2. First Order Logic 2.1. Expressions. Definition 2.1. A language L consists of a set LF of function symbols, a set LR of relation symbols disjoint from LF , and a function arity : LF [LR ! N. We will sometimes distinguish a special binary relation symbol =. 0-ary function symbols are called constant symbols. We will always assume that both LF and LR are countable. First-order logic will involve expressions built from symbols of our lan- guage together with additional symbols: • Infinitely many first-order variables, x0; x1;:::, • The logical connectives ^; _; !; ?, • Quantifiers 8 and 9, • Parentheses ( and ). We usually write x; y; z; : : : for first-order variables. Definition 2.2. The terms of L are given inductively by: • The variables x0; x1;::: are all terms, • If F 2 LF , so F is a function symbol, arity(F ) = n, and t1; : : : ; tn are terms, then F t1 ··· tn is a term. Note that 0-ary function symbols are permitted, in which case each 0-ary function symbol is itself a term; we call these constants. If R 2 LR, so R is a relation symbol, arity(R) = n, and t1; : : : ; tn are terms then Rt1 ··· tn is n atomic formula. The formulas of L are given inductively by: • Every atomic formula is a formula, •? is a formula, • If φ and are formulas then so are (φ ^ ), (φ _ ), (φ ! ), • If x is a variable and φ is a formula then 8xφ and 9xφ are formulas. Formulas and terms are collectively called expressions. When R is binary, we often informally write with infix notation: t1 < t2, 0 t1 = t2, etc.. We sometimes write Q; Q , etc. for an arbitrary quantifier. 2.2. Free Variables. Definition 2.3. If e is an expression, we define the free variables of e, free(e), recursively by: • free(x) = fxg, S • free(F t1 ··· tn) = i≤n free(ti), S • free(Rt1 ··· tn) = i≤n free(ti), • free(?) = ;, • free(φ ~ ) = free(φ) [ free( ), • free(Qxφ) = free(φ) n fxg. 1 2 Definition 2.4. Given t, we define the formulas φ such that t is substitutable for x recursively by: • If φ is atomic or ? then t is substitutable for x in φ, • t is substitutable for x in φ ~ iff t is substitutable for x in both φ and , • t is substituable for x in 8yφ, 9yφ iff either { x does not occur in 8yφ, { y does not occur in t and t is substituable for x in φ. If s is a term, we define s[t=x] by: • If y 6= x, y[t=x] is y, • x[t=x] is x, • (F t1 ··· tn)[t=x] is F t1[t=x] ··· tn[t=x]. When t is substitutable for x in φ, we define φ[t=x] by: • (Rt1 ··· tn)[t=x] is Rt1[t=x] ··· tn[t=x], •?[t=x] is ?, • (φ ~ )[t=x] is φ[t=x] ~ [t=x], • If y 6= x,(Qyφ)[t=x] is Qy(φ[t=x]), • (Qxφ)[t=x] is Qxφ. We only write φ[t=x] when it is the case that t is substitutable for x in φ. If we have distinguished some variable x we write φ(t) for φ[t=x]. Definition 2.5. We define the alphabetic variants of a formula φ recursively by: • If φ is atomic, φ is its only alphabetic variant, 0 0 0 • φ ~ is an alphabetic variant of φ~ iff φ is an alphabetic variant of φ and 0 is an alphabetic variant of , • Qxφ0 is an alphabetic variant of Qyφ iff there is a φ00 such that φ00 is an alphabetic variant of φ and φ is φ00[x=y]. Definition 2.6. We define the substitution instances of φ inductively by: • φ is a substitution instance of φ, • If is a substitution instance of φ and t is substitutable for x in then [t=x] is a substitution instance of φ. 2.3. Sequent Calculus. The sequent calculus for first-order logic is a direct extension of the cal- culus for propositional logic. We add four new rules: Γ; φ[y=x] ) Σ Γ ) φ[t=x]; Σ L9 R9 Γ; 9xφ ) Σ Γ ) 9xφ, Σ Where y does not appear free in ΓΣ Γ; φ[t=x] ) Σ Γ ) φ[y=x]; Σ L8 R8 Γ; 8xφ ) Σ Γ ) 8xφ, Σ Where y does not appear free in ΓΣ 3 In L9 and R8, the variable y is called the eigenvariable, and the condition that y not appear free in ΣΓ is the eigenvariable condition. When φ is a formula with a known distinguished variable x, it will be convenient to write φ(t) for φ[t=x]. Definition 2.7. Fc is the system consisting of the nine rules of Pc together with the four rules above. As before, Fi is the fragment of Fc where the right- hand part of each sequent consists of 1 or 0 formulas, Fm is the fragment of cf Fi omitting L?, and F is the fragment of F omitting Cut. Taking for granted that Fm ` φ ) φ for any formula, we can consider some examples. Example 2.8. φ(y) ) φ(y) φ(y) ) 9xφ(x) 8yφ(y) ) 9xφ(x) ) 8yφ(y) ! 