulas by reference to the truth or falsehood of simpler ones. Let’s restate the procedure in the new language. Suppose that P and Q are formulas of FOL. Then • W |= ¬P iff W 6|= P • W |= P ∧ Q iff both W |= P and W |= Q; • W |= P ∨ Q iff either W |= P, or W |= Q (or both); • W |= P → Q iff either W 6|= P or W |= Q (or both); • W |= P ↔ Q iff either W |= P and W |= Q, or W 6|= P and W 6|= Q. For example, let W be defined as above. Then have we W |= (F a → F b) ↔ ¬Rba? To settle this question, first apply the definition of truth for closed atomic formulas to determine the truth or falsehood at W of F a, F b, and Rba. Then use the truth-tables to chase the truth-values up the formation tree. 4.2.3
Quantification.
The definition of truth for formulas containing quantifiers is a much deeper problem.6 Consider, for example, the formula ∀xF x. Our earlier strategy was to reduce the question of its truth-value to the question of truth-values of its simpler subformulas. However, the only simpler subformula of (x)F x is F x. And in F x, the x occurs free. This means that F x is analogous to a sentence like “it is green” where the “it” is unspecified. Being not closed, the subformula F x ought not to have a truth-value at all. Thus, quantified formulas are not in general built from simpler closed formulas. So, we need a new idea. Suppose again that the domain of W is {0,1,2}. Then what should W be like in order that it make true a universally quantified formula ∀xA? Intuitively, that formula says “everything has the property expressed by A”, or that every object satisfies A. Now according to W, all the things that exist are simply 0, 6 The first solution to this problem was given by Alfred Tarski in 1933, “The concept of truth in formalized languages”. We’ll follow an approach from Joseph Shoenfield’s Mathematical Logic (1967).
11
1, and 2. So, the formula ∀xA should be true over W provided that each of 0, 1 and 2 satisfies A. So, the question we need to answer is this: • what is it for an object to satisfy a formula over a structure? Standard names and their denotations. We have reduced the concept of quantification to the concept of satisfaction. But what is satisfaction? Intuitively, the idea is that an object satisfies a formula over a structure if what the formula says is true of that object in that particular structure. But what is this relation “true of”? Here an explanation is quite natural: a formula is true of an object if the result of plugging into the formula a name of that object is simply true. Again, recall P[x/a] represents the result of substituting a for all free occurrences of x in P. Now, suppose that a is a name of object o. We would like to say that if P contains just the free letter x, then P is true of the object o just in case P[x/a] is true. This approach immediately confronts a difficulty: it is not in general the case that every object has a name. In the real world, not all ants have names, nor do all grains of sand. So, we need a trick to arrange that every structure has a “hardwired” name for each object in its domain. This is not too bad, because we are working with artificial worlds, whose domain can be supposed to consist entirely of numbers 0, 1, 2, . . .. And every such number does have a name. Indeed, every natural number has standard Hindu-Arabic decimal name. For example, • the standard name of 7 is “7”; • the standard name of 21 + 210 is “231”; • the name of 21 consists of the name of 2 followed by the name of 1, i.e., it’s “21”, etc. For any object o, let’s write o for its standard name. For example, 21 =“21”. The notation “o” might be pronounced “bar-oh” or “the name of o”. Thus, the bar represents a function which, applied to an object, returns its standard name. To give another example: suppose that the standard name of a human being is just their full personal name. Then, The US president in 2016 begins with the letter ‘B’, followed by the letters ‘a’, ‘r’, ‘a’, ‘c’, ‘k’, followed by a space, etc. If you have trouble remembering the names of people, a bar function would come in handy. We now stipulate that in any structure whatsoever, a standard name o denotes the object o. That is,
12
• oW = o for all W. W
So, for any W, we have that 7 = 7. Likewise 21 + 210 is a string of three W characters, but 21 + 210 is an integer greater than 230 and less than 232. Similarly, if Barack Obama were an element of the domain of W, then you could say that The US president in 2016
W
was born in Hawaii.
The definition of satisfaction, and the definition of truth for quantified formulas. We wanted to explain satisfaction of a formula by an object in terms of the result of plugging into the formula a name of an object. This idea can now be made precise. Recall that A[x/t] is the result of substituting t for all free occurrences of x in A. Suppose that A is a formula whose one free variable is x. Then, • for an object o to satisfy the formula A over W just is for the formula A[x/o] to be true over W. Recall the W from page 10. In that case, the object 0 satisfies F x over W, because W |= F 0. And because 1 doesn’t satisfy F x, therefore 1 does satisfy ¬F x. If bW = 1, then 0 satisfies F x ↔ ¬F b, but 1 does not. And so on. With that, our problem of defining truth for quantifiers is solved. We can say that ∀xA is true over W provided that every object in the domain of W satisfies A. And, ∀xA is true over W provided that every object in the domain of W satisfies A. More precisely, say that A is a formula whose one free variable x. Then • W |= ∀xA iff W |= A[x/o] for each object o in ∗W . • W |= ∃xA iff W |= A[x/o] for some object o in ∗W . And with that, the definition of truth is complete. Examples. Consider again the structure W from page 10. Have we W |= ∀xF x? This will be so just in case W |= F o whenever o is any of 0, 1, 2. Since W 6|= 1, therefore actually W 6|= ∀xF x. On the other hand we do have, for example, W |= ∃xF x, since W |= F 0. Thus, W |= ∃xF x ∧ ∃x¬F x. In contrast, it’s clear without checking the details of W that W 6|= ∃x(F x ∧ ¬F x). For otherwise, there’d be a o such that W |= F o ∧ ¬F o. And then we’d have that o both does and doesn’t belong to F W , which is impossible.
