Handout No. 2 Phil. 015 Zoltan Domotor Department of Philosophy, University of Pennsylvania February 12, 2016 SEMANTICS AND SYNTAX OF PREDICATE LOGIC Now that you are familiar with the syntactic and semantic structures of LSL1 (presented in terms of natural deduction rules and truth tables) and are aware of simple predications and quantification, we can devise a formal (artificial) language to represent logical forms in predicate logic. All we need to do is make certain modifications to sentential (truth-functional) logic. However, now we will need six kinds of symbols:
1. Individual Variables: x, y, z, .. . . Individual variables in the object language of LMPL2 are lower-case italic letters, taken from the end of the latin alphabet with or without primes or numerical subscripts. We use individual variables exclusively to formalize individual ‘variable words’ in a natural language or scientific discourse. Given Given a particular particular interpretatio interpretation, n, individual individual variables variables take their values precisely in the specified domain (universe of discourse) course) of a given given interpret interpretati ation. on. For exampl example, e, if the domain domain is the class of all humans, living in the past or present, then the individual variables stand for humans only. 2. Individual Constants: a, b, c, . . . . Individual (non-logical) constants are lower-case italic letters from the initial segment of the latin alphabet, with or without primes or numerical merical subscripts subscripts.. We use indivi individual dual constant constantss to symboliz symbolizee proper proper names names of indivi individual dualss in a natural natural or scien scientific tific language language.. Granted Granted an interpretation, individual constants refer to designated individuals in the specified domain of interpretation. 3. Monadic (unary) Predicate Constants: F , G , H , . . . , F , G , H , . . . . Predicate Predicate (non-logical) (non-logical) constants constants are primed capital italic letters, often ′
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Language of Sentential Logic Language of Monadic Predicate Logic
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with numerical subscripts to indicate the number of arguments. In general, we shall omit the primes (e.g., instead of F we shall write F ) and the subscripts as well. Thus, instead of F 1 , G1 , H 1 , dots, we write F , G , H , . . . . Monadic predicate constants formalize one-argument predicate expressions in a natural or scientific language. ′
4. Binary Predicate Constants: F , G , H , . . . , F 2 , G2 , H 2 , . . . Binary (dyadic, two-place) predicate (non-logical) constants are primed capital italic letters with a numerical subscript 2, indicating that the predicate has two arguments. As practiced in Modern Logic , we shall omit primes and subscripts. ′
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For the time being, we shall use only one-place (monadic, unary, or of degree one) and two-place (diadic, binary, or of degree two) predicate constants that apply only to, respectively, single individuals or pairs of individuals. For example, the sentence ‘Alma speaks both French and German’ is formalized as F a & Ga. Here a refers to Alma and the monadic predicate constants F and G symbolize the respective properties (attributes) of being able to speak French and German. Note that we write the predicate constant first and the individual constant next with no punctuation between. Likewise, ‘Alma loves Bill’ is formalized as Lab. Here a refers to Alma, b refers to Bill, and L stands for the love relationship. As expected, ‘Bill loves Alma’ is formalized as Lba. Ternary (triadic) relations, as in ‘Alma is between Bill and Carl’, are less frequent. This sentence is formalized as Bbac. In general, n-place predicates (with n ≥ 3) lead to additional enrichments of the predicate language.
