text:
First Degree Entailments
- Angel Lucas
- 3 years ago
- Views:
1 ANDERSOn, A. R., and N. D. B~L~AP, JR. Math. Annalen 149, (1963) First Degree Entailments By A. l~. ANDERSON and N. D. BELNAe, JR. in New Haven, Conn./USA Contents Page Preface I. The system LEQ II. Semantics III. Consistency IV. Completeness V. Undeeidabitity VI. Loewenheim-Skolem Theorem VII. LEQt and EQ VIII. First degree formulas Bibliography Preface This paper generalizes some results of BELNAP 1959b (presented more elaborately in A~DERSO~ and BEL~CAP 1962), to a non-classical system of quantification theory, based on an intensional relation of entailment. The fundamental idea behind the present quantificational treatment is most easily made clear by reference to the propositional calculus E (for which see ANDERSOI 1959, together with works cited above). Let A-+ B be an entailment in which only truth-functions occur in A and B. In the system E we can rewrite A in disjunctive normal form, and B in conjunctive normal form, obtaining an equivalent formula (in the sense of the double arrow*-~): Al v... v A,~-+ BI &... & B ~. In the system E, as classically, the validity of the foregoing formula reduces to the validity of A~-~ B~ for each i and j. Now classically there are three conditions under which we regard At -~ Be as valid. (1) Contradiction on the left, (2) excluded middle on the right, (3) sharing of an atom (i.e., a variable or negated variable). Xn the system E, however, only condition (3) assures validity: a necessary and sufficient condition for provability of A -* B in E is that each A~-+ Bj share an atom. The arrow of E thus secttres revelance of antecedent to consequent. The proof techniques for the generalization are essentially due to KA~GER 1957 (building on G~.NTZEI~ 1934). Such interest as the paper may have lies in the application of strong and interesting classical techniques to a nonclassical system of logic.
2 First Degree Entailments 303 I. The System LEQ1 A syntactical completeness theorem for the system EQ of entailment with quantification may be found in ANDERSON 1959, and a semantical completeness theorem for a certain fragment of E has been proved by BEL~AP 1959b. But the general problem of providing E Q with an adequate semantical theory has posed difficulties. As a partial solution to the problem, we here under~ke a discussion of first-degree entailments (i.e., entailments in which no arrow occurs within the scope of an arrow). Notation/or the system LE Q1. We will assume that propositional variables are specified, and that well-formed formulas ("wffs") are as usual in treatments of first order functional calculi (with negation, disjunction, and existential quantification as primitive), x, y, z, range over individual variables; F, (7, H, range over function variables; and A, B, C, D, range over wffs (all these with or without subscripts). A is the negation of A, A v B is the disjunction of A and B, and 3 x is an existential quantifier. A x is a formula in which x may occur free, and A y is the result of replacing every free occurrence of x in A x by y, in such a way that y is free in A y where ever x is free in A x. The system LEQ1 of first degree entailments consists of sequents (in the manner of GE~TZE~ 1934) of the form J-~ K (where J, K, L, M, N, and P, range over sequences of wits). Axioms have the form J,A,K~L,A,M, where A is an atom (i.e., A is either a propositional variable, or the denial of such, or else has the form F(x 1... %) or F(xl,..., x~)); and we have the following ten rules of inference. (R1) (R2) J,A,K---> L J, ~, K -+ L J --> K, A, L J-~ K,~,L J,A,K-~L J,B,K--->L (R3) J,A v B, K---> L J --~ K, A, B, L (R4) J-~ K,A v B,L J,A,B,K--> L (R5) J,A V B, K--> L J--~ K,A,L J--> K,~,L (R 6) --J -~ K, A~"B, L (R7) (R8) J, Ay, K--> L J, 3xAx, K---> L J -~ K, Ay, L, 3xAx J.--> K, 3x Ax, L (R9) J,~,K, 3x Ax---~ L J, 3xAx, K --~ L J-~ K, Ay, L (R10) J.-~ K,3x~,L ~fath. Ann
3 304 A.R. A~D~Bso~ and N. D. BELNAP, JR. : In (R 7) and (R 10) it is required that y not occur free in the conclusion; and we also require a rule for alphabetic change of bound variables. II. Semantics Our problem in this section is to provide appropriate semantics for LEQ1, relatively to which LEQ1 may be proved complete. To this end we begin by postulating the existence of propositions p, q, etc., and of classes X, Y, and Z, of such propositions. (We conceive of propositions as intensional entities.) Associated with each proposition 27 we postulate the existence of a unique proposition q which is the denial (negation) of _p, where q is such that the truth of ~ entails the denial of q, and the falsehood of _p entails the truth of q. We write p' for the denial of_p, and where _X is a class of propositions, X' is the class containing all and only the denials of members of X. (Warning: X' is not the complement of X.) Acting under the philosophical conviction that we have no semantic grounds for distinguishing between "positive" and "negative" propositions, we instead simply treat "is the negation of" as a symmetrical relation, and we reflect this view in the conditions that (i) if _p is the denial of q, then _q is the denial of p, and (ii) if X = _Y', then _X' = _Y(where "=" is class identity). Then from Y' = Y' it follows that _Y" = Y. We now introduce a relation between classes of propositions, "X cons _Y", to be read "the proposition that all the members of X are true has as a logical consequence the proposition that at least one of the members of Y is true." We postulate that "cons" satisfies conditions (1)--(5) for every X, _Y, and _Z: (1) If X and _Y share a member, then X cons _Y. (2) If X cons _Y, and ~} cons _Z for every _p in _Y, then _X cons _Z. (3) If X cons {p} for every ~ in Y, and _Y cons _Z, then X cons Z1). (4) If X cons _Y, then Y' cons X'. We also postulate that for each class Z of propositions, there is a proposition p satisfying the following condition: for every _X and _Y, (5') _X u {p} cons _Y iff for every ~ in Z, X ~ {_q} cons _Y. ("iff" abbreviates "if and only if.") We first show that, given _Z, the proposition _p of (5') is unique. For suppose that for every X and _Y, (5") ~ ~ {r} cons _Y iff for every q in _Z, X ~ {q} cons Y. Then we have immediately that, for every X and _Y, (6) X ~ {r} cons _Y iff _X ~ {2} cons Y. From which (taking X as void and _Y as (r} ~ _W, and then as {~} u _W), we get by (1) {~} cons {r} v W, and {r} cons {~} ~ W. Moreover, by (1) and the latter we have that for every ~ in {~} u W, {~} cons {r} ~ [~', and every in u w, cons w, 1) This property of "cons" is not independent; it follows from (2) and (4).
4 ' hence by (2)--(3), we have for every _X and _W, First Degree Entailments 305 (7) _z cons {_r} ~ W ~_x cons {~} w W. _p is then unique (in the sense that (6) and (7) both hold), and we introduce the notation _D-Z (read "disjointly, -Z") for p; so that (5') may now be rewritten as (5) X ~ (D-Z} cons _Y iff for every q in _Z, X ~j {_q} cons _Y. In consequence of (1)--(5), we have properties (8) --(10) below for D (which, incidentally, we may think of as a function taking a class of propositions X into a proposition _D_X)2). (S) _X cons Z w-z iff _X cons -y w {_D-Z}. For by (1), we have {q} cons Y ~J-Z for every q in Z; hence by (5) (with void), {pz} cons Z w-z. But this fact and (1) yield that {q} cons _Y ~JZ, for every q in Y ~ {_D-Z}. Hence by (2), if _X cons _Y ~/{D-Z} then _X cons _Y u-z. For the converse, suppose that X cons _Y w _Z. Now for every _q in _Y ~j-z, {_q} cons Y ~ {_D-Z}, since if q is in i Y, {q} cons _Y ~J (DZ} by (1), and if q is in~, {_q} cons Y ~J {_D_Z} by (5) and (1). Then by (2), X cons Y ~J {D-Z}, as required. (9) X Lt Y' cons -Z iff X w {D _Y}' cons -Z. For every q in X ~J -y', by (1) {q}'cons_x'~j Y, and hence by (8), {q}' cons X' ~ {_D -y}; so by (4) X ~ {_D -y}' cons {q}, for every q in _X ~J -y'. Hence by (3), if _X ~ -y' cons-z, then X ~J {_D Y}' cons-z. The converse is similar. And we state finally without proof, (10) X cons _Y '~ {_D-Z}' iff for every q in -Z, X cons -y w {q}'. We now introduce the notion of a/rame, i.e., a triple (P,!,_F) (where P is a non-empty class of propositions, / is a non-empty class of individuals, and F is the set of all functions taking n-tuples of individuals into members of P), such that P satisfies the following two conditions for every non-empty subset X of P: (11) if ~ ~q P, then X' _c P, and (12) if _X g P, then _D_X E P- An assignment of values for a wff A of LEQ I from (P,/, _-LF) consists of a function which (i) gives to each free individual variable of A an element of _/, (ii) gives to each function variable of A an element of F, and (iii) gives to each propositional variable of A an element of _P. (We let range over _P, i range over I, and _/range over one-argument functions in F; -/(i) denotes the value ill P assumed by/inf for the argument i in -/.) Then each wf part of A assumes a propositional value in P as follows: A propositional variable B assumes as value the proposition ~ in _P for the value ~ in P of B. ~) Dually, we could introduce a function CZ (read "conjointly, Z") satisfying the condition: X cons Y u {CZ} iff for every q in Z, X cons Y U {q}; and then continue the development in a similar (dual) way. 21"
5 306 A.R. AND]~RSOI~ and N. D. BELNAP, JR. : F(x~... %) assumes as value the proposition _p in P into which the function assigned to/~ takes the individuals assigned to x~..., %. If, for a system of values of the free variables of B, B assumes as value the proposition _p, then B assumes as value the denial of _p. If moreover the value assumed by C is q, then B v C assumes as value ill P the value _D {_p, q}. And if, for a system of values for the free variables of 3 x B, B assumes the value f in F, then ~ xb assmnes as value the proposition _D_X, where X is the range of values assumed by f (i.e., X has/(/) as a member for every i in!)- We see inductively that if values from (P, I, F~ are assigned to all free variables in J --> K, then for this assignment the wffs in J and K all assume propositions in P as values. For a given assignment of values, we let Xj [_Y~] be the class of propositions assumed by J [K], as values. We then call J -> K true in (P, I, F), for an assignment of values from (P,!, F), if _X~ cons _Y~. And J -~ K is valid in (P, I, F) if J -~ K is true in (P,!, F) for every assignment from (P,/, F). Finally, J-> K is valid, if it is valid in every frame. III. Consistency Theorem. LEQ 1 is consistent (i.e., if J->K is provable in LEQ1, then J -> K is valid). Proo t. The axioms of LEQ 1 are all valid in every frame (by (1)), and the rules all preserve this property. We consider four cases as examples. Case (R1). If J -+ K is a consequence of L -+ K by R1, then in every frame Xj = _XL; hence if _X L cons Y~:, then Xj cons -YK. Case (R6). Suppose that J-~ K, A, L and J-~ K, B, L are both valid. Then Xj cons Y~ ~J {_p}' Z L and Xj cons -YK ~ {q}' u_z~, where A [B] has the value _p[q]. Then for every f in (p, q}', _Xj cons -YK '~ (r} '~_ZL, so (10) applies, and we have Xjcons-YK ~J (D(~, q})' ~J~z. Since nothing in the argument depends on the choice of frame or assignment, we conclude that J -~ K, A v B, L is valid. Case (R7). Suppose that J, Ax, K -+ L is valid. Choose a frame, and an assignment of values to all variables except x. Since x does not occur in J, K, or L, this assignment assigns propositional values to every member of J, K, aald L, and moreover gives a value / to A x. Since J, A x, K -~ L is valid, J, A x, K -~ L is true for every assignment of values to x; i.e., Xj ~] ( } ~J -YK cons _Z z for every ~ in the range W of f. But then by (5), Xj ~/(DW} ~/_YK cons_zl; i.e., J, 3 x.4 x, K -+ L is valid. Case (R9). Suppose that J, Ay, K, 3x`4x--> L is valid. Then Xj ~ (_?}'~' -YK ~/{_D _W}' cons ZL, where.4 y assumes the value ~, and where _W is the range of values assumed by ] for every / in I, _] being the value of A x. But ~J ~ {P}' ~ -YK ~/{D _W}' cons ~L iff (by (9)) Xj ~/(~}' ~/~K ~ _W' cons ~, iff (since ~ E _W) _Xj ~ _W' ~/Y/~ cons Zz, iff (by (9) again) Xj~/(_DI V}' u -YK cons Z L. So J, 3 x A x, K -~ L is valid as required. Other eases are similar.