5
The method of truth-functional expansion
In truth-functional logic, the definition of truth is easy to apply: just chase the truth-value up the formation tree. Unfortunately, the situation is not so straightforward in first-order logic. In particular, if the domain is infinite, then 13
the truth-value of a generalization depends on the truth-value of infinitely many simpler formulas.7 Luckily, our main purpose is the assessment of deductive arguments. An argument, as ever, is invalid if it has a countermodel—that is, if there’s a world where all premises are true while the conclusion is false. And it turns out that most invalid arguments you’ll ordinarily encounter have countermodels which are very small, with just two or three elements in their domain. Let’s now develop a practical procedure for evaluating formulas in small structures. The key idea is that once the domain of a structure has been fixed as some finite set of objects, universal and existential quantifiers can be regarded as conjunctions and disjunctions. That is, suppose that the elements of the domain are o1 , . . . , ok . We’ll associate, to each quantified formula A, another formula Ao1 ,...,ok called the truth-functional expansion of A over o1 , . . . , ok . Over any structure whose domain contains just o1 , . . . , ok , the formulas A and Ao1 ,...,ok will have the same truth-value. So over a given finite structure, the method of truth-functional expansion reduces the evaluation of a quantified formula to the evaluation of an unquantified one. To state the method, it will help to have one further piece of terminology. Say that an W-instance of a generalization ∀xA or ∃xA is a formula A[x/o], where o is an element of ∗W . For example, if ∗W contains 1, then an instance of ∃x∃y(F y ∧ F x) is ∃y(F y ∧ F 1). Now the method for computing an expansion of a formula over a domain {o1 , . . . , ok } is simply this. • The expansion of a universal generalization is the conjunction of the expansions of its instances • The expansion of an existential generalization is the disjunction of the expansions of its instances • The expansion of a truth-functional combo of some formulas is just that combo of the expansions of those formulas. • The expansion of an atomic formula is just that formula. Examples. For example, suppose that 0, 1, 2 are the elements of ∗W . And consider a random formula A, like ∃xF x → ∀xF x. The expansion A0,1,2 of A over 0, 1, 2 now gets computed like this: A0,1,2
(∃xF x → ∀xF x)0,1,2 (∃xF x)0,1,2 → (∀xF x)0,1,2 (F 0 ∨ F 1 ∨ F 2) → (F 0 ∧ F 1 ∧ F 2).
7 For example, the natural numbers, together with addition, multiplication, and exponentiation can be considered to form a structure. Problems of classical number theory are simply questions whether a given formula is true in that structure.
14
We can use the expansion of A to compute the truth-value of A. Suppose, for example, that the W-extension of F is {0, 2}. Granted that ∗W is {0, 1, 2}, this reduces the relevant portion of W to the row of a truth-table! Namely, F0 T
F1 F
F2 . T
So it is clear that W |= F 0 ∨ F 1 ∨ F 2. Meanwhile, W 6|= F 0 ∧ F 1 ∧ F 2. Consequently, W 6|= (F 0 ∨ F 1 ∨ F 2) → (F 0 ∧ F 1 ∧ F 2). And that’s just to say W 6|= (∃xF x → ∀xF x)0,1,2 . Since, however, 0, 1, 2 are precisely the elements of W, it follows that A0,1,2 must, over W, have the same truth-value as A! Therefore, W 6|= ∃xF x → ∀xF x. Caution! It should be clear that first-order logic is much more expressive than truth-functional logic. This is because the quantifiers are a fundamentally new concept. Quantifiers let us say things we can’t say only with truth-functions. Consider, for example, the formula ∀xF x. This is certainly not equivalent to a conjunction of its instances. For no matter how many objects you consider, it’s logically possible for there to exist some other object besides those, and this object might not satisfy F x. Thus, ∀xF x does not follow from any conjunction of atomic formulas. For this reason, it should be clear that a quantified formula is not equivalent to its truth-functional expansion. Rather, the method says only that a formula and its expansion over a domain ∗W have the same truth-value over W. To bring this out, here are a couple of exercises. (i) Find a formula, together with two domains, such that the expansion-of-the-formula-over-one domain is not equivalent to its expansion-over-the-other-domain. (ii) Find a formula and a domain such that the formula does not imply its expansion-over-that-domain. Illustrating the method of expansions. The method of truth-functional expansion is useful for distinguishing the meanings of more complicated formulas. Consider, for example, ∀x∃yRxy and ∃y∀xRxy. Are these equivalent? If so, then they must have the same truth-value over all structures whatsoever. Do they? Consider first a structure whose domain is just {0}. In that case, the two formulas both expand into R00. So they must have the same truth-value over all one-element structures. 15
But consider instead the domain {0, 1}. Then the formulas expand like this. On the one hand, (∀x∃y(Rxy))0,1
(∃y(R0y))0,1 ∧ (∃y(R1y))0,1 (R00 ∨ R01) ∧ (R10 ∨ R11).