5. Sentential Letters: P, Q, R, . . . . Sentential letters (with or without numerical subscripts) are also included, and are taken from sentential logic LSL. For example, the sentence ‘It is raining’ is captured by R, without any reference to individuals. Thus, predicate logic can be viewed as a natural extension of truth-functional (sententional, propositional) logic. 6. Logical Constants: logical connectives & , ∨, →, ↔, ∼, and quanti fiers ∀, ∃. 7. Auxiliary Symbols: parantheses ( and ), brackets [ and ], and braces { and } , needed for unique and easy readability. Using these symbols we can generate an infinite set of formulas in the language of predicate logic of the form F a, ∃xF x, ∀x[F x → Gx], ∃ y {F y & ∀x[F x → Hx]}, ∀z [Hz → P ], . . . . Only formulas obtainable by obvious 2
repeated applications of logical connectives and quantifiers to given formulas are well-formed. Not surprisingly, just like in the world of natural numbers, not all well-formed formulas (e.g., formulas taking thousands of pages to write down) have transparent interpretations in a natural language. Predicate logic uses the same ten rules of inference as sentential calculus (and hence the same derived rules and theorems). But it also has additional introduction and elimination rules for quantifiers . We shall treat these rules after the semantic approach. A CRASH COURSE IN FORMAL SEMANTICS FOR PREDICATE LOGIC
Recall that in sentential (propositional, truth-fumnctional) logic we define an interpretation by assigning truth values to all sentential letters of our formal language LMPL. For example, in the case of the sentence Q → (P → Q) we can assign value T (true) to Q, value F (false) to P , and ‘calculate’ the resulting truth value of Q → (P → Q). We already know that the resulting value will be T under any truth-value assignment. That is, we can write |= Q → (P → Q), indicating that the foregoing formula is semantically valid in LSL. Unfortunately, this truth value-based semantic method does not work in predicate logic. To fully understand what is meant by “all” and “some”, we need to know what individuals constitute the subject of discourse. For example, on one occasion we can say that “All are rotten” (symbolized by ∀xRx) and be talking about just the oranges in a crate. On another occasion we can say the same “All are rotten” and be thinking about just the apples in a basket. On yet another occasion we may be talking about just the members of the Mafia. And so on. What differs here is what logicians call the universe of discourse or, for short, the domain . The domain is always a nonmepty set of individuals talked or reasoned about, needed in any interpretation of predicate formulas. Accordingly, an interpretation I of (or simply a model for) formulas in the language of predicate logic LMPL consists of (i) a nonempty domain D of individuals; (ii) an assignment that associates with each individual constant in LMPL a designated element of D; (iii) an assignment that associates with each one-place predicate F a designated subset of D (called the extension of F ), namely the subset of those elements in D that have property F ; 3
(iv) an assigment that associates with each two-place predicate R a designated subset of ordered pairs in the Cartesian product D×D (called the extension of R), namely the subset of those ordered pairs of elements in D that stand in relation denoted by R, and (v) an assignment of one of the truth-values T or F to each sentential letter. For example, the sentence ∀xF x is true under I if and only if for every element d in the chosen domain D, d belongs to the subset assigned to F . Similarly, under the same interpretation, the formula ∃xF x is true if and only if there exists an element d in the domain D such that it belongs to the subset assigned to F . For another example, given the universal conditional ∀x[F x → Gx], below we determine its truth value under interpretation I , specified as follows: (i) The domain is: D =df {1, 2, 3, 4}. I.e., the domain is specified by the set of the first four consecutive natural numbers. Here =df is short for “is equal by definition”; (ii) the monadic (one-place) predicate F is understood to refer to the property of being even , i.e., formally its interpretation is given by the subset F = {2, 4} of all even numbers in the chosen domain { 1, 2, 3, 4}. (iii) the monadic predicate G is understood to refer to the property of being a prime number, i.e., its interpretation is given by the subset G = {2, 3} of all prime numbers in the specified domain { 1, 2, 3, 4}. There are no other nonlogical constants to interpret. Remember, F refers to a linguistic entity that belongs to the alphabet of the object language of LMPL, and F denotes a set (a mathematical object). It is easy to see that under the foregoing interpretation – which is often denoted by the ordered triple D, F , G , the universal conditional says that all even natural numbers below 5 are prime. Since 4 is below 5 and is not prime, under the standing interpretation I the universal conditional ∀ x[F x → Gx] is false. Simply, it is not the case that all even numbers among 1, 2, 3, 4 are prime. However, under the same interpretation I , the existential sentence ∃x[F x & Gx] comes out true, because there exists an even number that is prime, namely 2. ′