6 First Degree Entailments 307 IV. Completeness Theorem. LEQ 1 is complete (i.e., if J -~ K is valid, then J -~ K is provable in LEQI). Proo]. After introducing some terminology, we begin by describing a uniform proof procedure for LEQ 1. If A, in a sequent J, A, K -~ L or J -~ K, A, L, is a propositional variable or a negated propositional variable, or has the form F (x I... x~) or F (x 1... %), then we call A an atom of the sequent; otherwise A is non-atomic. Without loss of generality (in view of the rule for alphabetic change of bound variables), we restrict attention to sequents in which no individual variable occurs both bound and free, and if J -~ K is a candidate for the proof procedure, then V(= (vl, v ) is the set of all variables (in alphabetic order) which do not occur bound in J -~ K. Now given a candidate sequent J -~ K of LEQ1, we attempt to construct a dendriform proof. The tree consists of branches, and a lull normal branch a) is defined as a sequence S o... Si... (possibly finite, possibly infinite) of sequents, satisfying the following conditions: S O is J -+ K. If S~ is an axiom, or if S~ contains no non-atomic formulas, then the branch terminates at S_/; otherwise we obtain Si+ 1 by an L-rule or an R-rule below (beginning with an L-rule, if possible, and alternating between applications of L-rules and R-rules as much as possible) : L-rules: If S/ has the form J, A, K-~ L, where A is the leftmost nonatomic formula in the antecedent, then (a) if A is B, then S_/+1 is J, B, K-~ L; (b) if A is BvC, then S~+ 1 may be either J, B,K~L or J,C,K~L; (e) if A is B v C, then Si +1 is J, B, C, K -~ L; (d) if A is 3 x Bx, then Si +1 is J, Bv~, K -~ L, where v~ is the first individual variable in V which occurs in no $i for ~ g/; and Bv$ is the result of writing v~ for free x throughout Bx; and (e) if A is 3 x Bx, then S i +1 is J, Bye, K, 3 x Bx ~ L, where v~ is the first individual variable in V such that Bv~ occurs in the antecedent of no S~ for _~./(where again Bv~ is the result of writing v~ for free x throughout Bx). R-rules. If Si has the form J-~ K, A, L, where A is the teftmost nonatomic formula in the consequent, then (a) if A is B, then Si+ 1 is J-> K, B, L; (b) if A is B v C, then Si+ I is J -~ K, B, C, L; (c) if A is B v C, then St+ 1 is J-~ K, ~, L, or J~K, C, L; (d) if A is 3x Bx, then Si+ 1 is J-~K, Bye, L, 3x Bx, where v~ is the first individual variable in V such that B v~ occurs in the consequent of no S~ for ~ _~ _/. (e) if A is ]xbx, then Si+ 1 is J-~K, Bv~,L, where [as in L-rule (d)]. 8) The terminology is that of KA~OER 1957, from whom the ideas of Lemm~s I and II below were adapted.
7 308 A.R. A~DEJ~so~ and N. D. BELNAP, JR. : We introduce the following notation. Where is a full normal branch for J-~ K, we use L( ) to refer to the class of formulas occurring in the anteeedents of sequents in, and R( ) for the class of formulas occurring in consequents. Lemma I. If is a full normal branch for J-~ K not terminating in an axiom, then there is no atom A which is in both L( ) and R( ). Proo]. The rules for constructing full normal branches guarantee that no atoms are lost in passing up the branch. Hence if L( ) and R( ) shared an atom, the branch would terminate in an axiom. Lemma II. If is a full normal branch not terminating in an axiom, and if (1).~, (2) A v B, (3) A v B, (4) 3xAx, (5) 3xAx occurs in L( ), then so does (1) A, (2) A [or B], (3) A and B, (4) Av~ for some v~, (5) Av~ for every v,; and if (1) A, (2) A vb, (3) A vb, (4) 3xAx, (5) 3xAx occurs in R( ), then so does (1) A, (2) A and B, (3) -4 [or B], (4),4 v~ for every v~, (5) A v~ for somev~. Proo/. By rules for branch construction. We come now to the principal Lemma (required for the proof of the completeness of LEQ1), which shows that validity depends in an essential way on the atoms in a full normal branch. Lemma III. Let be a full normal branch for J -~ K not terminating in an axiom, and consider an assignment of values to the variables in from a frame (P,!, F). Then if there exist subsets M of L( ) and N of R( ) such that X M cons _Y~r under this assignment, then there exist subsets M 0 of the atoms of L( ) and N o of the atoms of R( ) such that XM0 cons _rhr0, under the same assignment. Proo]. We define the grade o/ a ]ormula or class of formulas as follows. If a wff A is an atom, then it has grade zero; if A has grade n, then 3 xa has grade n ~- 1 ; if A has grade _n, then A has grade n -~ 1 (unless A is an atom) ; and if A has grade.n and B has grade m, then A v B has grade max (m, _n) ~- 1. And the ~Trade o/a subset of L( ) or R( ) is the maximum grade among the wffs of that subset. The proof is by induction, first on the grade n of N, and then on the grade _m of M. Assume that for M 0 of grade zero, and for every IV* of grade less than n, the Lemma holds. We show that it holds also where N is of grade n. The Lemma is trivial for IV of grade zero, and we consider IV of grade n > 0. Given IV, we define IV* as follows. For every A in N: (0) if A has grade less than n, A is in 2/*. Otherwise A has grade n, and (i) A has the form B. Then B is in IV*. (ii) A has the form B v C. Then both B and C are in IV*. (iii) A has the form B v C. Then if ~[~] is in R( ) (by Lemma II at least one wi~ be), then B [~] is in IV*. (iv) A has the form 3x Bx. Then By i is in IV* for every v i. (v) A has the form ~]x Bx. Then if Bvi is in R( ) (by Lemma II, By i is in R( ) for some vt), then By i is in IV*.
8 First Degree Entailments 309 We now establish that for every A in 37, _X{A} cons Y~,. For the following cases, we suppose that the value assumed by B [C] is _p[_q], and that for the assignment to all the free variables of Bx except x, Bx assumes the value ] (in_f). Moreover, for each vi in V, v~ assumes the value _/in!. Then we consider cases as follows. (0) A is in N*. Then _X{A) cons _Y~, by (1). For the remaining cases, A is in not N*. (i) A has the form B. Then _X{A} is p", i.e., _p. But B also has the value ~. and B is in AT* by the definition of 37* ; hence _X{A} cons -Yz, by (1). (if) A has the form B v C. Then _X{A} is D{p, _q}, and by the definition of AT*, both B and C are in 37*. Hence both _p and q are in Y~,, so by (1), _p cons -YN, and _q cons Y~,. But then by (5), _D{_p, q} cons _Y~,; i.e., _X(A) cons -Y~v,. (iii) A has the form B v C. Then X{A } is {_D{_p,_q}}', and by the definition of 37", either B or C is in 37*. Hence either p' or _q' is in -YN,. Then by (1), {_p',_q'} cons _Y~,; and by (9), {D{_p,_q}}' cons -YN,; i.e., _X{A} cons _Y~,. (iv) A has the form =ix Bx. Then X{A } is D_Z, where _Z is the range of values of the value _f of Bx. By the definition of 37", Bvi is in 37* for every vi. By (1), {/(_/)} cons ~iv, for every /(i) in Z; hence by (5), _D_Z cons _Y~,; i.e., _X{a} cons _Y~,. (v) A has the form 3 x Bx. Then _X{A) is {D_Z}', where _Z is as in (iv). By the definition of 37", Bvi is in 37* for some vi. By (1), Z' cons -YN,. Hence by (9), {_D_Z}' cons _Y~,; i.e., _X{A) cons -Ylv,. It follows then by (2) that if _XM cons -YN, then _XM cons _Y~,, where, by Lemma II, 37* is a subclass of R( ), and is of grade less than n. But by the inductive hypothesis, if _XM cons -Y~v,, then _XM cons -YN. for some subset 370 of the atoms of R( ); hence if _XM cons Y~, then _XM cons -Ylv. We wish now to establish that if X M cons _Y~-, then XM0 cons -Ytv,, for some subset M 0 of the atoms of L( ). Assume then that for every 37, and for every M* of grade less than m, the Lemma holds. We show that it holds also where M is of grade m. The previous induction secured the Lemma for M of grade zero, and we now consider M of grade m > 0. We define M* as follows. For every A in M: (0) if A has grade less than m, then A is in M*. Otherwise A has grade m, and (i) A has the form B. Then B is in M*. (if) A has the form B v C. Then if B[C] is in L( ) (by Lemma II at least one will be), then B [C] is in M*. (iii) A has the form B v C. Then both B and ~ are in M*. (iv) A has the form 3x Bx. Then if By i is in L( ) (by Lemma II it will be in L( ) for some vt), then Bv t is in M*. (v) A has the form 3x Bx. Then By i is in M* for every v i. Then by methods exactly dual to the foregoing, we establish that for every A in M, _XM, cons -Y{A). Then with the help of (3) and the two inductions, the Lemma is secured.
