quantificational logic motivation and syntax
Notes on quantificational logic, I: motivation and syntax Max Weiss 15 October 2015
We’ve developed a complete analysis of truth-functional logic: The truth-tables for the connectives give a theory of meaning, which yields a notion of semantic consequence. On the other hand, the method of natural deduction gives a system of proof. And as it turns out, the theories of meaning and of proof align, because the relation of semantic consequence coincides with the relation of derivability. However, truth-functional logic is not all of logic! Consider, for example 1. “all humans are mortal, Confucius is immortal, so Confucius is not human” 2. “all yoga teachers are narcissists, so all parents of yoga teachers are parents of narcissists” 3. “everybody loves my baby, my baby loves nobody but me; therefore, I am my baby.” 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 (1-3). The validity of the arguments (1-3) depends crucially on the concepts “all” and “some”. These concepts are known as quantifiers. We’re now going to study quantificational logic. Just as truth-functional logic is the logic of truth-functions, so quantificational logic is the logic of quantifiers.
The possible worlds of quantificational logic Recall that logic studies the way that the formal features of statements constrains what distributions of truth and falsehood are possible for them. An approach to this study is that of truth-functional logic. There, we isolate a certain class of formal features, those which comprise their truth-functionality. Thus, truth-functional logic studies the way that statements can be truth-functional combinations of simpler statements they contain. E.g., the truth or falsehood 1
of “either it’s raining or it’s sunny” is a function of the truth or falsehood 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 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 “it’s raining” 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. The analysis of such interdependencies leads to the notion of possible world, or “structure”: 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 structure 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 according to W , 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 . For example, to give a structure W you can, e.g., • choose three objects, Alice, Bob, Carol to form the domain ∗W , and • decide that Alice and Bob are happyW , that Bob is smilingW , and that Alice lovesW Bob while Bob lovesW Carol.
2
Or in other words, ∗ : {Alice, Bob, Carol} Happy : {Alice, Bob} W = Smiling : {Bob} Loves : {(Alice, Bob), (Bob, Carol)} About the notion of domain of a structure there are a couple of crucial provisos. A domain can be any collection of objects whatsoever provided that the collection is nonempty. This assumption is not absolutely essential, but it simplifies the technical machinery. Moreover, it belongs to the standard development of quantificational logic. After coming to grips with the standard approach, any concerned reader can pursue the exercise of rewiring the machinery to obviate the assumption.
Using quantifiers The logical purpose of a notion of structure is to generate a suitable notion of truth. So after introducing a notion of structure for quantificational logic, 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
3
• ¬Happy(Carol) • 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. 4
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. Here are two ways: • there is an x such that Alice loves x. • ∃x(Loves(Alice, x)) 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: • there is an x such that Alice loves x • there is an x such that Bob loves x Or formally: • ∃x(Loves(Alice, x)) • ∃x(Loves(Bob, x)) Thus, 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 • there is an x such that y loves x Or formally, in W each of Alice and Bob satisfies • ∃x(Loves(y, x) This commonality between Alice and Bob might suggest new generalizations. Now, the commonality is not universal: for it is not shared by the third object Carol. Let’s nonetheless 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: 5
• if Alice is happy, then there is an x such that Alice loves x • if Bob is happy, then there is an x such that Bob loves x • if Carol is happy, then there is an x such that Carol loves x Those three statements all exemplify the pattern • if y is happy, then there is an x such that y loves x which can be formalized by • Happy(y) → ∃x(Loves(y, x)) As before, Alice, Bob, Carol are all the elements of the domain of W . So that pattern holds universally in W . In other words, according to W the following is the case: • for all y, if y is happy, then there is an x such that y loves x Or in the official jargon, the following formula is true in W : • ∀y(Happy(y) → ∃x(Loves(x, y)))
Formal syntax Let’s now start making the previous discussion systematic.1 The first task is to specify a formal language, call it L20 , which serves to formalize the varieties of expression just introduced.2 Specifying the language L20 basically requires determining what are its formulas; but it will be useful to define some other notions too along the way. A statement 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. So in general, statements will contain terms. But a statement doesn’t just mention a bunch of objects; rather the statement says how the objects are. This is the role of signs of another kind, predicates. That is, a predicate is a sign which, when combined with terms for some objects, thereby says that the objects are in a certain way. 1 For