Now, suppose we change the interpretation to I , newly specified as follows: (i) The domain is: D =df {1, 3, 5, 7}. I.e., the domain is defined by the set of the first four odd natural numbers; ′
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(ii) the monadic predicate F is now understood to refer to the property of being even in the domain, i.e., its interpretation is given by the empty set F = ∅, which is of course a legitimate subset of the domain {1, 3, 5, 7}; ′
(iii) the monadic predicate G is understood to refer to the property of being a prime in the domain, i.e., its interpretation is given by the set G = {3, 5, 7} of all prime numbers in the specified domain {1, 3, 5, 7}. There are no other nonlogical constants to interpret. ′
It is easy to see that this time the universal conditional ∀x[F x → Gx] is true under interpretation I = D , F , G , and of course the existential sentence ∃x[F x & Gx] is false under I . The first sentence is true under I because the assumption of any number being even is always false in its domain and hence the conditional is true. The second is false because there are no even prime numbers among 1, 3, 5, 7. ′
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Now suppose we want to explore the interpretation of Ha → ∃yH y. The previous interpretations I and I are not suitable for this purpose because they do not include a meaning assignment to the individual constant a but contain a meaning assignment to G that we do not need. However, consider a brand-new interpretation J = D , a, H, specified by ′
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(i) D =df {1, 2, 3, 4}; ∗
(ii) a denotes a = 2; (iii) The meaning of H is to be even , i.e., given (as before) by the set H = {2, 4}. There are no other nonlogical constants to interpret. Then the conditional Ha → ∃yH y is clearly true under J . It now says that if 2 is even, then there are even numbers in the domain. Observe that the formula will come out true even if a denotes 1, because falsehood implies anything. If we mofify I into a new interpretationJ = D♯ , a, H, where D♯ = { 1, 3, 5, 7}, a denotes a = 3, and F expresses evenness, i.e., H = ∅, then the foregoing conditional still comes out true. As a matter of fact, the conditional Ha → ∃xHx is semantically valid because it comes out true under every possible interpretation. (This can be shown by assuming the opposite and then deriving a contradiction.) ′
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PROOFS IN PREDICATE LOGIC
This handout includes various proofs (also known as derivations) of logical theorems and argument-forms in predicate logic with identity (equality), using the usual ten basic rules of inference in sentential logic LSL , four basic inference rules for quantification in LMPL, two basic rules for identity in LPLE, and some previously proven theorems. Along traditional lines, we proceed from simple theorems and arguments in predicate logic to increasingly more complex ones that often depend on other theorems and arguments. Feel free to use any of these ‘solved problems’ in your work by quoting their labels PL0, PL1, PL2, etc. The basic inference rules – an introduction rule and a corresponding elimination rule for the universal and existential quantifier are symbolized in an argument form as follows:
1. Introduction and Elimination of the Universal Quantifier: Introduction of ∀, symbolized by I∀: ¿From any formula of the form Z (a), containing an individual constant ‘a’ not occurring in any assumption A or in any hypothesis H in effect at the line on which Z occurs, we may infer the formula of the universally quantified form ∀xZ (x) (or ∀yZ (y), etc.), where by a syntactic definition Z (x) = Z (a/x) is the result of replacing all occurrences of a in Z (a) with individual variable x, not already in Z (a). (If x occurs in Z (a), then we use another free variable, say y that is not in Z (a).) The requirement that a not occur in any assumption or in any hypothesis already in effect at the line on which Z occurs ensures that we assume nothing that distinguishes the individual designated by a from any other individual. The rule I∀ must be applied strictly as stated. Thus, intuitively speaking, for I ∀ to be applicable, the individual constant a must refer to a ‘generic’ (“the all American boy/girl”) individual. Obviously, from ‘Einstein is a brilliant physicist’ it certainly does not follow that everybody is a brilliant physicist. Stated in another way, the rule I∀ applies to Z (a) if constant a does not occur outside the subproof where it is introduced. (A subproof is a proof that occurs within the context of a larger proof. As with any proof, it may, and usually will, begin with an assumption or hypothesis, indicated by A,H respectively. In this case the subproof ends when the hypothesis is discharged. You get out of a subproof via conditionalization or
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RAA.) Another important point: all rules apply only to sentences in their entirety and not to subsentences. Z (a) ∴ ∀xZ (x) where a does not occur outside the subproof where it is introduced.