9 310 A.R. ANDERSON and N. D. BELNAP, JR, : Returning now to the completeness of LEQ1, we first note that, if J-~ K is unprovable, then some branch of the full nornml tree for J-~ K fails to terminate in an axiom. Then we want a frame (P, I, F) from which we can assign values to variables in J -~ K, in such a way that _Xj o~ -YK (i.e., it is not the case that Xj cons -~K). Where is a full normal branch for J-~ K not terminating in an axiom, it suffices to find an assignment such that for every subset M 0 of the atoms of L ( ) and N o of the atoms of R ( ),_XM. c~ _Y. For by Lemma III, it will follow that for all subsets M of L( ) and N of R( ), _X M ~ _Y~; and in particular _Xj ~ -Yr. We now introduce the notion of an atomic/tame, defined as follows. If _Y is a class of propositions, and P is the closure of -Y under ' and _D, we will say that P is generated by -Y. We postulate the existence of atomic classes -Y of propositions, satisfying the condition that for all subsets _X and _W of -Y w ', _X cons I_ if and only i/_x r~ W_ ~ O. ~) If P is generated by an atomic class of propositions, we say that (P,!, _F) is an atomic/tame. Now let be a full normal branch, which does not terminate in an axiom, and assign values to the constants and variables in as follows. We choose an atomic frame (-P1,!1, _Fs), where -P1 is generated by an atomic class {_p,_q, _r} of propositions; -/1 is the natural numbers (and F 1 is a set of functions taking n-tuples of natural numbers into -P1). Then to each individual variable v i we give as value the natural number i. If A is a propositional variable, then: (i) If neither A nor.4 occurs in R( ), give A the value p~_ (ii) If A is not given a value by (i), and if neither A nor A occurs in L( ), give A the value q. (iii) If A occurs in L( ) and A occurs in R( ), give A the value _r. (iv) If A occurs in L( ) and A occurs in R( ), give A the value r'. And to each function letter F we assign as value that function / taking the n-tuple -il... i. into (i) ~, if neither E(vtl..., v~) nor its negate occurs in R( ); (ii) _q, if F(vil..., vi~ ) has not been given a value by (i), and if neither F(v h..., vi~ ) nor its negate occurs in L( ); ( iii) r_, if F (v h..., v i~ ) occurs in L ( ) and F (v ll..., via ) occurs in R ( ) ; and (iv) r', if ~V(vil... via ) occurs in L( ) and F(v h... vt, ) occurs in R( ). (The possibility of this assignment is guaranteed by Lemma I.) 4) The atoms of an atomic frame are thus logically independent; as an example we mention the frame generated by (the proposition that MICHELANGELO designed the Sistine Chapel, the proposition that SCOTT was the author of Waverly}. No one really believes that there is any logical connection between these two propositions, though those who believe that material "implication" is an implication relation must maintain that "If I~CHBLA~OELO designed the Sistine Chapel, then SCOTT was the author of Waverly," is true. We think it is false, and self-evidently so. Certainly there i8 a sense in which the two propositions are logically independent of one another--and that is the sense invoked here. For further consideration of the underlying philosophical issues, see A~DE~SO~ and B~.L~AP 1960b.
10 First Degree Entailments 311 Then for this assignment all the atoms in L( ) take _p, p', or r as values' and all the atoms of R ( ) take the values q, _q', or r'. Hence if Z is any subset of _XM, and W is any subset of _Y~, then _Z c~ _W by the atomicity of (-P1, 11, -F1) ; hence by Lemma III, _Xj c~ -YK. And the completeness of LEQ 1 is then secured. If every branch terminates in an axiom, then J-> K is provable, the tree constituting a proof. And if some branch fails to terminate in an axiom, then J -~ K is invalid in the atomic frame generated by {_p,_q, r}. V. Undeeidability Theorem. LEQ1 is undecidable. Any decision procedure for LEQ1 could be made to yield a decision procedure for the classical first order functional calculus, as follows. BnLNXe 1959a provides a proof technique for formulas C of the first-order functional calculus, in which all rules are entailments, and all axioms are of the form HA, where//is a sequence (possibly void) of universal quantiflers, and A is a (perhaps multiple) disjunction containing atoms B and B as disjuncts. Moreover, the positive atoms B are all subformulas of the formula C; hence if C is a valid formula of the first order predicate calculus, then the positive atoms B occurring in axioms leading to C are positive atomic subformulas of' C. Now H(B v B)-~ H' (D 1 v B v D~ v B v D3) is a valid entailment in L E Q1, where the Di are arbitrary, and H' contains all the quantifiers in//(and perhaps some others). It follows that the conjunction ]II(B 1 v B1) &''" &H~(B~ v B~) containing Hi(B~ v/~) (where //_/ contains a universal quantifier for each variable x in Bi) for each atom B~ in C, entails a conjunction of all the axioms leading to C in BEL~AP'S treatment. And since the arrow of J --> K in LEQ~ is transitive when J and K each contain a single wff, it follows that the conjunction above entails the candidate C, if C is valid. Hence C is provable in the first order functional calculus iff v v c is provable in LEQ1, and the decision problems are equivalent. Hence LEQ1 is undecidable. VI. L~wenheim-Skolem Theorem In the preceding section we defined true and valid for sequents J-~ K independently of the truth or falsity of the formulas in J and the formulas in K, a fact which reflects the intensional character of first degree entailments: the truth or falsity of such an entailment depends on a connection of meaning between antecedent and consequent, and is not simply a truth-function of them. But of course it is also possible to define truth for wffs, and, with a view to providing an appropriate version of the LSwenheim-Skolem theorem for LEQ1, we now define true and valid for wffs (as opposed to sequents) of the system.
11 312 A.R. A~DEr~So~ and N. D. B~L~P, JR. : Given a frame (P, L F), we assume that for each _p in P, _p is true if and only if it is not false; hence ~ is true iff ~' is false. Moreover, _DX is true iff some member of _X is true. We say that A is true in (_P, I_, F_), for an assignment of values to the variables in A from (_P,_/,_F), if and only if the proposition assumed by A as value (under the valuation of section II) is true. A is valid in (_P,!, F) if A is true in (_P,!, F) for every assignment from (P,!, F); and A is valid if A is valid in every frame. We see at once that this definition coincides with the classical one for special frames (P~,!, F~), where / may be arbitrary, but -P2 is generated by a class of propositions containing just/_){~, _p'} (for some ~). Since {D{_p, ~'}}' cons{_d{~, _p'}}, it is easy to see inductively that every proposition in the resulting P, is equivalent (in the sense of "cons") either with _D{p, ~'} or with (_D{_p, ~'})'; hence -P2 contains exactly two propositions, one of which is true, and. the other false. (We note that for an assignment of values from (-P2, -/, -F2), J-+ K is true iff the conjunction of J materially "implies" the disjunction of K; that is, Xj cons -YK if~ D{~Xj, \J ~YK).) Lemma. If A is true [false] for a given assignment Q of values from an arbitrary (_P,/, F), then it is true [false] in (-P2,!,_F2) for the following assignment Q*: (i) if Q assigns a true proposition to a variable B, then Q* assigns _D{_p,_p'} to B, and otherwise Q* assigns (D{~, p'})' to B; (ii) if Q assigns the individual i to the variable x, then Q* does likewise ; and (iii) if Q assigns the function / to the n-ary function variable F, then Q* assigns the function/* taking -/1...,/~ into _D{~, _p'} if/(-/1... _/,) is true, and into (_D{~, ~'})' otherwise. Proof. Obvious from truth.conditions for atoms, and for formulas of the form B, B v C, and 3 x B. The relation between validity in general, and validity in (P,, I, F~ is then stated in the following Theorem. A is valid if and only if A is valid in every frame (_P~,/, F~). Proo]. It is trivial that if A is valid, it is valid in (_P~,/, F~), and the converse follows immediately from the Lemma. Our object now is to show that if a set of sequents is simultaneously satisfiable, it is so in a frame with denumerably many individuals, and for this we need definitions of satisfiability for J -~ K. This in turn requires us to place an additional condition on the "cons" relation, to wit: (13) If all the propositions in ~ are true, and all the propositions in _Y are false, then ~: ~ _ys). Then we say that J ~ K is satisfiable if there exists an assignment from a frame (_P,!, ~), such that for this assignment ~j cons ~K; and J -+ K is ltnsatisfiable if for every assignment from every frame, Xj c~ ~K. 5) For present purposes, (13) could be weakened to (13'): if each member p of X is such that {p}' cons{10 } and each member q of Y is such tha~ {q} cons{q}', then X cons Y.
12 First Degree Entailments 313 L6wenheim-Skolem theorem/or LEQ1. Let S be a set of sequents, simultaneously satisfiable in (P,!,_F). Then the members of S are simultaneously satisfiable in (_P,!1, if), where Is is denumerable. Proo[. For a satisfying assignment in (P,!, F), each sequent J -~ K in S must have either a false A in J or a true B in K, by (13). Consider the set T containing the negation _4 of every A such that A is a false member of some antecedent of a sequent in S, and containing also every B such that B is a true member of the consequent of some member of S. All members of T are true under the assignment. Then choose a proposition _p in P, and consider the sub-frame (_P*,!, _F*) of (/tp,-/, F), generated by D _ {p, _p' }. By the Lemma, the members of T are simultaneously satisfiable in (P~,!, _Fe*)- But then (choosing _D{_p, ~'} as classical/alsehood), the sequents of T are simultaneously satisfiable in the classical sense, and the classical Lbwenheim-Skolem theorem applies. Hence the members of T are simultaneously satisfiable in a frame (-P*, -/1, F*) with denumerable!1. But if X and _Y are classes of propositions from P*, and either _X contains (_D{f, _p'})' or _Y contains _D{_p,_p'}, it is easy to see that X cons _IT. Hence in (P~, -/1, F~), for every J -~ K in S, _Xj cons -YK; and the sequents of S are simultaneously satisfied in (-P2*,-/1, F*) -- hence also in (_P, -/1,_F). By similar arguments it can also be shown that: Theorem. J ~ K is unsatisfiable iff J is valid and K is unsatisfiable. (We remark that then J-~ K is unsatisfiable if and only if the material "implication" from the members of J taken conjointly, to the members of K taken disjoint]y, is also unsatisfiable.) VII. LE01 and EQ The system EQ (ANDERSON 1959) has the following axioms (with wffs as usual, and with dots replacing parentheses in accordance with the conventions of C~VRCH 1956): (1) A-~A~B~B (2) A -+ B-~. B-.'. C-~. A ~ C (3) (A -~. A -~ B) -~. A -~ B (4) A & B~ A (5) A & B-~ B (6) (A -> B) & (A -> C) -~. A -~ (B & C) (7) NA&NB-~N(A&B) [NA=~fA~A~A.] (8) A-~ A v B (9) B ~ A v B (10) (A -~ O) & (B -~ C) -~. (A v B) -~ G (11) A & (B v C) -~ (A & B) v C (12) A -~ A-+ (13) A ~ B-~. B-~A