further discussion see §§4.1-4.2 of Holbach. language L20 is a simplification of Halbach’s L2 . The difference is that L20 does not contain any constants. We will consider the addition of constants later on. 2 The
6
Now, the language L20 is designed to mimic this logical function of statements. So, to build up the language of L20 , we begin by introducing for it a basic collection of terms and predicates. In general, quantificational languages contain two kinds of term: constants and variables. However, we will begin by introducing variables only, and consider constants later on. • a variable of L2 is any of the signs x, y, z, x0 , y0 , z0 , x1 , . . .. Now the notion of term of L20 is defined like this: • the terms of L20 are just the variables of L20 . The system of predicates is just a tiny bit fussier. Note that a predicate can apply to single things, saying of a thing e.g. that it is happy or smiling; such predicates are said to be one-place. In contrast, two-place predicates apply to ordered pairs of things, saying of the pair that e.g. the first loves the second. Similarly there are three-, four-, and in general n-place predicates. Just for fun we will also allow zero-place predicates; these correspond to the atomic formulas of L1 . The predicates of L20 are now defined as follows: • a zero-place predicate is any of the signs P 0 , Q0 , R0 , P00 , Q00 , R00 , P10 , . . .; • a one-place predicate is any of the signs P 1 , Q1 , R1 , P01 , Q10 , R01 , P11 , . . .; • a two-place predicate is any of the signs P 2 , Q2 , R2 , P02 , Q20 , R02 , P12 , . . .; etc. In the context of a single problem, we will tend use a predicate like P 1 not with a predicate like P 2 , but rather with a predicate like Q2 instead. This means that we can simply write P and Q instead, relying on context to determine the intended number of argument-places. As promised, the terms and the predicates now can be said to combine to yield atomic formulas: • an atomic formula of L20 is an n-place predicate followed by n occurrences of terms. 6 x0 x1 x2 x0 x1 x5577 , So, for example, atomic formulas include P00 , Q13 x, R1325123 etc. But in practice we will often simply write P0 , Q3 x, R1325123 x0 x1 x2 x0 x1 x5577 , taking the superscript of the predicate to correspond to the length of the subsequent string of terms.
We now need to explain how to construct further formulas from the atomic ones. This requires an inductive definition which is almost exactly the same as was given for L1 . It goes as follows. The set of formulas of L20 is the smallest set X which satisfies the following four conditions. 7
• every atomic formula of L20 is an element of X; • if φ is in X, then so is (¬φ), • if φ and ψ are both in X, then so are (φ ∧ ψ), (φ ∨ ψ), (φ → ψ), and (φ ↔ ψ); • if φ is in X and v is a variable, then (∀vφ) and (∃vφ) are in X. Recall that for L1 we introduced a conventional “order of operations” to allow judicious dropping of parentheses. This convention extends easily to L20 : any quantifier ∀v or ∃v should be taken to govern the smallest formula which immediately follows it. Thus for example ∃xP x∧Qy is to be rewritten ((∃xP x)∧Qy). For later developments it will be useful to have set up a bit more syntactical terminology now. Notice that any occurrence of a quantifier in a formula is immediately followed by a formula φ; that occurrence of φ is said to be the scope of the quantifier. For example, the scope of the first occurrence of ∃x in ∃x(P x ∧ Qx) → ∃xP x is the following occurrence of (P x ∧ Qx); and the scope of the second occurrence of ∃x is the following occurrence of P x. 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 of a variable in an atomic formula is free. A variable occurs free in a truth-functional combo ¬φ, φ ∧ ψ, etc., just where it occurs free in φ or in ψ. The free occurrences of variables other than v in ∀vφ or in ∃vφ are precisely the free occurrences of variables other than v in φ. But wherever v occurs free in φ, then it is bound by the quantifier ∀v or ∃v in ∀vφ or ∃vφ. You should think of a variable as something like a pronoun. Thus, for example, if the predicate Q expresses the property of being square, then you should think of the formula Qx as saying “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 ∃xQx 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. Finally, we’ll need one more syntactical notion, which is a special kind of substitution on expressions. Roughly speaking, substitution is just a “copy-and-paste” operation which replaces all 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 φ is a formula, then φ[v/t] is the result of substituting a term t for all free occurrences of v in φ. Here it’s probably useful to consider some examples. 8
1. If φ is Rxy, then φ[x/z] is Rzx. 2. If φ is ∃yRxy, then φ[y/z] is ∃yRxy, which is just φ. 3. If φ is ∃yRxy ∧ Rxy, then what is φ[y/x]?
