 38 results  books.google.com Then I=sn(t=S)lIInll iff =s.rt?.1=st?.1 (15.11) iff =s¢<tt?.1>=¢<st7.1) (ifby 11
mess) iff I:,qt[Ii1'>,...,i,'I']] = s[[if',...,i;f (by(1)) iff I:R(t:s)[Ii1¢,...,i,'II (by 1.5.17) (3): It is
an easy exercise to see that if J5 and W have the property claimed, then so do —
J5 ... 

 books.google.com (ii) For every Sformula p: 3 (= F iff T' (= w. This will complete the proof. (i) can
easily be proved by induction on terms, (ii) is proved by induction on formulas <p
simultaneously for all assignments 0 in 21. We only treat the case of atomic ... 

 books.google.com (T) “S” is true iff, S We get an instance of schema (T) by plugging in an actual
sentence for S. e.g. “snow is white” is true iff, snow is white”. But what is schema (
T) good for? Tarski says however you define truth; you should guarantee that
every ... 

 books.google.com By definition of A#, x e A# iffr(x, x) e A. Also, r(x, x) e A iff (1) Ely(r(x, x) I WW 6 A)(
Here y e A is not, strictly speaking, a formula of Elementary Arithmetic. But see
below how it can be taken to be an abbreviation for an arithmetic formula, indeed
of ... 

 books.google.com For the formula 1111 we have: at F 41;/I/1] iff 21 F1;/I/1] by(1.5.0.7) iff 21' F1;/I/1']
by ind. hyp. ... For the fonnula (II/1 /\ II/2) we have, analogously: Pl F (11/1A 11/2)l/
11 iff (Q1 F 11/1 ihl and 91 F 11/21/1 l) by (1508) iff (21' F 1;/1 I/1'] and 21' F 1 ... 

 books.google.com A is iff condition holds notation: satisfied in V iff ̂V(A) = 1 V = A not satisfied in V
iff ̂ V (A) = 0 V = A valid/tautology iff for all V : V = A = A falsifiable iff there is a V
: V = A = A satisfiable iff there is a V : V = A unsatisfiable/contradiction iff for all ... 

 books.google.com iff (by induction) for all T that differ from only in what they assign the variables in
the prefix at (2): iff for all that differ from G only in what they assign the variables in
the prefix at (2): iff for all that differ from only in what they assign the variables in ... 

 books.google.com Peter B. Andrews. Clearly a wff which is valid in the general sense is valid in the
standard sense, though we shall see in Chapter 7 that the converse of this
statement is false. We shall show that a wffo of Qo is a theorem iff it is valid in the
... 

 books.google.com PI— H iff H is Ab_l_derivab]e from Ax u f u I£ iff F u Td_ \— H And we write also
F N= H iff H is Avalid in all identity interpretations A for which all the wff of T are A
valid. From the definition of identity interpretation by straightforward calculations
... 

 books.google.com The forcing relation is usually denoted by l. Definition of forcing by induction on
sentences in 17*: (1) pIIxéy iff xéy (2) pllw=y iff w=y (3) pli G(w) iff PCw)=1 (4)
PH ¢/\<b iff Pll' ¢> and PH ¢ (5) pl! 3w¢(w) iff it (pli <I>(w)) (6) 10 ll —14> iff
For ... 

 books.google.com Walter Alexandre Carnielli, Luiz Paulo de Alcantara, Sociedade Brasileira Matemática  1988  Preview  More editions have the same germ at 0 iff rem 4.4 we can identify L/J. tion to {0} x[o,1]( f =g on
some open set containing 0 . By Theo .. and £ . Each element of £^ is the restric
of a function ?e£,. Since f(o)e{o,i}, it follows from the linearity properties of ... 

 books.google.com We deal with the rule and the axiom together, by showing that if x £ FV(ip) then
Ex a <p —> \}> is satisfied iff cp — > Vx : Al is. Now foLx. ^[Vxi//!^, =/7A(M°t,,) iff W
^r/^W^r,, iff fl<p aExJ^M,. Since <p — » i/» is satisfied iff l<pi* EC, Hall, we are ... 

 books.google.com For this semantics, we now consider v∅ ([b∪c](p→q)) = t where p and q are
distinct propositional variables, and b and c are distinct atomic sequences. Then,
we obtain two different interpretations: 1. v∅([b ∪c](p→q)) = t iff v∅([b](p→q))) ... 