On the other hand, (∃y∀x(Rxy))0,1
(∀x(Rx0))0,1 ∨ (∀x(Rx1))0,1 (R00 ∧ R10) ∨ (R01 ∧ R11).
Can you find a structure with domain {0, 1} which makes one true and the other false? Finding such a structure amounts to filling in the question marks in R00 ?
R01 ?
R10 ?
R11 ?
This is an exercise in truth-functional logic. You should be able to find a W with ∗W = {0, 1} so that W |= ∀x∃yRxy while W 6|= ∃y∀xRxy.8 This will demonstrate that ∀x∃yRxy does not imply ∃y∀xRxy.
8 For
example, it will work to choose W so that
RW = {(0, 0), (1, 1)}.
16
R00 T
R01 F
R10 F
R11 . That is to say, T
1
The possible worlds of first-order logic
You can think of logic as investigating the way that the formal features of statements constrain what distributions of truth and falsehood are possible for them. An approach to this study is that of truth-functional logic. There, we study truth-functional combinations of statements they contain. E.g., the truthvalue of “either it’s raining or it’s sunny” is a function of the truth-value of “it’s raining” and “it’s sunny”. However, not all logically complex statements are truth-functions of simpler statements they contain. Consider, e.g., “all humans are mortal”. To determine whether this is true or false, you have to consider each thing x in the universe, and check whether or not it is human but immortal. Thus it depends on every statement of the form “if x is human then x is mortal”. And there are way 1 According to that method, a proof of a formula is just a closed tableau beginning with the formula labelled F . 2 So, sometimes it’s called “quantificational logic”.
1
too many of these to write down in a single sentence. Moreover, in knowing how to write them down, we’d have to know what are all the objects that exist; but that goes beyond what can be determined by logic alone. In other words, it sometimes comes about that the truth or falsehood of a statement depends essentially on what objects there are, and on what those objects are like. Now, the truth and falsehood of statements like “Socrates is human” is not a matter for logic to decide. Logic doesn’t in general determine what is true or false, but rather determines the way in which truth and falsehood of statements is interdependent. Such interdependencies can be clarified using the notion of possible world: for example, two statements are incompatible if there is no possible world in which both are true. So it remains to find a suitable notion of structure for the logic of quantification. As we’ve already concluded, the truth or falsehood of quantified formulas depends on what objects there are and on what those objects are like. This suggests that a notion of possible world W which is suitable for the logic of quantification can be given by choosing • which objects exist in W, and • how those objects happen to behave in W, i.e., – which objects have which properties, and – which objects stand to each other in which relations. The set of objects which exist in W is called the domain of W. The set of objects which have a property F in W is called the extension of F ; the set of pairs of objects standing to each other in some relation R is called the extension of R. We will write the domain of W as ∗W , and write the extension of a property F , or of a relation R, as F W or as RW . A possible world for first-order logic is called a structure. Defining a structure W is a bit like playing creator. You simply • choose some objects, say Alice, Bob, Carol to form the domain ∗W , and • decide e.g. that Alice and Bob are happyW , that Bob is smilingW , and that Alice lovesW Bob while Bob lovesW Carol. It is usual to insist that the domain of a structure must not be empty. Therefore in any possible world, at least one thing exists. You may or may not find this weird, but it simplifies many technical details. We’ll present structures in the following format: ∗ : {Alice, Bob, Carol} Happy : {Alice, Bob} W= Smiling : {Bob} Loves : {(Alice, Bob), (Bob, Carol)} A few comments about notation may be in order here. An expression {o1 , o2 , . . . , } represents the set whose elements are the objects o1 , o2 , . . .. The 2
most fundamental fact about sets is this: that no two sets have precisely the same elements. Therefore, for example, {2, 1 + 1, 1} = {2, 2, 1} = {2, 1} = {1, 2}. It is also sometimes useful to keep in mind that a set may have no elements. From what I’ve said already, it follows that there is only one such empty set. Second, an expression (o, p) represents the ordered pair whose first element is o and whose second element is p. Ordered pairs are identical just in case they have the same first and second elements. So, (1, 2) = (1, 1+1) but (1, 2) 6= (2, 1). The first and second elements of an ordered pair may be the same object, as in for example (1, 1). A set can contain any objects whatsoever. This includes other sets, or also ordered pairs. For example, if you decide to build a structure in which Alice loves Bob, then you are putting the pair (Alice, Bob) into the set of all pairs whose first element loves the second.3
2
Using quantifiers
The logical purpose of a notion of possible world is to generate a suitable notion of truth. So after introducing the notion of structure, our main task becomes to spell out the conditions under which a quantificational formula is true or false. But let’s not plunge into that right away. Rather, let’s begin by trying to describe the structure W we just built, using English sentences which would be true if the actual world were like W. The applicability of the intuitive notion of truth to these English sentences will motivate the formal account of truth for quantificational formulas. To begin with, suppose you have built the above world W and somebody asks you what W is like. That is, they are asking you to make some statements which are true in W. In response, some things you might say are pretty much baked into W. You might say • Alice is happy, • Bob loves Carol,. . . . Getting a little more creative, these baked-in truths can be combined using the resources of truth-functional logic. E.g., • Carol is not happy • Alice is happy or Carol loves Alice if and only if neither Carol nor Alice loves Alice. Of course, these can be formalized as in • ¬Happy(Carol) 3 In fact, it is possible to define the concept of ordered pair using the concept of set. This is left as an exercise.