Elimination of ∀, symbolized by E∀: From any universally quantified sentence ∀xZ (x) we may infer any sentence of the form Z (a) (or Z (b), etc.) which results from replacing each occurrence of the free variable x in Z with an individual constant a of your choice, whether or not it has been used elsewhere in the proof. That is, we have Z (a) = Z (x/a). Of course, the same rule applies to any individual variable, including ∀ yZ (y), ∀ zZ (z ), etc.
∀xZ (x) ∴ Z (a) 2. Introduction and Elimation of the Existential Quantifier: Introduction of ∃, symbolized by I∃: From any formula of the form Z (a) we may infer the existential formula of the form ∃ xZ (x) (or ∃yZ (y), etc.), where Z (x) = Z (x//a) is the result of replacing all or some occurrences of a in Z with the individual variable x, not already in Z . Thus, from F a & Ga we may infer both ∃y[F y & Gy] and ∃y[F y & F a] (or ∃y[F a & Gy]). In addition, we can also use other variables, as in ∃x[F a & Gx] or ∃z [F a & Gz ]. Observe that the rule I∃ places no restrictions on previous occurrences of the individual constant a. The individal constant a may occur legitimately in an undischarged hypothesis or in an assumption, as above. Z (a) ∴ ∃xZ (x) Elimation of ∃, symbolized by E∃: Given an existentially quantified sentence ∃ xZ (x) and a derivation of some conclusion Y from an existential hypothesis Z (a) = Z (a/x) (the result of replacing each occurrence of the variable x in Z (x) with an individual constant a not already occurring in Z), we may discharge the existential hypothesis Z (a) and reassert Y . Here the following restriction 7
applies: individual constant a may not occur in Y , nor in any assumption A, nor in any hypothesis H that is in effect at the line at which E∃ is applied. Alternatively, the restriction states that constant a is not allowed to occur outside the subproof where it is introduced. We may think of the name a introduced at the beginning of the subproof as the formal counterpart of the English “Let a be an arbitrary individual (e.g., John Doe) such that Z (a).” The restriction is handled automatically by introducing an individual constant a that has not been used in any of the assumptions or hypotheses before. Intuitively, in any proof, the rule E∃ allows us to put ∃xZ (x) on hold and work instead in the subproof with the temporary existential hypothesis Z (a) about a “John Doe” a, until the desired result is obtained. At that point, by the rule E∃ we can return to the use of ∃xZ (x) and at the same time discharge the temporary existential hypothesis Z (a).
∃xZ (x) Z (a) H Y ∴ Y where constant a does not occur in Y or any assumption, or hypothesis. 3. Introduction and Elimation of the Identity Relation: Special logical (unary, binary, etc.) predicates may be added to predicate calculus for special purposes. One of the most useful binary logical predicate constants is the classical identity =, where a = b expresses the fact that individual a is identical to individual b or a is the same thing as b, or simply a is equal to b. For example, if ‘a’ denotes Mark Twain and ‘b’ denotes Samuel Clemens, then a = b expresses the fact that Mark Twain is the same individual as Samuel Clemens. The identity predicate = is special because like all logical symbols and unlike all predicate constants, its interpretation is fixed . It always means “it is identical to.” Specifically, given an interpretation with a domain D in which individual constants a and b are interpreted as suitable designated elements in D , say a and b respectively, then a = b is true under the extant interpretation just in case a = b, i.e., if and only if a and b are the same objects in D . Also, note that = is syntactically peculiar in that unlike all nonlogical binary predicate constants, it is written between the individual constants or variables (as in a = b or x = y) to which it applies rather than in front of them (as in Lab or Lxy). The so-called Polish notation 8
= ab is hard to read. Of course, the identity requires a customary pair of rules of inference.