 books.google.com is (simultaneously) satisfiable iff there exists an interpretation IU such that vIU (Ai)
= T for all i. The satisfying interpretation is a model of U. U is valid iff for every
interpretation IU, vIU (Ai)=Tfor alli. The definitions of unsatisfiable and falsifiable ... 

 books.google.com The following statements hold intuitively: (1.6) A set is enumerable iff it is void or
the range of a computable function. (1.7) A 1place function is computable, iff the
2place relation R is enumerable, where R holds for y and x iff y=f(x). (1.8) A set ... 

 books.google.com 1ft is f(t1, . . . , in), then x is free in t iff it is free in at least one of the 11,. The case
of formulae: Atomic case: (1) x is not free in any of .L, T, p. (2) x is free in t = 5 iff it
is so in at least one oft or s. (3) x is free in (15(t1, . . . , tn) iff it is so in at least one ... 

 books.google.com Proof. Let A = C°°CIR )/I , and $ = {Z(f)fel} the corresponding filter in TF , as
above. To prove (i) , let a , b be represented by functions £ a(x) , b(x) : 3R > R (
depending only on a finite set D c E of coordinates). Then B(a) c B(b) iff for every
prime ... 

 books.google.com I:Q1 EIx01[s] iff I:Q1 —Vx —01[s], iff big, Vx —r01[s], iff it is not the case that for
all d in IQtI, lI21 —I<1lS(Xld)], iff it is not the case that for all d in IQtI, F521 0l[S(X
ld)], iff for some d in IQtI, I:Q1 o1[s(x I d)]. Logical Implication We now have the ... 

 books.google.com Let E', F' come from E, F by interchanging & with V.“ Then: (a) IfF 1E, then F E'. (b
) IfF E, then F 1E'. (c) IfFE~F,thenFE'~F'. (d) 1fFEDF,thenFF'DE'. '° Although by '31
the association is ordinarily immaterial to us, we can for definiteness regard ... 

 books.google.com For instance, suppose that we start from a premise (p and use the axioms (1)(16)
, the axioms of SF, Modus Ponens, and Generalization to derive a conclusion iff.
Generalization should not be applied to free variables in (p, of course. Then, we ... 

 books.google.com Then the closure of S in the topology j is the pullback (S,6r) c> (A,ST) Hence, we
have s ={((o,/3),a) E J x A  e((a,/3)(a,/3)) = <5r(a,o),Xs(a) = (a,/3)} = = {a € .4  (Off,
r(a)) C J for a ^ 5 or (r(a),r(a)) E J for a E S). 7.2 Category of Qfuzzy sets The ... 

 books.google.com A. R. D. Mathias, H. Rogers  1973  Snippet view  More editions We let ( ) be any binary Z function from L(k) oneone onto L(<). Let ((x) ,(x) ) = x.
Do not confuse this with Definition 6.5. DEFINITION 6.16. We define R^focp.g), for
x e L(k), cp a formula, g e L^)"1, as follows: l) R1/,2(x,an£ am,g) iff g(n) e g(m) ii) ... 

 books.google.com Proposition 3.7. Let F := (M, n, □,£,<*>) be a Kripke frame. Then 1. Bl is valid in F
iff (x □ y)C\(x □ z) < x □ (y D z) for all x,y,z£ M, 2. B2 is valid in F iff for all x,y,z €
M with x > z, y > z andxDy < z, there exist u, v £ M such that x < u, y < v and uf)v ... 

 books.google.com We first prove that (1) for all t e Tx, \t \ e N iff [BE']? e X. V(B) For any t e Tx, \t\ e N
iff 9Jt0f((B) = 1 by definition of N, iff [B]E' f e X by induction hypothesis, iff [BE']? e
X Theorem 1.2(H) (p. 14). Consequently <BE',x> is a representative of iV, ... 

 books.google.com Mod3Ptr..tr iff 3(P) fits 3(t1),...,3(tr) iff 3(P) fits t1,...,tr by (***) iff Ptj...t € M* by (*) (b)
or is an equation, i.e. tr = t. = t_. We have Mod 3^ =t2 iff 3(tj) =3(t2) iff tj = t2 by (***)
iff tj ~ t2 iff tj = t2 6 TO* (c) a is a negation, i.e. asiP. Then R(P)<R(a), and we ... 

 books.google.com ... follows: (i) if <p is a sentence letter, 9, then 8(9) = T iff 8 assigns T to <p. (ii) if y
is a negation, i0, then S(i0) = T iff 8(0) = F. (iii) if *> is a conditional, (0 — > x) ,
then B(0 _> x) = T iff either 8(0) = F or 8(*) = T (or both) . (iv) if <p is a conjunction,
... 