3
• Happy(Alice) ∨ Loves(Carol, Alice) ↔ ¬Loves(Carol) ∧ Loves(Alice, Alice). So far there’s nothing really new here. But then you might also notice the obtaining or nonobtaining of patterns and want to express this. For example, W makes true all of the following: • Smiling(Alice) → Happy(Alice) • Smiling(Bob) → Happy(Bob) • Smiling(Carol) → Happy(Carol) These statements all exemplify a certain pattern, namely • Smiling(x) → Happy(x) But moreover, Alice, Bob, Carol are all of the objects that exist in W. Hence, the pattern holds universally in W. That is to say, the following is true in W: • every object x is such that: Smiling(x) → Happy(x) This observation can be fully formalized with the help of a new symbol ∀x, a so-called universal quantification of x: • ∀x(Smiling(x) → Happy(x)) It could have been expressed idiomatically: • everything smiling is happy In a similar vein, you might notice that although in W not everything happy is smiling, nonetheless something is. That is to say, • there is an x such that x is happy and x is smiling. This can be rephrased using the new symbol ∃x, the so-called existential quantification of x: • ∃x(Happy(x) ∧ Smiling(x)) Or idiomatically, the same thought is • something happy is smiling. Let’s continue developing new forms of observation about W. In particular, note that patterns can themselves involve objects. Consider, for example, the pattern • Alice loves x. It is a fact about W that this pattern is satisfied by at least one thing (which?). This fact-in-W can itself be stated, like this: • ∃x Loves(Alice, x)
4
Just as we found a pattern involving Alice, there is a similar pattern involving Bob instead: • Bob loves x And indeed, this pattern too is satisfied by at least one thing according to W. Hence in W the following statements both hold: • ∃x Loves(Alice, x) • ∃x Loves(Bob, x) Now it’s clear that Alice and Bob themselves have a commonality: you get a truth-in-W by taking either of them to be the value of y in • ∃x Loves(y, x) This commonality between Alice and Bob might suggest new generalizations. Unfortunately, the commonality is not universal: it is not enjoyed by the third object Carol. But, let’s try to make some generalization in the vicinity. First, note that Alice and Bob are distinguished from Carol by being happy. Thus, the following are all true: • Happy(Alice) → ∃x Loves(y, x) • Happy(Bob) → ∃x Loves(y, x) • Happy(Carol) → ∃x Loves(y, x) What we’ve observed, then, is that every object in the domain satisfies the following condition: • Happy(x) → ∃x Loves(y, x) Thus, the following formula is true in W: • ∀y(Happy(y) → ∃x Loves(x, y)) How would you express this fact in English?
3
Formal syntax
Let’s now make the previous discussion systematic. The first task is to specify a formal language, call it FOL, which serves to formalize the varieties of expression just introduced. Specifying the language FOL basically requires determining what are its formulas; but it will be useful to define some other notions too along the way. A sentence of FOL is supposed to say how things are. This has two components: mentioning objects, and saying something about them. The way to mention objects is to use a sign which refers to them. A sign which refers to objects is a term. On the other hand, a predicate is a sign which, when combined with terms for some objects, thereby says that the objects are in a certain way. So, to build up the language of FOL, we need to introduce terms and predicates. These combine to yield formulas of the simplest kind, atomic formulas. Further formulas will result from formulas by applying truth-functional connectives and quantifiers. 5
Atomic formulas. The language of first-order logic contains two kinds of term: constants and variables. A constant is like a proper name, in that as part of its meaning, there is some fixed object it denotes. A variable, on the other hand, is like a pronoun; its meaning is rather just to denote whatever was denoted by some previous occurrences of terms. More formally, • a constant of FOL is any of the signs a, b, . . . , w, a0 , b0 , . . . , a1 , . . .. • a variable of FOL is any of the signs x, y, z, x0 , y0 , . . .. • the terms of FOL are the constants and variables of FOL. The system of predicates is just a tiny bit fussier. The simplest kind of predicate just applies to single things, saying of a thing e.g. that it is happy or smiling; such predicates are said to be one-place, or monadic. But some predicates are to be applied not to one thing but to more than one; such predicates are said to be polyadic. Specifically, a dyadic predicates applies to an ordered pair of things, saying of the pair e.g. that the first element loves the second element. Similarly there are triadic, tetradic, and in general n-place predicates. We will also allow zero-place “predicates”; these were the atomic formulas of FOL. Many textbooks begin by introducing predicate logic first with monadic predicates only. Here, I’ll present the grammar of the whole system. Typographically, monadic predicates should be distinguished from dyadic, triadic, etc. So, we’ll use accents to indicate the number of argument places. The predicates of FOL are now defined as follows: • • • •
a zero-place predicate is any of the signs A, B, . . . , Z, A0 , B0 , . . . , A1 . . .; a one-place predicate is any of the signs A0 , B 0 , . . . , Z 0 , A00 , B00 . . . , . . . A01 . . .; a two-place predicate is any of the signs A00 , B 00 , . . . , Z 00 , A000 , B000 , . . . , A001 , . . ., and more generally an uppercase letter with n primes is an n-place predicate.