9xφ(x) Example 2.9. :φ(x); φ(x) )? :φ(x); 8yφ(y) )? 9x:φ(x); 8yφ(y) )? 9x:φ(x) ) :8yφ(y) ) 9x:φ(x) ! :8yφ(y) Example 2.10. ) φ, :φ ) 9xφ, :φ ) 9xφ, 8x:φ ? ) 9xφ :8x:φ ) 9xφ ) :8x:φ ! 9xφ Example 2.11. φ(x); φ(y) ) φ(y); 8yφ(y) φ(x) ) φ(y); φ(y) ! 8yφ(y) φ(x) ) φ(y); 9x(φ(x) ! 8yφ(y)) φ(x) ) 8yφ(y); 9x(φ(x) ! 8yφ(y)) ) φ(x) ! 8yφ(y); 9x(φ(x) ! 8yφ(y)) ) 9x(φ(x) ! 8yφ(y)) cf Lemma 2.12. For any φ, Fm ` φ ) φ. Proof. For atomic φ this is an axiom, and for φ ~ we have covered this case in propositional logic. The only new cases are the two quantifiers. 4 IH φ ) φ φ ) 9xφ 9xφ ) 9xφ Similarly, in the 8 case, we have: IH φ ) φ 8xφ ) φ 8xφ ) 8xφ 2.4. Properties of Intuitionistic Logic. cf Theorem 2.13 (Generalized Subformula Property). If F ` Γ ) Σ then every formula appearing in the deduction is a substitution instance of a subformula of some formula in either Γ or Σ. cf Theorem 2.14 (Existence Property). If Fi ` 9xφ then there is a term t cf such that Fi ` φ[t=x]. Proof. The last inference rule of this deduction must be R9, and therefore must have concluded φ[t=x]. cf Theorem 2.15. If Fi ` 8x9yφ then there is a term t (possibly containing cf x free) such that Fi ` 8xφ[t=y]. Proof. The last inference rule of this deduction must be R8, and concluded 9yφ, and so the previous line must have been φ[t=y]. Therefore by R8, 8xφ[t=y]. 2.5. Double Negation. Definition 2.16. As before, we define φ∗ recursively by: ∗ • (Rt1 ··· tn) is (Rt1 ··· tn) _?, •?∗ is ?, ∗ ∗ ∗ • (φ ~ ) is φ ~ , • (Qxφ)∗ is Qx(φ∗). Theorem 2.17. For any φ, ∗ (1) Fi ` φ $ φ , ∗ (2) Fm ` ? ! φ , ∗ (3) If Fi ` φ then Fm ` φ . Proof. Again, we prove only the third part. We show by induction on de- ductions ∗ ∗ Suppose Fi ` Γ ) Σ. If Σ = fφg then Fm ` Γ ) φ , and ∗ if Σ = ; then Fm ` Γ )?. 5 The induction from the propositional case works for all rules of Pi, and the four new rules are all rules of Fm which go through unchanged. Definition 2.18. We define the double negation interpretation of φ, φN , inductively by: •?N is ?, • pN is ::p, N N N • (φ0 ^ φ1) is φ0 ^ φ1 , N N N • (φ0 _ φ1) is :(:φ0 ^ :φ1 ), N N N • (φ0 ! φ1) is φ0 ! φ1 , • (8xφ)N is 8xφN , • (9xφ)N is :8x:φN . Again ΓN = fγN j γ 2 Γg. N N Lemma 2.19. Fm ` ::φ ) φ . Proof. By induction on φ. We have already handled all cases except when φ is Qx . For φ = 8x , we have N ) N 8x N ) N : N ; 8x N )? : N ) :8x N IH ::8x N ) :: N :: N ) N ::8x N ) N ::8x N ) 8x N For φ = 9x , we have 8x: N ; :8x: N )? 8x: N ) ::8x: N :::8x: N ; 8x: N )? :::8x: N ) :8x: N N N Lemma 2.20. If Fm ` Γ; :φ )? then Fm ` Γ ) φ . N Theorem 2.21. If Fc ` φ then Fm ` φ . Proof. Again, we prove by induction on deductions N N If Fc ` Γ ) Σ then Fm ` Γ ; :Σ )?. The inductive steps from the propositional case work unchanged, so we need only deal with the four new rules. If the last inference of the original deduction was R8 then we have :8xφN in :ΣN , so we have: 6 IH ΓN ; :ΣN ; :φN (y) )? 2.20 ΓN ; :ΣN ) φN (y) ΓN ; :ΣN ) 8xφN ΓN ; :ΣN ; :8xφN )? If the last inference of the original deduction was L8 then we have 8xφN in ΓN , and so: IH ΓN ; φN (t); :ΣN )? ΓN ; 8xφN ; :ΣN )? If the last inference is R9 then we have ::8x:φN in :ΣN , and so: IH ΓN ; :ΣN ; :φN (t) )? 2.19 ΓN ; :ΣN ; 8x:φN )? ::8x:φN ) 8x:φN ΓN ; :ΣN ; ::8x:φN )? If the last inference is L9 then we have :8x:φN in ΓN , and so IH ΓN ; φN (y); :ΣN )? ΓN ; :ΣN ):φN (y) ΓN ; :ΣN ) 8x:φN 8x:φN ; :8x:φN )? ΓN ; :8x:φN ; :ΣN )? We also have Theorem 2.22.