 books.google.com First suppose that x is R*rfe0* ..•l«*• .or eachy' < m let x> = ^fj)k0, . .., v^u). Con
ce (9, xo Xm) We have, with S = pH, %, fn, . . . , fkipn  „) e R<* iff there is a ) such
that g e aP,,, A> and 93 N ?[A:?oS, s as above, then /e/. at, M,[pr<og]} I 6 / : 91, ... 

 books.google.com Ayda Ignez Arruda, R. Chuaqui, Newton C. A. Da Costa  1980  Snippet view A theory T (for LQ) is quantiileA complete iff both rich and saturated; iVuught iff
prime and quantifiercomplete; and adequate iff straight and regular. T is no n
degenerate (n. d.) iff T is neither null nor universal (i.e. contains every wff). 

 books.google.com (i) 03 is a substructure of some model of T iff every finitely generated substructure
(Exercise 3.3.39) of 03 is a substructure of a model of T; (ii) if T is universally
axiomatizable, then 03 is a model of T iff every finitely generated substructure of
53 ... 

 books.google.com Again by the Correspondence, me have: Theorem 3** is modal axiomatic over K*
iff J is modal axiomatic over K. # This and Goldblatt St Thomason's Theorem 8
give necessary & sufficient conditions for a class of K*frames to be modal ... 

 books.google.com R is symmetric on D iff (Vx)(Vy)(Dx a Dy  > (fay * Ryx)). 4. P is asymmetric on D
iff (Vx)(Vy)(Dx a Dy * (Rxy * ~\Ryx)). 5. R is antisymmetric on D iff (Vx)( Vy)(Dx a
Z>y  > (fay a Ryx * x = y)). 6. P is transitive on D iff (Vx)(Vy)(Vz)(Dx a Dy a Dz ... 

 books.google.com If A is pa i . . . a„, where p is not = , we let a(A) = T iff pa(a(a ,),..., a(a„)) (i.e., iff the
Htuple (ct(ai), . . . , a(a„)) belongs to the predicate pa). If A is ~1B, then a(A) is i/n(
a(B)). If A is B V C, then «(A) is #v(a(B), a(C)). If A is 3xB, then a(A) = T iff a(Bx[i]) ... 

 books.google.com Prove: (i) A point p is isolated — i.e., {p} is open — iff n(p) = {p}. (Use 4.1.) (ii) X is
a T0 space — i.e., if p,q are distinct points then />$ {q}~ or 4$ {p}~ — iff for any
distinct points p,q we have q$n(p) OTp$fi(q). (Use 4.2.) (hi) X is a Tx space — i.e.,
... 

 books.google.com [Ber.Mon.]) An extensional formula ,p € *i is of the second kind iff one of the
following conditions is satisfied: (i) <pv induces a pos. precomplete numeration.
iff (ii) There exists a rec. fct. / making the partial rec. fct. 0* defined as follows: {the
least ... 

 books.google.com Let A be an arbitrary class of atype models i.e. A cm and let d> e F be arbitrary .A
satisfies $ iff a T a each model of A satisfies it,i.e.A*=i> iff for any 91 eA 91 M
holds. Let Th(A)^{ij! e F  A »=<»>} be the set of true formulas in A . Let ZcF . 

 books.google.com H. Arnold Schmidt, Kurt Schütte, Helmut J. Thiele  1968  Snippet view  More editions ... all aeStlnd(i) such that ha is shown SATURATED INTUITTONISTIC THEORIES
s. 1) Not »,□ lhp(«i .,«n) iff 2) Mav iff M and 3) Mvy iff M or 4) hft+y iff implies for
all 5) lhV*a(x) iff for some aelnd(/); 6) lh,Axa(x) iff Ih,a(«) for all and all aelndQ). 

 books.google.com Wilfrid Hodges. if L U <p then for some a e Xn/c, L = <p. /c is IIS reflecting if k is
nfireflecting on On. 2)3 reflecting ordinals are defined similarly. The proof of the
following is straightforward. PROPOSITION 1. (i) k is ng reflecting iff k is nj ... 

 books.google.com Iff(x)=0 be any logical equation involving the class symbol x, with or without other
class symbols, then will the equation /(l)/(0)=0 be true, independently of the
interpretation of x; and it will be the complete result of the elimination of x from the
... 

 