In practice, it is annoying to write 0 all the time. But if we agree not to use predicates like A0 and A00 together in the same context, then no ambiguity will arise.4 As promised, the terms and the predicates now can be said to combine to yield atomic formulas: • an atomic formula of FOL is an n-place predicate followed by n occurrences of terms. Officially, then, atomic formulas include things like D, B 0 a, A0 x, E 00 ax, F cxbxy, etc. But in practice we will often simply write D, Ba, Ax, Eax, F cxby, etc., taking the superscript of the predicate to correspond to the length of the subsequent string of terms. 00000
4 It is sometimes considered good manners to use letters like F, G, H . . . as monadic predicates, and R, S, T . . . for polyadic predicates.
6
Formulas. We now need to explain the general notion of formula. Rather than just mentioning particular formulas, we are now going to generalize about them. Recall that in arithmetic when you mention particular integers, you do so with expressions like ‘0’, ‘1’, ‘2394’, etc. But when you want to generalize, then instead you use variables x, y, z . . . which range over all integers. And similarly to generalize about formulas, we need variables which range over all formulas, or all predicates, terms or whatever. I will use boldface letters for this purpose. In the new notation we can restate the definition of atomic formula like this: • An atomic formula is an expression of the form Fa1 · · · ak , where F is a k-place predicate and a1 , . . . , ak is a series of k terms. Now the first part of the definition of formula is easy. • Every atomic formula of FOL is a formula of FOL. How about more complicated formulas? As before, you can construct some using truth-functional connectives. So provided that P and Q are formulas of FOL, then also • ¬P, (P ∧ Q), (P ∨ Q), (P → Q), (P ↔ Q) are formulas of FOL. But the key new ingredient is construction of formulas with quantifiers. Let’s say that x is a variable of FOL, so that it is one of the letters x, y, z, x0 , . . .. Now provided that P is a formula of FOL, then also • ∀xP and ∃xP are formulas of FOL. So, formulas of FOL include for example F a, ¬Rab, ∀xF x → ¬∃yRxy, etc. Order of operations, a review. Recall that for truth-functional logic we adopted a conventional order of operations. Those conventions remain in force for first-order logic. Moreover, we now stipulate that • The scope of a quantifier ∀x or ∃x is the smallest complete formula immediately to its right. So the precedence order for connectives of first-order logic is ∃x, ∀x, ¬, ∧, ∨, →, ↔ . For example, in ∀y(F y → Gd), the ∀y governs F y → Gd, while in ∀yF y → Gd it governs just F y. You should now be able to draw the formation tree of any formula of first-order logic. Draw for youself the tree for ∀x(¬∀yF y → Rxy) → ∃zRxz. For later developments we’ll need a few more syntactical ideas, so let’s develop them now. A variable can occur in a formula in one of two mutually exclusive ways: either as free, or as bound by a quantifier. Every occurrence 7
of a variable in an atomic formula is free. A variable occurs free in a truthfunctional combo ¬P, P ∧ Q, etc., just where it occurs free in P or in Q. The free occurrences of variables other than x in ∀xP or in ∀xP are precisely the free occurrences of variables other than x in P. But wherever x occurs free in P, then it is bound by the indicated occurrences of ∀x and ∃x in ∀xP or ∃xP. For example, in the formula ∃xF x → (F x ∧ F y) the variable y has one occurrence and that occurrence is free. The quantifier ∀x, on the other hand, binds the first subsequent occurrence of x but not the last one, so that the last, and only the last, occurrence of x is free. You should think of a variable as something like a pronoun. Thus, for example, if the predicate F expresses the property of being square, then the formula F x says something like “it is square”. The proper response to that formula is “which thing?” In contrast, a quantification with respect to x, say ∃x, completes the thought: thus ∃xF x amounts to “there is some thing or other such that: it is square.” In that case the meaning of the pronoun is fixed by the antecedent noun phrase “some thing or other”. Thus, you may think of a formula as “expressing a complete thought” when no variable occurs free in it. A formula in which no variable occurs free is said to be closed. Only a closed formula, then, represents a complete thought. The concept of truth for predicate logic will apply to closed formulas only. Finally, we’ll need one more syntactical notion, which is a special kind of substitution on expressions. Roughly speaking, substitution is a sort of “copyand-paste” operation which replaces occurrences of one expression with another. So, for example, the result of substituting “t” for “b” in “babble” is “tattle”. There are also notions of substitution which are more specific. For example, the result of substituting “r” for all occurrences of “b” which are not adjacent to another “b” in “babble” is “rabble”. One such more specific notion we’ll use repeatedly. • If P is a formula, t is a term, and x is a variable, then P[x/t] is the result of substituting t for all free occurrences x in P. Suppose, for example, that P is the formula ∃xF x → (F x ∧ F y). Since only the last occurrence of x in P is free, therefore P[x/a] is ∃xF x → (F a ∧ F y). Underlying this special concept of substitution derives from the fact that quantifiers express generalizations. A formula P[x/t] is said to be an instance of a universal or existential generalization ∀xP or ∃xP. Thus, we will be able to say that a universal generalization implies each of its instances, while an existential generalization is implied by each of its instances.