13 314, A. R. A~D~I~SON and N. D. BELI~AP, JR. : (I4) (15) (16) (17) (18) (19) (20) A-~ A (x) Ax ->- A y (x) (B ~ A) 4. B -+ (x)a (x) (A v B) -+. (x)a v B (x) (A ~ C) -~. (x)a -~ (x)c (x) A & (x) C -+ (x) (A & C) (x) NA ~ N(x)A And finally, if A is an axiom, so is (x)a. (In (15)--(20), A may contain x free, Ay is like Ax except that all free occurrences of x in Ax are replaced by free occurrences of y in A y, B does not contain x free, and C is any wff.) The rules are modus l~mens (for the arrow) and ad]unction. The formulation above takes entailment, conjunction, disjunction, negation, anct universal quantification as primitive; but the axioms for conjunction are redundant, since conjunction may be defined in terms of disjunction and negation in the usual way, and shown to have the right properties. And existential quantification may also be defined in EQ. Similarly in LEQ1, which takes entailment, negation, disjunction, and existential quantification as primitive, we may define conjunction and universal quantification. We shall assume that all these definitions have been made, and that J--> K in LEQ1 is to be interpreted in EQ as A -+ B, where A is a conjunction of all the wits in J, and B is a disjunction of all the wffs in K. (Since C & D [C v D, (C & D) & E, (C v D) v E] is intersubstitutablc with D & C [D v C, C & (D & E), C v (D v E)] in both systems, it makes no difference how parentheses are associated in A and B.) Theorem. Where A is a conjunction of all wffs in J, and B is a disjunction of all wffs in K, A ~ B is provable in EQ iff J -+ K is provable in LEQ r Proo]. It is easy to show that all the axioms and rules of LEQ~ are, under the correspondence between wffs of EQ and sequents of LEQ 1, theorems and rules of E Q; hence E Q contains LE Q1- For the proof of the converse we require the matrices on p. 315 (BEL~AP 1959), in which + values are designated. The proof then proceeds as follows. We first show that, under an evaluation procedure to be explained, all theorems of EQ take designated values for all assignments from the matrices. Then we show that if a sequent J ~ K is unprovable in LEQ1, then it can be made to assume an undesignated value from the matrix. It will follow then that if A -+ B is provable in EQ then it is also provable in LEQ1; and the equivalence of the two systems (in the required sense) will be established. Consider then a quasi-frame <_P,/, _F> where _P contains the eight "propositions" f the matrices on p. 315, I is the set of non-negative integers, and _F is the set of functions taking n-tuples of members of / into P. Given a wff A, we may then assign values from P to the propositional variables in A, values from! to the individual variables in A, and values from _F to the function variables in A, and then compute the value in _P of A as follows.
14 First Degree Entailments A-+B A A -1 &B AvB t ~ } ' If A is a propositional variable, A assumes the value given to A under the assignment; and if A is F (xl,..., x~), then A assumes as value the value into which the function assigned to F takes the individuals assigned to x 1... %. If A has the form B, B & C, By C, or B~* C, then the value of A is determined from the values of B and C by the matrices above. If x is an individual variable, and A a wff in which x occurs free, then for a system of values for the free variables of (x) A, (x) A assumes as value the value of the conjunction of those values assumed by A for all values of x. (And if x does not occur free in A, the value of (x)a is the same as the value of A.) (We remark that in calculating values of formulas in EQ, it follows from the definitions of disjunction and universal quantification, and from the condition above, that (~x)a assumes as value the value of the disjunction of the values assumed by A for the various values of x.) We leave to the reader the tedious but straightforward task of verifying that, under this evaluation procedure, all the axioms of EQ take designated values for every assignment to their variables, and that the rules of EQ preserve this property. It follows that all theorems of E Q satisfy the matrices. Now let J -~ K be an unprovable sequent in LEQ I. Then some full normal branch for J-~ K fails to terminate in an axiom. To the variables of this branch we assign values from the quasi-flame <P,/, F> as follows (the possibility of this assignment being guaranteed by Lemma I of section IV): If A is a propositional variable, then
15 316 A.R. ANDERSON and N. D. BELNAP, JR. : (i) if A occurs in L( ) but not in R( ), give A the value +1; (ii) if A occurs in R( ) but not in L( ), give,4 the value -~2; (iii) if A occurs in L( ) and A occurs in R( ), give A the value 3; and (iv) if A occurs in L( ) and A occurs in R( ), give A the value - 3. To each individual variable v~ give the value _/from I. To each function letter F we assign as value that function f taking the n-tuple -/1,., -/~ into (i) 1, iff(x 1..., %) occurs in L( ) but not in R( ); (ii) ~-2, iff(x 1..., %) occurs in R( ) but not in L( ); (iii) ~-3, if F(xl,..., %) occurs in L( ) and F(x I... %) occurs in R( ); and (iv) -3, if F(x I... x~) occurs in L( ) and F(x 1... %) occurs in R( ). Now we claim that for this assignment, J-~ K takes an undesignated value. For (a) every member of L( ) takes a value 3, 1, - 1, or - 0; and (b) every member of R( ) takes the value 2, 0, -2, -3, J-~ K therefore taking an undesignated value. We prove (b), leaving (a) to the reader. As a basis for the induction, all atoms in R ( ) assume values from 2, ~- 0, -2, -3. Now suppose all members of R( ) of length less than the length of.4 take one of those values. Then reference to the tables above shows that if A has the form B [B v C; B v C], then A has one of the values 2, 0, - 2, or -3. Hence by Lemma II and the inductive hypothesis, A has one of these values. And if A is :4x Bx, then Bv~ occurs in R( ) for every k (Lemma II); hence Bv~ takes only values ~-2, Jr 0, -2, -3; hence by the table, 3 x Bx is confined to the same values. Lastly, if A is 3x Bx, then for some _k, Bv~ assumes 2, ~- 0, - 2, - 3. Then Bv~ has a value - 2, - 0, ~- 2, 3, whence 3x Bx has one of the latter values. Then :Ix Bx has 2, ~-0, -2, -3, as required. Hence if J-> K is unprovable in LEQ1, it assumes an undesignated value, and is therefore unprovable in EQ; and the equivalence (in the appropriate sense) of the systems is established. VIIL First Degree Formulas A first det'ee ]ormula of EQ is any wff in which no arrow occurs within the scope of an arrow. More generally, we define the degree of a formula of EQ as follows: any atom is of degree zero, and if A and B are of degree m and n respectively, then A, 3xA, and (x)a are of degree ~, A v B and A & B are of degree max{~, n) and A-> B is of degree max{m, n)~-1. In view of the difficulty of providing appropriate semantics for EQ, it seems plausible that start toward generalizing the results of the preceding sections might be made by considering first degree formulas. But even for this fragment of E Q difficulties arise almost at once, and our purpose in this section is to state some partial results, and raise some questions. We define validity for first degree formulas as follows.
16 First Degree Entailments 317 Given a frame (_P,!, F), and an assignment from (P, L _F~ to the free variables in a first degree formula A, we will say that A is true (in (_P,/, F), for the given assignment) under the following conditions: If A is a propositional variable, then A is true iffthe value p assigned from _P to A is true. If A has the form F(x 1... x~), then A is true iff the proposition into which the function / assigned to F takes the individuals -/1,.,./~ assigned to the variables x 1... x~, is true. If A has the form B, then A is true iff B is not. If A has the form B & C [B v C], then A is true iff both B and D are [either B or C is]. If A has the form (x) Bx [3 x Bx], where x is an individual variable occurring free in Bx, then for the assignment to the free variables of (x)bx [3x Bx], (x)bx [~x Bx] is true ifffor all values of x [for some value of x], Bx is true. (And if x does not occur free in Bx, then (x) Bx [3x Bx] is true if[ Bx is true.) Finally, if A has the form B-~ C, then A is true iff X B cons -Yc, where --XB [-Yc'] is he (unit) class of propositions assumed as values by B [C]. Then A is valid in (_P,!, F_) if A is true in (_P,!, _F) for every assignment, and A is valid if A is valid in every frame. We conjecture that a first degree formula is provable in EQ just in case it is valid under the definition above, but no proof has thus far been forthcoming. However, previous results enable us to establish the conjecture for various fragments of the system of first degree formulas. Theorem. The fragment of EQ consisting of (perhaps multiply) disjoined first degree entailments is complete and consistent. Proof. We treat each A1 --> Bi in the n-termed disjunction A 1 -> B 1 v v A,-~ B~ separately, using the proof methods of section IV. If all the branches of the tree for some A l -~ Bi terminate in axioms, then Ai -~ Bi is provable (and valid); hence the n-termed disjunction is provable (and valid). Now we show that if an _n-termed disjunction of first degree entailments is unprovable, then it is falsifiable in an atomic frame generated by 3_n atoms, from which it follows that the fragment of E Q in question is complete. Let A 1 -~ B 1 v- v A~ -> B~ be unprovable. Then each Ai -~ Bi has a tree containing at least one branch which fails to terminate in an axiom; let these branches be ~..., ~. We now divide the 3_n atoms into three groups: p-atoms ~1..., p,; _q-atoms _ql... q~ ; and r-atoms r I... r~. We first assign values to variables of 1, as in the proof in section IV, in such a way that all the atoms in L (~) take _Pl, p~, or r 1 as values, and those in R (~) take ql, ql, and _r~, thus falsifying A 1 -~ B 1 (by Lemma III of section IV). This assignment may have occasioned assignments to some of the atoms of Ca, but we may be sure that these assignments do not satisfy A~ -~ B~, in view of Lemma I of sectioniv. Hence we may assign values ~, q~, and r~. to variables of Ca in a similar way, with the result that all atoms not previously assigned of L(q~z) take ~z, ~, or r2, and all unassigned atoms of/~ (Ca) take _qg, _q2 or _r~. Hence A 2 -~ B 2 is also falsified; and obviously the procedure may be continued.
17 318 A.R. ANDERSON and N. D. BELNAt', JR. : We note the following corollaries (which of course do not hold for material or strict "impheation") : Corollary: If S is a set of falsifiable first degree entailments, then the members of S are simultaneously falsifiable. Corollary: A 1 -+ B 1 v v A n -+ B~ is provable just in case Ai -+ B~ is provable for some _/e). Theorem. The fragment of EQ consisting of (perhaps multiply) disjoined negated first degree entailments is complete and consistent. Proo/. Let (1) AI-+ B 1 v'.. vau-+ Bu be a formula of the required sort, and consider (2) (A 1 & B1) v.-. v (A, & B,). Suppose (1) is unprovable. Then since (2) entails (1) by A & B-+ A --> B and properties of disjunction, (2) is unprovable. But then, since EQ contains the classical first order functional calculus (B~L~AP 1959a), (2) is unprovable classically, and hence invalid. Then (2) has a falsifying assignment in the classical two-valued frame. Hence if we replace classical truth by _D{_p, 2'} and classical [alsehood by (_D{_p, ~'})', then the formula has a falsifying assignment in the frame <-P2,!, _F2> of section VI. Hence to each A~ & Bi, this assignment either gives (D{_p, _p'})' to A~ or gives D{E, _p'} to B/. But in both cases we have X{ai} cons -Y{Bi} ; hence A t -~ Bg is true, and Ai -+ B/ is false. Then A 1 -+ B 1 v v A~ ~ B~ is invalid7). It follows that the fragment is complete. For consistency, assume that (1) is invalid. Then by an obvious generalization of the Lemma of section VI, (1) is falsifiable in <-P2, -/, _F~>. But then (1) is falsifiable in <-P2, -/, -F2> with the arrow interpreted as material "implication". But every theorem of EQ is valid in <_P2,/, _F~> if the arrow is interpreted in this way, as is easily verified. Hence if (1) is invalid, it is unprovable. Note that neither of the corollaries above hold for this case: the set {A -+ A, A -+ A} and the formula A --> A v A -+ A are counter-examples to both. But attempts to generalize further have thus far failed. We therefore list some open problems, solutions to which may provide steps toward finding satisfactory semantics for the full system E Q. Let A be a quantifier-free formula of the first degree fragment of E Q. (1) Is it true that A is valid if and only if it is valid in every atomic frame? (2) Is A valid iff A is valid in some frame <P,/, _F>, with P finite (and of course / and F empty)? 0) Compare GSDEL'S conjecture (proved in McKINsEY and TA~SKI 1948) that NAv NB is provable in $4 iff either A or B is provable in $4. T) As is clear from the method of the proof, the fragment of EQ considered in the theorem could be expanded to include disjunctions of negated entailments together with expressions of degree zero.