9
We’ve developed a complete analysis of truth-functional logic: The truth-tables for the connectives give a theory of meaning, which yields a notion of semantic consequence. On the other hand, the method of natural deduction gives a system of proof. And as it turns out, the theories of meaning and of proof align, because the relation of semantic consequence coincides with the relation of derivability. However, truth-functional logic is not all of logic! Consider, for example 1. “all humans are mortal, Confucius is immortal, so Confucius is not human” 2. “all yoga teachers are narcissists, so all parents of yoga teachers are parents of narcissists” 3. “everybody loves my baby, my baby loves nobody but me; therefore, I am my baby.” 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 (1-3). The validity of the arguments (1-3) depends crucially on the concepts “all” and “some”. These concepts are known as quantifiers. We’re now going to study quantificational logic. Just as truth-functional logic is the logic of truth-functions, so quantificational logic is the logic of quantifiers.
The possible worlds of quantificational logic Recall that logic studies the way that the formal features of statements constrains what distributions of truth and falsehood are possible for them. An approach to this study is that of truth-functional logic. There, we isolate a certain class of formal features, those which comprise their truth-functionality. Thus, truth-functional logic studies the way that statements can be truth-functional combinations of simpler statements they contain. E.g., the truth or falsehood 1
of “either it’s raining or it’s sunny” is a function of the truth or falsehood 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 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 “it’s raining” 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. The analysis of such interdependencies leads to the notion of possible world, or “structure”: 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 structure 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 according to W , 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 . For example, to give a structure W you can, e.g., • choose three objects, Alice, Bob, Carol to form the domain ∗W , and • decide that Alice and Bob are happyW , that Bob is smilingW , and that Alice lovesW Bob while Bob lovesW Carol.
2
Or in other words, ∗ : {Alice, Bob, Carol} Happy : {Alice, Bob} W = Smiling : {Bob} Loves : {(Alice, Bob), (Bob, Carol)} About the notion of domain of a structure there are a couple of crucial provisos. A domain can be any collection of objects whatsoever provided that the collection is nonempty. This assumption is not absolutely essential, but it simplifies the technical machinery. Moreover, it belongs to the standard development of quantificational logic. After coming to grips with the standard approach, any concerned reader can pursue the exercise of rewiring the machinery to obviate the assumption.
Using quantifiers The logical purpose of a notion of structure is to generate a suitable notion of truth. So after introducing a notion of structure for quantificational logic, 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
3
• ¬Happy(Carol) • 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. 4
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. Here are two ways: • there is an x such that Alice loves x. • ∃x(Loves(Alice, x)) 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: • there is an x such that Alice loves x • there is an x such that Bob loves x Or formally: • ∃x(Loves(Alice, x)) • ∃x(Loves(Bob, x)) Thus, 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 • there is an x such that y loves x Or formally, in W each of Alice and Bob satisfies • ∃x(Loves(y, x) This commonality between Alice and Bob might suggest new generalizations. Now, the commonality is not universal: for it is not shared by the third object Carol. Let’s nonetheless 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: 5
• if Alice is happy, then there is an x such that Alice loves x • if Bob is happy, then there is an x such that Bob loves x • if Carol is happy, then there is an x such that Carol loves x Those three statements all exemplify the pattern • if y is happy, then there is an x such that y loves x which can be formalized by • Happy(y) → ∃x(Loves(y, x)) As before, Alice, Bob, Carol are all the elements of the domain of W . So that pattern holds universally in W . In other words, according to W the following is the case: • for all y, if y is happy, then there is an x such that y loves x Or in the official jargon, the following formula is true in W : • ∀y(Happy(y) → ∃x(Loves(x, y)))
Formal syntax Let’s now start making the previous discussion systematic.1 The first task is to specify a formal language, call it L20 , which serves to formalize the varieties of expression just introduced.2 Specifying the language L20 basically requires determining what are its formulas; but it will be useful to define some other notions too along the way. A statement 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. So in general, statements will contain terms. But a statement doesn’t just mention a bunch of objects; rather the statement says how the objects are. This is the role of signs of another kind, predicates. That is, a predicate is a sign which, when combined with terms for some objects, thereby says that the objects are in a certain way. 1 For