Introduction of Identity =, symbolized by I=: For any individual constant a, b, etc., we may assert a = a, b = b, etc., at any line of the proof.
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a = a
Observe that rule I= introduces formulas into proofs without deriving them from previous lines. In particular, the proof of the law of reflexivity of identity ⊢ ∀xx = x consists of two simple steps: I= 1 a = a 2 ∀xx = x 1, I∀ Note that the use of I∀ in step 2 is legitimate because the proof does not use any assumptions or hypotheses containing a. This should be clear, since a is generic. We could have started with b = b, and so on.
Elimation of Identity =, symbolized by E=: ¿From a formula Z (a) (containing individual constant a) and a = b or b = a we may infer Z (b), the result of replacing one or more occurrences of a in Z by b. (i) (ii) Z (a) Z (a) a = b b = a ∴ Z (b//a) ∴ Z (b//a) where the expression Z (b//a) denotes the formula that results from Z (a) upon replacing one of more occurrences of a in Z with b. Clearly, the inference rule E= is the traditional substitution law : If a and b are really one and the the same (i.e., a = b), then anything true of one of them must be true of the other as well. Specifically, Z (b//a) must be true if obtained from the ‘truth of’ Z (a) by substituting equals for equals. A proof or derivation in predicate logic with identity is a finite sequence of consecutively numbered lines, each consisting of a sentence belonging to the language of predicate logic together with its line number (on the left) and all rules used (on the right), such that on each line the sentence is a given assumption or a temporary hypothesis, or it follows from previous lines in accordance with any of the 14 rules of sentential, 4 rules of predicate and two rules of identity in quantificational logic, itemized above and in Handout 9
No.1. If based on assumptions X 1 , X 2 , . . . , formula Y appears on the last line in which all premises belong to X 1 , X 2 , . . . , then the sequence of sentences under consideration is called the proof (derivation) of Y from X 1 , X 2 , . . . . If such a proof (derivation) exists, we write X 1 , X 2 , · · · ⊢ Y . SOLVED PROBLEMS IN PREDICATE LOGIC WITH IDENTITY
Here are the problems solved earlier.
M0
∼F a ⊢ ∼∀xF x Proof.
1 2 3 4 5
∼F a ∀xF x Fa F a & ∼F a ∼∀F x
A H (for RAA) 2, E∀,H 1,3 I & ,H 2-4 RAA,H
Observe that here we have not used I∀ at all.
M1
∀x[F x → (Gx ∨ Hx)], ∀ x∼Gx ⊢ ∀x[F x → H x] Proof.
1 2 3 4 5 6 7 8 9
∀x[F x → (Gx ∨ Hx)] ∀x∼Gx F a → (Ga ∨ Ha) ∼Ga Fa Ga ∨ Ha Ha F a → Ga ∀x[F x → H x]
A A 1, E∀ 2, E∀ H (for C) 3,5 MP, H 4,6 T, H 5-7 C,H 8, I∀
Observe that constant a does not occur in any of the assumptions. Its occurrence in the hypothesis does not matter because the hypothesis with its sub-subproof is discharged before the use of I∀.
M2
∃xF x ∨ ∃xGx ⊢ ∃x[F x ∨ Gx] Proof.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14
∃xF x ∨ ∃xGx ∃xF x Fa F a ∨ Ga ∃x[F x ∨ Gx] ∃x[F x ∨ Gx] ∃xF x → ∃x[F x ∨ Gx] ∃xGx Ga F a ∨ Ga ∃x[F x ∨ Gx] ∃x[F x ∨ Gx] ∃xGx → ∃x[F x ∨ Gx] ∃x[F x ∨ Gx]
A H (for C) H’ (for E∃) 3, I∨, H ,H’ 4,I∃,H,H’ 2,3-5, E∃,HH 2-6 C, H H (for C) H’ (for E∃) 9, I∨, H ,H’ 10,I∃,H,H’ 8,9-11,E∃,HH 8-12, C, H 1,7,13 E∨
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Additional Problems, with new labels, are solved below.