18 First Degree Entailments 319 (3) Is there some effective function of some effective property of A (e.g., of the number of variables in A), such that A is valid iff A is valid in a frame generated by n atomic propositions (where n is the value of the function for A)? Two other related questions also remain open: (4) Is the propositional fragment of EQ decidable? (5) Does the rule (~) of ACKER~tANN 1956 hold for EQ; i.e., is it the ease that whenever A and A v B are both provable, B is Mso provable? We notice that if the first degree formula fragment of EOis complete, then (y) holds for the fragment; but the more general problem remains untouched. Bibliography ANDERSOn, ALAN ROSS: Completeness theorems for the systems E of entailment and EQ of entailment with quantification. Technical Report No. 6, Office of Naval Research, Group Psychology Branch, Contract SAR/Nonr-609 (16), New Haven. Also in the Z. mathem. Logik u. Grundlagen Mathematik 6, (1959). --, and N~EL D. BELnAP jr. : A simple proof of GSdel's completeness theorem [abstract]. J. symbolic Logic 24, 320 (1960a) Tautological entailments. Philosophical studies, 13, (1962) A simple treatment of truth functions. J. symbolic Logic, 24, (1959) The pure calculus of entailment. J. symbolic Logic, forthcoming (1960b). BELnAP, :NUEL D., jr. : EQ and the first order functional calculus, appendix to Anderson (Also in the Z. mathem. Logik u. Grundlagen Mathematik 6, (1959a).) -- Entailment and relevance. J. symbolic Logic, 25 1~ 146 (1960a). --- Tautological entailments [abstract]. J. symbolic Logic 24, 316 (1959b). -- A formal analysis of entailment. Technical Report No. 7, Office of :Naval l~esearch, Group Psychology Branch, Contract SAR/:Nonr-609 (16), :New Haven (1960). CHURC, H, ALONZO: Introduction to mathematical logic, vol. I, Princeton: Princeton University Press, GENTZEN, GERHARD: Untersuchungen fiber das logisehe Schliel3en. Math. Z. 39, , (1934). KAnGER, STm: Provabitity in logic. Stockholm: Almqvist & Wiksell, McK~sEY, J. C. C., and ALFRED TA~SKI: Some theorems about the sententim calculi of Le~ds and Heyting. J. symbolic Logic 13, (1948). (Received May 16, 1961) Math. Ann
CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimesMore information
Fundamentals of Mathematics Lecture 6: Propositional Logic
Fundamentals of Mathematics Lecture 6: Propositional Logic Guan-Shieng Huang National Chi Nan University, Taiwan Spring, 2008 1 / 39 Connectives Propositional Connectives I 1 Negation: (not A) A A T FMore information
-123- A Three-Valued Interpretation for a Relevance Logic. In thi.s paper an entailment relation which holds between certain
Dr. Frederick A. Johnson Department of Philosophy Colorado State University Fort Collins, Colorado 80523-23- A Three-Valued Interpretation for a Relevance Logic In thi.s paper an entailment relation whichMore information
! " # The Logic of Descriptions. Logics for Data and Knowledge Representation. Terminology. Overview. Three Basic Features. Some History on DLs
,!0((,.+#$),%$(-&.& *,2(-$)%&2.'3&%!&, Logics for Data and Knowledge Representation Alessandro Agostini email@example.com University of Trento Fausto Giunchiglia email@example.com The Logic of Descriptions!$%&'()*$#)More information
Section 7.1 First-Order Predicate Calculus predicate Existential Quantifier Universal Quantifier.
Section 7.1 First-Order Predicate Calculus Predicate calculus studies the internal structure of sentences where subjects are applied to predicates existentially or universally. A predicate describes aMore information
Handout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty orMore information
Chapter I Logic and Proofs
MATH 1130 1 Discrete Structures Chapter I Logic and Proofs Propositions A proposition is a statement that is either true (T) or false (F), but or both. s Propositions: 1. I am a man.. I am taller thanMore information
Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets.
Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets. Logic is the study of reasoning. A language of propositions is fundamental to this study as well as trueMore information
The set consisting of all natural numbers that are in A and are in B is the set f1; 3; 5g;
Chapter 5 Set Theory 5.1 Sets and Operations on Sets Preview Activity 1 (Set Operations) Before beginning this section, it would be a good idea to review sets and set notation, including the roster methodMore information
Predicate Logic Review
Predicate Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Grammar A term is an individual constant or a variable. An individual constant is a lowercase letter from the beginningMore information
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,More information
Correspondence analysis for strong three-valued logic
Correspondence analysis for strong three-valued logic A. Tamminga abstract. I apply Kooi and Tamminga s (2012) idea of correspondence analysis for many-valued logics to strong three-valued logic (K 3 ).More information
Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statementMore information
Deductive Systems. Marco Piastra. Artificial Intelligence. Artificial Intelligence - A.A Deductive Systems [1]
Artificial Intelligence Deductive Systems Marco Piastra Artificial Intelligence - A.A. 2012- Deductive Systems 1] Symbolic calculus? A wff is entailed by a set of wff iff every model of is also model ofMore information
We would like to state the following system of natural deduction rules preserving falsity:
A Natural Deduction System Preserving Falsity 1 Wagner de Campos Sanz Dept. of Philosophy/UFG/Brazil firstname.lastname@example.com Abstract This paper presents a natural deduction system preserving falsity. This newMore information
Inference Rules and Proof Methods
Inference Rules and Proof Methods Winter 2010 Introduction Rules of Inference and Formal Proofs Proofs in mathematics are valid arguments that establish the truth of mathematical statements. An argumentMore information
SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Automated Theorem Proving av Tom Everitt 2010 - No 8 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLMMore information
Mathematical Induction
Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) HowMore information
(LMCS, p. 317) V.1. First Order Logic. This is the most powerful, most expressive logic that we will examine.
(LMCS, p. 317) V.1 First Order Logic This is the most powerful, most expressive logic that we will examine. Our version of first-order logic will use the following symbols: variables connectives (,,,,More information
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainlyMore information
Likewise, we have contradictions: formulas that can only be false, e.g. (p p).
CHAPTER 4. STATEMENT LOGIC 59 The rightmost column of this truth table contains instances of T and instances of F. Notice that there are no degrees of contingency. If both values are possible, the formulaMore information
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2012 1 p. 43 48 ON FUNCTIONAL SYMBOL-FREE LOGIC PROGRAMS I nf or m at i cs L. A. HAYKAZYAN * Chair of Programming and InformationMore information
A Propositional Dynamic Logic for CCS Programs
A Propositional Dynamic Logic for CCS Programs Mario R. F. Benevides and L. Menasché Schechter {mario,firstname.lastname@example.com Abstract This work presents a Propositional Dynamic Logic in which the programs areMore information
MATHS 315 Mathematical Logic
MATHS 315 Mathematical Logic Second Semester, 2006 Contents 2 Formal Statement Logic 1 2.1 Post production systems................................. 1 2.2 The system L.......................................More information
Discrete Mathematics, Chapter 5: Induction and Recursion
Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Well-foundedMore information
MAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to takeMore information
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule forMore information
Continuous first-order model theory for metric structures Lecture 2 (of 3)
Continuous first-order model theory for metric structures Lecture 2 (of 3) C. Ward Henson University of Illinois Visiting Scholar at UC Berkeley October 21, 2013 Hausdorff Institute for Mathematics, BonnMore information
INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28More information
First-Order Logics and Truth Degrees
First-Order Logics and Truth Degrees George Metcalfe Mathematics Institute University of Bern LATD 2014, Vienna Summer of Logic, 15-19 July 2014 George Metcalfe (University of Bern) First-Order LogicsMore information
CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduceMore information
LOGICAL INFERENCE & PROOFs. Debdeep Mukhopadhyay Dept of CSE, IIT Madras
LOGICAL INFERENCE & PROOFs Debdeep Mukhopadhyay Dept of CSE, IIT Madras Defn A theorem is a mathematical assertion which can be shown to be true. A proof is an argument which establishes the truth of aMore information
Introduction to Proofs
Chapter 1 Introduction to Proofs 1.1 Preview of Proof This section previews many of the key ideas of proof and cites [in brackets] the sections where they are discussed thoroughly. All of these ideas areMore information
Predicate Logic. PHI 201 Introductory Logic Spring 2011
Predicate Logic PHI 201 Introductory Logic Spring 2011 This is a summary of definitions in Predicate Logic from the text The Logic Book by Bergmann et al. 1 The Language PLE Vocabulary The vocabulary ofMore information
Foundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard, Bernhard Nebel and Martin Riedmiller Albert-Ludwigs-Universität Freiburg Contents 1 AgentsMore information