further discussion see §§4.1-4.2 of Holbach. language L20 is a simplification of Halbach’s L2 . The difference is that L20 does not contain any constants. We will consider the addition of constants later on. 2 The
6
Now, the language L20 is designed to mimic this logical function of statements. So, to build up the language of L20 , we begin by introducing for it a basic collection of terms and predicates. In general, quantificational languages contain two kinds of term: constants and variables. However, we will begin by introducing variables only, and consider constants later on. • a variable of L2 is any of the signs x, y, z, x0 , y0 , z0 , x1 , . . .. Now the notion of term of L20 is defined like this: • the terms of L20 are just the variables of L20 . The system of predicates is just a tiny bit fussier. Note that a predicate can apply to single things, saying of a thing e.g. that it is happy or smiling; such predicates are said to be one-place. In contrast, two-place predicates apply to ordered pairs of things, saying of the pair that e.g. the first loves the second. Similarly there are three-, four-, and in general n-place predicates. Just for fun we will also allow zero-place predicates; these correspond to the atomic formulas of L1 . The predicates of L20 are now defined as follows: • a zero-place predicate is any of the signs P 0 , Q0 , R0 , P00 , Q00 , R00 , P10 , . . .; • a one-place predicate is any of the signs P 1 , Q1 , R1 , P01 , Q10 , R01 , P11 , . . .; • a two-place predicate is any of the signs P 2 , Q2 , R2 , P02 , Q20 , R02 , P12 , . . .; etc. In the context of a single problem, we will tend use a predicate like P 1 not with a predicate like P 2 , but rather with a predicate like Q2 instead. This means that we can simply write P and Q instead, relying on context to determine the intended number of argument-places. As promised, the terms and the predicates now can be said to combine to yield atomic formulas: • an atomic formula of L20 is an n-place predicate followed by n occurrences of terms. 6 x0 x1 x2 x0 x1 x5577 , So, for example, atomic formulas include P00 , Q13 x, R1325123 etc. But in practice we will often simply write P0 , Q3 x, R1325123 x0 x1 x2 x0 x1 x5577 , taking the superscript of the predicate to correspond to the length of the subsequent string of terms.
We now need to explain how to construct further formulas from the atomic ones. This requires an inductive definition which is almost exactly the same as was given for L1 . It goes as follows. The set of formulas of L20 is the smallest set X which satisfies the following four conditions. 7
• every atomic formula of L20 is an element of X; • if φ is in X, then so is (¬φ), • if φ and ψ are both in X, then so are (φ ∧ ψ), (φ ∨ ψ), (φ → ψ), and (φ ↔ ψ); • if φ is in X and v is a variable, then (∀vφ) and (∃vφ) are in X. Recall that for L1 we introduced a conventional “order of operations” to allow judicious dropping of parentheses. This convention extends easily to L20 : any quantifier ∀v or ∃v should be taken to govern the smallest formula which immediately follows it. Thus for example ∃xP x∧Qy is to be rewritten ((∃xP x)∧Qy). For later developments it will be useful to have set up a bit more syntactical terminology now. Notice that any occurrence of a quantifier in a formula is immediately followed by a formula φ; that occurrence of φ is said to be the scope of the quantifier. For example, the scope of the first occurrence of ∃x in ∃x(P x ∧ Qx) → ∃xP x is the following occurrence of (P x ∧ Qx); and the scope of the second occurrence of ∃x is the following occurrence of P x. 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 of a variable in an atomic formula is free. A variable occurs free in a truth-functional combo ¬φ, φ ∧ ψ, etc., just where it occurs free in φ or in ψ. The free occurrences of variables other than v in ∀vφ or in ∃vφ are precisely the free occurrences of variables other than v in φ. But wherever v occurs free in φ, then it is bound by the quantifier ∀v or ∃v in ∀vφ or ∃vφ. You should think of a variable as something like a pronoun. Thus, for example, if the predicate Q expresses the property of being square, then you should think of the formula Qx as saying “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 ∃xQx 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. Finally, we’ll need one more syntactical notion, which is a special kind of substitution on expressions. Roughly speaking, substitution is just a “copy-and-paste” operation which replaces all 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 φ is a formula, then φ[v/t] is the result of substituting a term t for all free occurrences of v in φ. Here it’s probably useful to consider some examples. 8
1. If φ is Rxy, then φ[x/z] is Rzx. 2. If φ is ∃yRxy, then φ[y/z] is ∃yRxy, which is just φ. 3. If φ is ∃yRxy ∧ Rxy, then what is φ[y/x]?
9