4
Semantics of FOL
Having specified the syntax of a language, it’s time to explain what makes its formulas true or false. In the case of first-order logic, this is a rather interesting problem which takes a couple of steps. I’ll start by defining an appropriate notion of world, and then turn to the concept of truth. 8
4.1
The worlds of FOL, again
To begin with, we have to define a notion of world which is suitable for FOL. This just makes precise the idea informally sketched in §1. Whereas a possible world of truth-functional logic is a row of a truth-tables, a world of FOL is called a structure. A structure W consists two things • a nonempty set ∗W , called the domain of W • an interpretation of the constants and predicates of FOL. In what follows, we’ll mainly consider domains whose elements are among the natural numbers 0, 1, 2, . . .. This is just convenient, since everybody knows what those objects are.5 It remains, then, to stipulate how the constants and predicates of FOL should be interpreted by a structure. A constant is supposed to be like a name, in simply standing for an object. Thus • the W-interpretation aW of a constant a is an element of ∗W . Let’s now consider the case of predicates. First of all, a monadic predicate is combined with a term for an object, in order to say the object has a certain property. The collection of objects which have the property is called its extension. So, a monadic predicate can be interpreted by specifying its extension. • the W-interpretation FW of a monadic predicate F is an set of elements of ∗W . Similarly, a dyadic predicate is combined with a pair of terms for objects, in order to say that the two terms, in that order, stand in a certain relation. The extension of a relation is the set of ordered pairs of objects which, in that order, bear the relation. • the W-interpretation RW of a dyadic predicate R is an set of ordered pairs of elements of ∗W . More generally, • the W-interpretation of an n-place predicate is an set of ordered n-tuples of elements of ∗W . At last, and as in truth-functional logic, • the W-interpretation of zero-adic predicate is a truth-value.
4.2
Truth
Having defined for FOL the concepts of formula and of possible world, it remains to define the concept of truth. Since FOL contains a fair number of moving parts, it will be easiest to proceed in stages. 5 You could instead use, for example, marbles, but it’s hard to talk about particular marbles because few of them have names. Also, the number of numbers is importantly greater than the number of marbles.
9
4.2.1
Closed atomic formulas.
In FOL, an atomic formula consists of an n-place predicate followed by n terms. And a formula is closed if no variable occurs free in it. But any occurrence of a variable in an atomic formula is free in that formula. So, a closed atomic formula must consist of n-place predicate followed by n constants. Now, consider a structure W of FOL. And consider, for example, a closed atomic formula F a. What determines whether F a is true or false in W? Recall that informally, a formula F a represents a statement that a is F . This statement is true just in case the object denoted by a falls in the extension of the predicate F . Fixing denotations of constants and extensions of predicates is the job of a structure. For example, consider ∗ : {0, 1, 2} a : 0 . W= b:1 F : {0, 2} R : {(0, 1), (1, 2)} Then aW = 0, while F W = {0, 2}. So according to W, the object denoted by a does fall into the extension of the predicate F . Therefore, according to W, the formula F a ought to be true. On the other hand, the formula F b ought to be false (why?). Similarly, the ordered pair (0, 1) falls into the W-extension of R. This means that according to W, 0 bears R to 1. Again because a, b refer to 0, 1 therefore the formula Rab ought to be true. On the other hand, the ordered pair (1, 0) does not belong to RW , so the formula Rba ought to be false. More generally, suppose that F is any monadic predicate, and that c is any constant. Then • the formula Fc is true in W iff the object cW belongs to the set PW . Now a bit of notation. Write W |= A to mean that A is true in W. Also, write W 6|= A if it is not the case that W |= A . In the new notation, the clause for handling monadic predicates can be restated like this: • W |= Fc iff cW belongs to FW . So, given the above choice of W, we then have W |= F a but W 6|= F b. Let’s now turn to the case of dyadic predicates. In this case, a predicate is combined with two names, and the result should be true iff the extension of the predicate contains the pair of denotations of those names. That is, suppose R is a dyadic predicate and c, d are constants. Then • W |= Rcd iff (cW , dW ) belongs to RW . Then for example, given the same W from 10, we get W |= Rab but W |6 = Rba. The treatment of general polyadic predicates is similar. If P is an n-place predicate, then 10
W W • W |= Pc1 , . . . , cn iff (cW 1 , . . . , cn ) belongs to P .
Zeroadic predicates are handled just as in truth-functional logic: • W |= P iff PW = T. 4.2.2
Truth-functional combinations.
The logic FOL includes the connectives of truth-functional logic. And they’re evaluated just as usual. That is, we exploit truth-functionality to determine the truth or falsehood of complicated formulas by reference to the truth or falsehood of simpler ones. Let’s restate the procedure in the new language. Suppose that P and Q are formulas of FOL. Then • W |= ¬P iff W 6|= P • W |= P ∧ Q iff both W |= P and W |= Q; • W |= P ∨ Q iff either W |= P, or W |= Q (or both); • W |= P → Q iff either W 6|= P or W |= Q (or both); • W |= P ↔ Q iff either W |= P and W |= Q, or W 6|= P and W 6|= Q. For example, let W be defined as above. Then have we W |= (F a → F b) ↔ ¬Rba? To settle this question, first apply the definition of truth for closed atomic formulas to determine the truth or falsehood at W of F a, F b, and Rba. Then use the truth-tables to chase the truth-values up the formation tree. 4.2.3
Quantification.