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, calledMore information
The Foundations: Logic and Proofs. Chapter 1, Part III: Proofs
The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build ArgumentsMore information
MA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki CounterexamplesMore information
INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The fullMore information
Math 3000 Running Glossary
Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (More information
Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of usMore information
Chapter 1 LOGIC AND PROOF
Chapter 1 LOGIC AND PROOF To be able to understand mathematics and mathematical arguments, it is necessary to have a solid understanding of logic and the way in which known facts can be combined to proveMore information
Foundational Proof Certificates
An application of proof theory to computer science INRIA-Saclay & LIX, École Polytechnique CUSO Winter School, Proof and Computation 30 January 2013 Can we standardize, communicate, and trust formal proofs?More information
Tatsuya Shimura Nobu-Yuki Suzuki SOME SUPER-INTUITIONISTIC LOGICS AS THE LOGICAL FRAGMENTS OF EQUATIONAL THEORIES 1 Kripke frame semantics is used as a concise and convenient apparatus for the study ofMore information
DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available freeMore information
MATH 55: HOMEWORK # SOLUTIONS ERIC PETERSON * 1. SECTION 1.5: NESTED QUANTIFIERS 1.1. Problem 1.5.8. Determine the truth value of each of these statements if the domain of each variable consists of allMore information
CS510 Software Engineering
CS510 Software Engineering Propositional Logic Asst. Prof. Mathias Payer Department of Computer Science Purdue University TA: Scott A. Carr Slides inspired by Xiangyu Zhang http://nebelwelt.net/teaching/15-cs510-seMore information
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largelyMore information
2. The Language of First-order Logic
2. The Language of First-order Logic KR & R Brachman & Levesque 2005 17 Declarative language Before building system before there can be learning, reasoning, planning, explanation... need to be able toMore information
Chapter 1. Logic and Proof
Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were knownMore information
6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )
6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are pointsMore information
Beyond Propositional Logic Lukasiewicz s System
Beyond Propositional Logic Lukasiewicz s System Consider the following set of truth tables: 1 0 0 1 # # 1 0 # 1 1 0 # 0 0 0 0 # # 0 # 1 0 # 1 1 1 1 0 1 0 # # 1 # # 1 0 # 1 1 0 # 0 1 1 1 # 1 # 1 BrandonMore information
Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.
Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of theMore information
MATH20302 Propositional Logic. Mike Prest School of Mathematics Alan Turing Building Room
MATH20302 Propositional Logic Mike Prest School of Mathematics Alan Turing Building Room 1.120 firstname.lastname@example.com April 10, 2015 Contents I Propositional Logic 3 1 Propositional languages 4 1.1 PropositionalMore information
Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.
MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But weMore information
Review for Final Exam
Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in ChaptersMore information
1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will mostMore information
Problems on Discrete Mathematics 1
Problems on Discrete Mathematics 1 Chung-Chih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission fromMore information
Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.
M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also considerMore information
Climbing an Infinite Ladder
Section 5.1 Climbing an Infinite Ladder Suppose we have an infinite ladder and the following capabilities: 1. We can reach the first rung of the ladder. 2. If we can reach a particular rung of the ladder,More information
MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:
MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,More information
Summary Last Lecture. Automated Reasoning. Outline of the Lecture. Definition sequent calculus. Theorem (Normalisation and Strong Normalisation)
Summary Summary Last Lecture sequent calculus Automated Reasoning Georg Moser Institute of Computer Science @ UIBK Winter 013 (Normalisation and Strong Normalisation) let Π be a proof in minimal logicMore information
Course 421: Algebraic Topology Section 1: Topological Spaces
Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............More information
3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)More information
Partitioning edge-coloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and PoliticalMore information
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,More information
Rigorous Software Development CSCI-GA 3033-009
Rigorous Software Development CSCI-GA 3033-009 Instructor: Thomas Wies Spring 2013 Lecture 11 Semantics of Programming Languages Denotational Semantics Meaning of a program is defined as the mathematicalMore information
Chapter 8. Validity Testing
Chapter 8 Validity Testing In the first three chapters various methods have been introduced to decide the validity of different sorts of inferences. We have discussed truth-tables and the update methodMore information
Double Sequences and Double Series
Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine E-mail: email@example.com Abstract This research considers two traditional important questions,More information
13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385
mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infiniteMore information
Jaakko Hintikka Boston University. and. Ilpo Halonen University of Helsinki INTERPOLATION AS EXPLANATION
Jaakko Hintikka Boston University and Ilpo Halonen University of Helsinki INTERPOLATION AS EXPLANATION INTERPOLATION AS EXPLANATION In the study of explanation, one can distinguish two main trends. OnMore information
Encoding Mathematics in First-Order Logic
Applied Logic Lecture 19 CS 486 Spring 2005 Thursday, April 14, 2005 Encoding Mathematics in First-Order Logic Over the past few lectures, we have seen the syntax and the semantics of first-order logic,More information
Odd induced subgraphs in graphs of maximum degree three
Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standingMore information
Non-deterministic Semantics and the Undecidability of Boolean BI
1 Non-deterministic Semantics and the Undecidability of Boolean BI DOMINIQUE LARCHEY-WENDLING, LORIA CNRS, Nancy, France DIDIER GALMICHE, LORIA Université Henri Poincaré, Nancy, France We solve the openMore information
2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.
2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true thenMore information
Linear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.More information
Classical BI. (A Logic for Reasoning about Dualising Resources) James Brotherston Cristiano Calcagno
Classical BI (A Logic for Reasoning about Dualising Resources) James Brotherston Cristiano Calcagno Dept. of Computing, Imperial College London, UK {jbrother,email@example.com Abstract We show how toMore information
Satisfiability Checking
Satisfiability Checking First-Order Logic Prof. Dr. Erika Ábrahám RWTH Aachen University Informatik 2 LuFG Theory of Hybrid Systems WS 14/15 Satisfiability Checking Prof. Dr. Erika Ábrahám (RWTH AachenMore information
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specializeMore information
NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004
NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras............................... 7 1.2 Generated σ-algebras.........................More information
Chapter 1. Informal introdution to the axioms of ZF.
Chapter 1. Informal introdution to the axioms of ZF. 1.1. Extension. Our conception of sets comes from set of objects that we know well such as N, Q and R, and subsets we can form from these determinedMore information
4 Domain Relational Calculus
4 Domain Relational Calculus We now present two relational calculi that we will compare to RA. First, what is the difference between an algebra and a calculus? The usual story is that the algebra RA isMore information
Local periods and binary partial words: An algorithm
Local periods and binary partial words: An algorithm F. Blanchet-Sadri and Ajay Chriscoe Department of Mathematical Sciences University of North Carolina P.O. Box 26170 Greensboro, NC 27402 6170, USA E-mail:More information
A Beginner s Guide to Modern Set Theory
A Beginner s Guide to Modern Set Theory Martin Dowd Product of Hyperon Software PO Box 4161 Costa Mesa, CA 92628 www.hyperonsoft.com Copyright c 2010 by Martin Dowd 1. Introduction..... 1 2. Formal logic......More information
SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. ExamplesMore information
CS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-orderingMore information
Mathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which areMore information
1. Truth-value gaps vs. multivalues
Logique & Analyse 125-126 (1989). 3-14 ARE ALL TAUTOLOGIES TRUE? Philip HUGLY and Charles SAYWARD More specifically, we are asking: are all tautologies true in a language with truth-value gaps? Our answerMore information
Row Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasionMore information
Mathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,More information
Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All RightsMore information
Continued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents IntroductionMore information
1.3 Induction and Other Proof Techniques
4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.More information
Propositional Logic. A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both.
irst Order Logic Propositional Logic A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both. Are the following sentences propositions? orontoMore information
ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, sinceMore information
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linearMore information
Semantics for the Predicate Calculus: Part I
Semantics for the Predicate Calculus: Part I (Version 0.3, revised 6:15pm, April 14, 2005. Please report typos to firstname.lastname@example.com.) The study of formal logic is based on the fact that the validityMore information
Metric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergenceMore information
Sequences and Convergence in Metric Spaces
Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {sMore information
Mathematical Induction. Lecture 10-11
Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reachMore information