∀x[F x → Gx], ∀xF x ⊢ Ga
PL0
Proof.
1 2 3 4 5
∀x[F x → Gx] ∀xF x F a → Ga Fa Ga
A A 1, E∀ 2, E∀ 3,4 MP
∀x∀yLxy ⊢ Laa
PL1
Proof.
1 ∀x∀yLxy 2 ∀yLay 3 Laa
A 1, E∀ 2, E∀
∀x[F x & Gx] ⊢ ∀xF x & ∀xGx
PL2
Proof.
1 2 3 4 5 6 7
∀x[F x & Gx] F a & Ga Fa Ga ∀xF x ∀xGx ∀xF x & ∀xGx
A 1, E∀ 2, E & 2, E & 3, I∀, a generic 4, I∀, a generic 5,6 I &
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∀x[F x ∨ Gx] ⊢ ∃xF x ∨ ∃xGx]
PL3
Proof.
1 2 3 4 5 6 7 8 9 10 11
∀x[F x ∨ Gx] F a ∨ Ga Fa ∃xF x ∃xF x ∨ ∃xGx] F a → ( ∃xF x ∨ ∃xGx) Ga ∃xGx ∃F x ∨ ∃xGx Ga → ( ∃F x ∨ ∃xGx) ∃xF x ∨ ∃xGx
A 1, E∀ H (for C 3, I∃, H 4,I∨,H 3-5, CH H (for C) 7, I∃,H 8,I∃,H 7-9,C 2,6,10,E∨
∼∃xF x ⊢ ∀x∼F x
PL4
Proof.
1 2 3 4 5 6
∼∃xF x Fa ∃xF x ∃xF x & ∼∃xF x ∼F a ∀x∼F x
A H (for RAA) 2, I∃,H 1,3,I & ,H 2-4,RAA, H (a is now outsdide H) 5, I∀
∀xLax, ∀x∀y[Lxy → M yx] ⊢ ∀xMxa
PL5
Proof.
1 2 3 4 5 6 7
PL6
∀xLax ∀x∀y[Lxy → M yx] Lab ∀y[Lay → M ya] Lab → M ba Mba ∀xMxa
A A 1, E∀ 2, E∀ 4,E∀ 3,5 MP 6, I∀ (b is generic)
⊢ ∀ x[F x → F x] Proof.
1 Fa H (for C) 2 F a → F a 1,1 C, H 3 ∀x[F x → F x] 2, I∀ Here in step 2 we apply conditionalization of F a to itself. Notice that on line 2 individual constant a is generic, because it falls outside the scope of hypothesis H. Remember that H is discharged in step 2. We 12
could have started with F a → F a on line 1, because it is a substitution instance of P → P , known to be logically valid.
⊢ ∼ [∀xF x & ∃x∼F x]
PL7
Proof.
∀xF x & ∃∼F x ∀xF x ∃x∼F x ∼F a Fa P & ∼P P & negP ∼[forallxFx & ∃x∼F x] Solved problems for Identity:
H (for RAA) 1,E & , H 1,E & ,H H’ (for E∃),H 2,E∀,H, H’ 4,5, T7, H, H’ 3,4-6,E∃ H, H 1-7, RAA, H
1 2 3 4 5 6 7 8
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F a, ∼F b ⊢ ∼a = b
PL9
Proof.
1 2 3 4 5 6
Fa ∼F b a = b Fb F b & ∼F b ∼a = b
A A H(for RAA) 1,3 E=, H 2,4 I & , H 3-5,RAA, H
⊢ ∀ x∀y[x = y → y = x]
PL10
Proof.
1 2 3 4 5 6
a = b a = a b = a a = b → b = a ∀y[a = y → y = a] ∀x∀y[x = y → y = x]
H (for C) I−, H 1,2 E= 1,3 C, H 4 I∀, b is generic 5 I∀
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