The definition of truth for formulas containing quantifiers is a much deeper problem.6 Consider, for example, the formula ∀xF x. Our earlier strategy was to reduce the question of its truth-value to the question of truth-values of its simpler subformulas. However, the only simpler subformula of (x)F x is F x. And in F x, the x occurs free. This means that F x is analogous to a sentence like “it is green” where the “it” is unspecified. Being not closed, the subformula F x ought not to have a truth-value at all. Thus, quantified formulas are not in general built from simpler closed formulas. So, we need a new idea. Suppose again that the domain of W is {0,1,2}. Then what should W be like in order that it make true a universally quantified formula ∀xA? Intuitively, that formula says “everything has the property expressed by A”, or that every object satisfies A. Now according to W, all the things that exist are simply 0, 6 The first solution to this problem was given by Alfred Tarski in 1933, “The concept of truth in formalized languages”. We’ll follow an approach from Joseph Shoenfield’s Mathematical Logic (1967).
11
1, and 2. So, the formula ∀xA should be true over W provided that each of 0, 1 and 2 satisfies A. So, the question we need to answer is this: • what is it for an object to satisfy a formula over a structure? Standard names and their denotations. We have reduced the concept of quantification to the concept of satisfaction. But what is satisfaction? Intuitively, the idea is that an object satisfies a formula over a structure if what the formula says is true of that object in that particular structure. But what is this relation “true of”? Here an explanation is quite natural: a formula is true of an object if the result of plugging into the formula a name of that object is simply true. Again, recall P[x/a] represents the result of substituting a for all free occurrences of x in P. Now, suppose that a is a name of object o. We would like to say that if P contains just the free letter x, then P is true of the object o just in case P[x/a] is true. This approach immediately confronts a difficulty: it is not in general the case that every object has a name. In the real world, not all ants have names, nor do all grains of sand. So, we need a trick to arrange that every structure has a “hardwired” name for each object in its domain. This is not too bad, because we are working with artificial worlds, whose domain can be supposed to consist entirely of numbers 0, 1, 2, . . .. And every such number does have a name. Indeed, every natural number has standard Hindu-Arabic decimal name. For example, • the standard name of 7 is “7”; • the standard name of 21 + 210 is “231”; • the name of 21 consists of the name of 2 followed by the name of 1, i.e., it’s “21”, etc. For any object o, let’s write o for its standard name. For example, 21 =“21”. The notation “o” might be pronounced “bar-oh” or “the name of o”. Thus, the bar represents a function which, applied to an object, returns its standard name. To give another example: suppose that the standard name of a human being is just their full personal name. Then, The US president in 2016 begins with the letter ‘B’, followed by the letters ‘a’, ‘r’, ‘a’, ‘c’, ‘k’, followed by a space, etc. If you have trouble remembering the names of people, a bar function would come in handy. We now stipulate that in any structure whatsoever, a standard name o denotes the object o. That is,
12
• oW = o for all W. W
So, for any W, we have that 7 = 7. Likewise 21 + 210 is a string of three W characters, but 21 + 210 is an integer greater than 230 and less than 232. Similarly, if Barack Obama were an element of the domain of W, then you could say that The US president in 2016
W
was born in Hawaii.
The definition of satisfaction, and the definition of truth for quantified formulas. We wanted to explain satisfaction of a formula by an object in terms of the result of plugging into the formula a name of an object. This idea can now be made precise. Recall that A[x/t] is the result of substituting t for all free occurrences of x in A. Suppose that A is a formula whose one free variable is x. Then, • for an object o to satisfy the formula A over W just is for the formula A[x/o] to be true over W. Recall the W from page 10. In that case, the object 0 satisfies F x over W, because W |= F 0. And because 1 doesn’t satisfy F x, therefore 1 does satisfy ¬F x. If bW = 1, then 0 satisfies F x ↔ ¬F b, but 1 does not. And so on. With that, our problem of defining truth for quantifiers is solved. We can say that ∀xA is true over W provided that every object in the domain of W satisfies A. And, ∀xA is true over W provided that every object in the domain of W satisfies A. More precisely, say that A is a formula whose one free variable x. Then • W |= ∀xA iff W |= A[x/o] for each object o in ∗W . • W |= ∃xA iff W |= A[x/o] for some object o in ∗W . And with that, the definition of truth is complete. Examples. Consider again the structure W from page 10. Have we W |= ∀xF x? This will be so just in case W |= F o whenever o is any of 0, 1, 2. Since W 6|= 1, therefore actually W 6|= ∀xF x. On the other hand we do have, for example, W |= ∃xF x, since W |= F 0. Thus, W |= ∃xF x ∧ ∃x¬F x. In contrast, it’s clear without checking the details of W that W 6|= ∃x(F x ∧ ¬F x). For otherwise, there’d be a o such that W |= F o ∧ ¬F o. And then we’d have that o both does and doesn’t belong to F W , which is impossible.