text:
SoyaVente Essence Sprays are a brand extension of the original line of SoyaVente soy candles. These 100% natural essences are infused with pure aromatherapy mood scents to promote different feelings. Ideal when used with aromatherapy, massage, yoga and meditation.
Brand Concept, Identity, & Copy: Cliff Shaluta
Graphic Designer: Kelley Coppinger

subset: commoncrawl


text:
In this 8×10 painting, I really wanted to push the warmth in the bushes to contrast all the cold in this scene. That part went pretty well, but I really struggled with the reflections in the ice. This is how it looks after 3 attempts at getting it right. At this point I have decided to let it dry before I do it again. It kind of amazes me sometimes, the things I struggle with. It certainly isn't predictable. I am hoping that I can get it right on the fourth go round. we shall see. Anyone with helpful tips here is welcome to chime in ( =
Tag Archives: white
Juneau, AK miniatures
More of my series of Juneau, AK in miniature. I decided these photograph better when I put them in groups of 4 and do not have to get so close to them. Each one of these is just an every day scene in Juneau. It is a small community and the mountains are in your face no matter where you go. It is such a gorgeous place! And I really love the low winter light. It is wonderful for filming or painting.
These are all the same mountains that surround the Mendenhall glacier, but each from a different part of town. It is almost like you cannot get away from it. These views are everywhere!
Red Evening Light
The intense pink on the white peaks here shows how red the evening light can be. Everything else around was almost black and white but the light reflecting on the mountain is red. I see this often here in Juneau on the mountains, and I wanted to get it down on canvas so I remember the strength of the color.
Tree Study
I love painted trees! I need to practice them close up because they are very tricky to paint. So in this I combined the tree practice with the strong warm light practice. It is just a small 5×7, but I think it has good impact. When studying a subject close up, the things you learn can be easily applied to a more suggestive painting later on. This is another thing that I need to remember.
Strong, Warm Light
This is a little (5×7) study that I did to practice the color, temperature and value of the light.The sun hits the mountain peaks like this in the winter here in Juneau. (on one of the maybe 10 sunny days we get) It is so beautiful! I want to be able to capture the extreme impact that it has. I think this is quite close to how it looks. I'm am going to do several more this week. It will be my focus.
White Exercises
These are my first attempts at painting a white coffee cup. One of the exercises in The Daily Painting book, suggested doing 4 paintings that consisted of a cup and a couple different pieces of colored papers. These each took about an hour a piece and I tried not to be too fussy or to go back over them after my time was up. Overall I think there is progress. So I'm going to call it good!


text:
(the problem is how to do that... I've seen many a good group waste away like that.... I guess we just have to get new members, the old ones grow up and don't have time for CR anymo )
ragged they kill..
Forum Mod
The problem is, we can't bring back people who don't have anything to do at the forum.... and I think the creator pretty much bowed out after a month...
That is why the "class enrollment" button on the front page doesn't work and same with the student IDs.....
Every now and then people would come and post something on the wall but that's pretty much it... No one really goes to the classes anymore besides us and a few would maybe add a reply to a game or something else.
And the problem with new people is that, if we don't get the original people back, how can we get the new people to join in? We're the only class that is actually active in a way. Also, even if I am a mod, my powers are limited. I can't change anything on the front page. The only power I have really is in the topics.
We're pretty much stuck when it comes to this group...
Either everyone comes back later on because they were all on vacation or because they had classes, if not this group died a long time ago.
I know that sounds pretty negative but it sounds worse when I say it positively...
We can try to get new people or try to get the other members back but it really all depends on them.
yeah you're right.... well then I guess the only thing is to either give up on this group, or stick to it.... I haven't been on CR that long, but much of the time once these groups get, er, abandoned they never really revive.... what should we do? leave or stay? transfer to another group maybe?
cuz if we decide to stick to it, maybe we could both ask Teardrops (who seems the be the creator now? *shrugs*) and ask to make us (powerful) mods or something, and try to revive this ghost town?
ragged they kill..
Forum Mod
Honestly, I don't know anything about avii making or things like that but if someone who wants to revive the group and knows how to do all that then let's go for it.
This is pretty much my first time being a mod in anything because I never really stay in groups for a long time. This is pretty much a first.
If we have all the rights and power to do it, I say go for it. I'll help in any way I can until I have to pay more attention to school. Which should be around mid-September. :P
But yeah. If you want to, and if we can get some help, let's go for it. I mean if it doesn't work, at least we tried right? :D
And if it does work that would be great. ^^
If you want, I can even make a topic asking for people to help.
I have at least that much power
yeah, ok! I am pretty so-so with graphics, so if you promise to help, we'll get SAS going again!
Well at least until school starts--even I won't be able to be on CR that much then. Hopefully by then we'll have found some other less busy mod to take over I'll ask a mod then to make me mod ...
ragged they kill..
Forum Mod
okay then I'll see if we can get anymore people that want to help revive this group ^^
Oh and if the creator doesn't want to take part then I think we pretty much need to start a new group but I think I have the requirements to do it. :]
I asked HazzelX to make me mod but so far she hasn't replied... :(
I already have a group of my own, if you want you could come join me in it... it's not a school though -_- it's called Anime Cyber Cafe
anyways i'll do my best with SAS
ragged they kill..
Forum Mod
I think for the most part, this group is dead.
We can't really do anything if no one actually replies.
yeah, alright. We did our best oh well.
ragged they kill..
Forum Mod
Off to ACC~!
ragged they kill..