5
The method of truth-functional expansion
In truth-functional logic, the definition of truth is easy to apply: just chase the truth-value up the formation tree. Unfortunately, the situation is not so straightforward in first-order logic. In particular, if the domain is infinite, then 13
the truth-value of a generalization depends on the truth-value of infinitely many simpler formulas.7 Luckily, our main purpose is the assessment of deductive arguments. An argument, as ever, is invalid if it has a countermodel—that is, if there’s a world where all premises are true while the conclusion is false. And it turns out that most invalid arguments you’ll ordinarily encounter have countermodels which are very small, with just two or three elements in their domain. Let’s now develop a practical procedure for evaluating formulas in small structures. The key idea is that once the domain of a structure has been fixed as some finite set of objects, universal and existential quantifiers can be regarded as conjunctions and disjunctions. That is, suppose that the elements of the domain are o1 , . . . , ok . We’ll associate, to each quantified formula A, another formula Ao1 ,...,ok called the truth-functional expansion of A over o1 , . . . , ok . Over any structure whose domain contains just o1 , . . . , ok , the formulas A and Ao1 ,...,ok will have the same truth-value. So over a given finite structure, the method of truth-functional expansion reduces the evaluation of a quantified formula to the evaluation of an unquantified one. To state the method, it will help to have one further piece of terminology. Say that an W-instance of a generalization ∀xA or ∃xA is a formula A[x/o], where o is an element of ∗W . For example, if ∗W contains 1, then an instance of ∃x∃y(F y ∧ F x) is ∃y(F y ∧ F 1). Now the method for computing an expansion of a formula over a domain {o1 , . . . , ok } is simply this. • The expansion of a universal generalization is the conjunction of the expansions of its instances • The expansion of an existential generalization is the disjunction of the expansions of its instances • The expansion of a truth-functional combo of some formulas is just that combo of the expansions of those formulas. • The expansion of an atomic formula is just that formula. Examples. For example, suppose that 0, 1, 2 are the elements of ∗W . And consider a random formula A, like ∃xF x → ∀xF x. The expansion A0,1,2 of A over 0, 1, 2 now gets computed like this: A0,1,2
(∃xF x → ∀xF x)0,1,2 (∃xF x)0,1,2 → (∀xF x)0,1,2 (F 0 ∨ F 1 ∨ F 2) → (F 0 ∧ F 1 ∧ F 2).
7 For example, the natural numbers, together with addition, multiplication, and exponentiation can be considered to form a structure. Problems of classical number theory are simply questions whether a given formula is true in that structure.
14
We can use the expansion of A to compute the truth-value of A. Suppose, for example, that the W-extension of F is {0, 2}. Granted that ∗W is {0, 1, 2}, this reduces the relevant portion of W to the row of a truth-table! Namely, F0 T
F1 F
F2 . T
So it is clear that W |= F 0 ∨ F 1 ∨ F 2. Meanwhile, W 6|= F 0 ∧ F 1 ∧ F 2. Consequently, W 6|= (F 0 ∨ F 1 ∨ F 2) → (F 0 ∧ F 1 ∧ F 2). And that’s just to say W 6|= (∃xF x → ∀xF x)0,1,2 . Since, however, 0, 1, 2 are precisely the elements of W, it follows that A0,1,2 must, over W, have the same truth-value as A! Therefore, W 6|= ∃xF x → ∀xF x. Caution! It should be clear that first-order logic is much more expressive than truth-functional logic. This is because the quantifiers are a fundamentally new concept. Quantifiers let us say things we can’t say only with truth-functions. Consider, for example, the formula ∀xF x. This is certainly not equivalent to a conjunction of its instances. For no matter how many objects you consider, it’s logically possible for there to exist some other object besides those, and this object might not satisfy F x. Thus, ∀xF x does not follow from any conjunction of atomic formulas. For this reason, it should be clear that a quantified formula is not equivalent to its truth-functional expansion. Rather, the method says only that a formula and its expansion over a domain ∗W have the same truth-value over W. To bring this out, here are a couple of exercises. (i) Find a formula, together with two domains, such that the expansion-of-the-formula-over-one domain is not equivalent to its expansion-over-the-other-domain. (ii) Find a formula and a domain such that the formula does not imply its expansion-over-that-domain. Illustrating the method of expansions. The method of truth-functional expansion is useful for distinguishing the meanings of more complicated formulas. Consider, for example, ∀x∃yRxy and ∃y∀xRxy. Are these equivalent? If so, then they must have the same truth-value over all structures whatsoever. Do they? Consider first a structure whose domain is just {0}. In that case, the two formulas both expand into R00. So they must have the same truth-value over all one-element structures. 15
But consider instead the domain {0, 1}. Then the formulas expand like this. On the one hand, (∀x∃y(Rxy))0,1
(∃y(R0y))0,1 ∧ (∃y(R1y))0,1 (R00 ∨ R01) ∧ (R10 ∨ R11).
On the other hand, (∃y∀x(Rxy))0,1
(∀x(Rx0))0,1 ∨ (∀x(Rx1))0,1 (R00 ∧ R10) ∨ (R01 ∧ R11).
Can you find a structure with domain {0, 1} which makes one true and the other false? Finding such a structure amounts to filling in the question marks in R00 ?
R01 ?
R10 ?
R11 ?
This is an exercise in truth-functional logic. You should be able to find a W with ∗W = {0, 1} so that W |= ∀x∃yRxy while W 6|= ∃y∀xRxy.8 This will demonstrate that ∀x∃yRxy does not imply ∃y∀xRxy.
8 For
example, it will work to choose W so that
RW = {(0, 0), (1, 1)}.
16
R00 T
R01 F
R10 F
R11 . That is to say, T