text:
Thank you for using Teaching Treasures online games and activities.
There are many species of barn owls. They are birds of prey and come out at night. Sometimes called monkey-faced owls because of their heart shaped facial disks and absence of ear tufts. Growing around 30 to 40 cm long and white to grey or yellowish to brownish orange in colour. Their eyes are dark and small in comparison with those of other owls. Barn owls mainly hunt small rodents (like mice) for food and nest in buildings, towers and old hawks nests. They mainly inhabit open grasslands.
Did you know.... that the common barn owl is found world wide except Antarctica and Micronesia. Australia's most common barn owl is the common grass owl.
An online interactive knowledge test about the barn owl. Scores answers with the option of correcting if a mistake was made. Simple true - false quiz which can be played alone or with a friend. Ideal for comprehension testing and reinforcing what was learned.... back to aussie animals main page
Thank you for using Teaching Treasures online games and activities.


text:
Find Air Conditioning in Wapping
Complete the form to get quotes from local contractors for the service requested as per our terms and conditions. We only use the information provided for this purpose. The prices you receive will be without obligation.
Get Your FREE Air Conditioning Quotation
Air Conditiong in Wapping
Home Extensions in Wapping
- Whitechapel - 0.5 miles away
- Fenchurch Street - 0.7 miles away
- Shadwell - 0.7 miles away
- London Bridge - 0.9 miles away
- Canada Water - 1 miles away
- Liverpool Street - 1 miles away
- Rotherhithe - 1.1 miles away
- Cannon Street - 1.1 miles away
- Moorgate - 1.2 miles away
- Shoreditch High Street - 1.3 miles away
Air conditioning companies are available accross the UK, including Wapping. Installers vary dramatically though in terms of price and workmanship, and it's important to find the best one you can rather than leave it to chance.
If you choose to use us today, we'll put you in touch with air conditioning companies in Wapping that you can rely on for quality and value for money. All our installers are carefully checked and vetted so you don't have to bother.
Whether you need air conditioning for your home or even your office in Wapping we can help, so give our fast quote service a try – you might be glad you did!


text:
Our destination this time was to The Chocolate Garden of Ireland.
A great surprise awaited us as we stepped into their premises for an Aladdin's Cave of mouth watering confectionary blinded our eyes.The warm welcome from the proprietor plus her immediate offer to sample slender leaves made from delicious dark, milk and white chocolate warmed our hearts.
After a relaxing coffee in the garden and a delicious organic ice cream each, we finally left with a selection of boxes and bags of chocolate. We shall return to The Chocolate Garden of Ireland very soon!


text:
OregaPet Bed & Body Spray
External infestation and infection defence for:
Fleas, Ticks, Mites, Lice
Fungal & Viral Infection
Ringworm, Chiggers
Bacterial Infection
Mosquitoes & Sand Flies
Parasitic Infection
or Infestations & Infections:
Apply spray to bed and whole body daily for 21 days. Treat all pets. Apply while sheilding your pets face to avoid eyes and irritation. Spray into disposable non-latex glove to apply to face and head.
Use before and after outdoor activity in high risk areas or when exposed to high risk animals.
Spray bed & body regularly.
Litter-boxes, doghouses and crates.
Combats bacterial, viral, fungal and parasitic infections. Antioxidant and anti-inflammatory.
Alcohol Denatured, purified water, oil of oregano in a proprietary blend of essential oils.
Copyright © 2018 Abaco Health. All rights reserved. 3818 Gordon Dr, Kelowna, BC V1W 3G8


text:
Marine protected areas in a welfare-based perspective
Permanent lenkehttp://hdl.handle.net/10037/6323
TypeJournal article
Peer reviewed
ForfatterReithe, Siv; Armstrong, Claire W.; Flaaten, Ola
Assuming a broad set of fisheries management goals, this paper analyzes the implementation of a marine protected area (MPA) together with open access outside, applying a bioeconomic model that ensures unchanged growth post-MPA. Taking into account that conservation and restoration, food security, employment and social surplus are amongst the objectives that many managers include in fisheries management, it is found that this broader welfare economic approach to MPAs may well recommend them to a greater degree than espoused in the more common resource rent focused studies carried out to date. It is shown that for overfished stocks, an MPA may yield resource protection, maximize harvests and increase consumer and producer surplus, as well as give higher employment. This, however, is less apparent for moderately overfished as well as highly migratory stocks. Resource protection and enhancement implicitly improves ecosystem services.
ForlagElsevier Science
SiteringMarine Policy 49(2014) s. 29-36
MetadataVis full innførsel
Følgende lisensfil er knyttet til denne innførselen:


text:
We will see Norman Cornett at McGill again when we reclaim our academic freedoms. Slawomir Poplawski's article raises a couple of valid questions. First, we learn of the "undue haste" in the "decision to fire Cornett" – "he was given only a half day's notice to clear his office." As correctly noted by Poplawski, the haste reminds of "wartime executions…carried out for treasonous behaviour." One might ask, Who was the enemy Cornett had teamed up with?
It is true that Cornett's "growing popularity among students" might have made some administrators nervous. It is also true that the calibre of Cornett's guests might have triggered undue resentment to the spotlight he shared with no one. But, as valid as they seem, these reasons could not have provoked the tacit claim that made McGill an "exclusion zone" for Cornett. The "exclusion order" – to use the term commonly associated with the internment of Japanese-Canadians or Japanese-Americans following imperial Japan's attack on Pearl Harbor – is an indication of a much larger obstruction that aims at restraining independent thought in academia. While Cornett's hasty dismissal is the symptom of this obstruction, it also became the best proof of the current practice of curtailing open-minded dialogue in educational institutions.
McGill might take pride in the latest results of QS World University Rankings – it occupies the 19th spot, down from the 18th last year. But curbing academic freedoms, witnessed by Cornett's hasty dismissal, will have adverse effects in the long run. Institutions of higher learning should be at the forefront of all new ideas, experiments, investigations, discoveries. True creativity flourishes in an atmosphere of total freedom; and open discussion, currently under attack in academia, should provide the foundation for all inspired learning.
Anait Keuchguerian
U3 Theology
B. Mus. Theory (Honours)
M. A. Music Theory


text:
Having a HCISPP certificate validates a practitioner's core knowledge and experience in security and privacy controls for personal health information. People holding this certificate are the front-line defense in protecting health information.
To qualify for the HCISPP certification, you must have in one or more knowledge areas of the HCISPP Common Body of Knowledge (CBK) that includes security, compliance and privacy and a minimum of two years of cumulative, paid, full-time work experience.


text:
← Creating a Rockstar Portfolio
Is Back
May 7th (previous entry)
Is Back
May 7th (previous entry)
(next entry) I Agreed to What?!
July 29th →
July 29th →
A Day with Stephan Franck
Visual Storytelling and more..
June 3rd
June 3rd 10am – 5pm A Day in the World of Stephan Franck "Visual Storytelling and Story Structure and Making Comics Independently – From Storytelling to Distribution" $55 (Student discounts 10%)
From my early childhood devouring comics in my parent's book store to a 27-year career as a professional animator/writer/director/comics creator for major studios and international publishers, I have never stopped learning. I have compiled and organized the key concepts and skills acquired on my own storytelling journey into a 6-hour masterclass: Visual / Storytelling.
Limited to 25
Lunch is provided.
Check us out on FACEBOOK https://www.facebook.com/events/1233877776731804/


text:
Greetings to our RRS Community!
Torea would like to welcome our new Year 7 students and the returning senior Year 8 students! 2018 is a full on year with our big event in Term 3 – Snow Camp. It is important that all students stay focused and informed as we build towards the camp in the second semester.
We have 2 new teachers in Torea this year. I will be on study leave from Term 1 until part way through Term 4 working on a Masters of Applied Practice Digital Technology. I will be on site throughout the year working alongside teachers, staff and community around Digital e-learning.
Our Year 7 & 8 teaching team for 2018:
- Rm 24 William Matautia
- Rm 25 Kona Palmer
- Rm 26 Udal Singh
- Rm 23 Vaipouli Siliga (PoM Samoan Bilingual/Semeatu Iosefa (Team Leader)
- Specialist Technology Teacher Vanita Narsai
Our program for 2018 involves e-learning and will require our students to use devices at school. More information around this will be forthcoming.
Specialist Technology equipment – students required to have closed in shoes and hair tied back during sessions. These are safety precautions.
We look forward to meeting you all in the new year.
If you have any queries please contact Assistant Principal Paulette Thompson.
Thank you and God bless.
Roz Tuitama
Torea Syndicate Leader


text:
Making 5 Web site Analysis Report. Developing An Exploration Essay case study writing services canada
Exploration old fashioned paper publishing takes on a huge role towards virtually virtually every work aspiration, rendering it necessary for master styles talent being a copywriter.custom case studies All analysis cardstock uses a unique procedure becoming taken to make sure the best is sent plus the strategy and design will in most cases depend upon the research old fashioned paper web page number. [Read more...]


text:
Our fully submersible dip meters are available in a range of lengths from 15 – 300 metres.
Each dip meter has an adjustable internal potentiometer which is often required when measuring saline conditions and materials.
Every dip meter is now also available with an accompanying buzzer and light attachment for even greater accuracy when taking water level measurements.
Our extensive range of water monitoring equipment are the perfect tools for gaining accurate water and fluid measurements, please contact us for more information.