I installed Logic Pro when it was version 8. I upgraded to version 9 and after the upgrade I ran the application. It asked for a serial number, I entered my new serial number which the app said was not valid (this must prove the app is working in this respect).
I then entered the version 8 serial number and the application thought about it for a few seconds then popped up a error box stating Logic Pro quit unexpectedly. And then asked to either ignore or send a crash report.
logic 9.1 5 serial number
Download File: https://reitoratechk.blogspot.com/?zx=2vFhcy
Ultimately second-order logic is just another formal language. It canbe given the standard treatment making syntax, semantics and prooftheory exact by working in a mathematical metatheory which we chooseto be set theory. It would be possible to use second-order logicitself as metatheory, but it would be more complicated, simply becausesecond-order logic is less developed than set theory.
of natural numbers. This, together with the axioms \(\forallx\,\neg(x^+=0)\) and \(\forall x\,\forall y(x^+=y^+\to x=y)\)characterizes the natural numbers with their successor operation up toisomorphism. In first order logic any theory which has a countablyinfinite model has also an uncountable model (by the UpwardLöwenheim Skolem Theorem). Hence (\refind) cannot be expressedin first order logic. Another typical second-order expression is theCompleteness Axiom of the linear order \(\le\) of the realnumbers:
This, together with the axioms of ordered fields characterizes theordered field of real numbers up to isomorphism. In first order logicany countable theory which has an infinite model has also a countablemodel (by the Downward Löwenheim Skolem Theorem). Hence(\refcomp) cannot be expressed in first order logic.
A vocabulary in second-order logic is just as a vocabulary infirst order logic, that is, a set L of relation,function and constant symbols. Each relation andfunction symbol has an arity, which is a positive naturalnumber.
Second-order logic has several kinds of variables. It hasindividual variables denoted by lower case letters\(x, y, z, \ldots\) possibly with subscripts. It has propertyand relation variables denoted by upper case letters\(X,Y,Z,\ldots\) possibly with subscripts. Finally, it hasfunction variables denoted by upper case letters \(F, G, H,\ldots\) possibly with subscripts. Sometimes function variables areomitted as, after all, functions are special kinds of relations. Eachrelation and function variable has an arity, which is a positivenatural number.
The terms of second-order logic are defined recursively asfollows: Constant symbols and individual variables are terms. If\(t_1,\ldots,t_n\) are terms, U is an n-ary functionsymbol and F is an n-ary function variable, then\(U(t_1,\ldots,t_n)\) and \(F(t_1,\ldots,t_n)\) are terms. Note thatterms denote individuals, not relations or properties. Thus Xalone is not a term but x is.
The atomic formulas of second-order logic are defined fromterms as follows: If t and \(t'\) are terms, then \(t = t'\) isan atomic formula. If R is an n-ary relation symbol and\(t_1,\ldots,t_n\) are terms, then \(R(t_1,\ldots,t_n)\) is an atomicformula. If X is an n-ary relation variable, then also\(X(t_1,\ldots,t_n)\) is an atomic formula. The intuitive meaning of\(X(t_1,\ldots,t_n)\) is that the elements \(t_1,\ldots, t_n\) are inthe relation X or are predicated by X. We donot allow atomic formulas of the form \(X = Y\), although they wouldhave an obvious meaning. We achieve the same effect by usingquantifiers.
Definition 1 The formulas of second-orderlogic are defined as follows: Atomic formulas are formulas. If\(\phi\) and \(\psi\) are formulas, then \(\neg\phi\), \(\phi\land\psi\), \(\phi\lor \psi\), \(\phi \to \psi\) and \(\phi\leftrightarrow \psi\) are formulas. If \(\phi\) is a formula,x an individual variable, X a relation variable andF a function variable, then \(\exists x\phi\), \(\forallx\phi\), \(\exists X\phi\), \(\forall X\phi, \exists F\phi\) and\(\forall F\phi\) are formulas.
It is interesting to note that in second order logic we can actually define the identity \(t=t'\) as \(\forall X(X(t)\leftrightarrow X(t'))\) and prove the familiar axioms of identity from properties of the implication.An important special case is monadic second-order logic whereno function variables are allowed and the relation variables arerequired to be monadic (a.k.a. unary), i.e., of arity one.
as a substitute for \(X=Y\). The advantage of taking \(X=Y\) as abasic atomic formula would be that using (\refXY) brings along extraquantifiers. Sometimes it is interesting to minimize the number ofquantifiers in a formula. Also, (\refXY) gives the identity \(X=Y\) an extensional flavour in contrast to a possibly different intensional construal.
The concepts of a free and bound occurrenceof a variable in a formula are defined in the usual way. A formula iscalled a sentence if it has no free variables. The concept ofa term being free for a variable in a formula isdefined as in first order logic.
We use the same concept of an L-structure (or equivalently, anL-model) as in first order logic. That is, anL-structure \(\mm\) has a domain M,which is any non-empty set, an interpretation \(c^\mm \in M\)of any constant symbol c of L, an interpretation \(R^\mm\subseteq M^n\) of any n-ary relation symbol R ofL, as well as an interpretation \(H^\mm : M^n \to M\) of anyn-ary function symbol H of L.
Now that the semantics of second-order logic is defined we can definewhat it means for a formula \(\phi\) to be valid and for twoformulas \(\phi\) and \(\psi\) to be logically equivalent. Asin logic in general, we say that \(\phi\) is (logically) valid if\(\mm\models_s\phi\) holds for all \(\mm\) and all s. Likewise,we define \(\phi\) and \(\psi\) to be logically equivalent,\(\phi\equiv\psi\), if \(\phi\leftrightarrow\psi\) is valid. Twomodels \(\mm\) and \(\mn\) are said to be second-orderequivalent, in symbols \(\mm\equiv_L^2\mn\) if for allsentences \(\phi\) we have \(\mm\models\phi\iff\mn\models\psi\).
When we use set theory as the metatheory for the semantics offirst order logic, the reliance on the metatheory is of alower degree than when we use set theory as the metatheory for thesemantics of second order logic. The central concept, whichexplains the difference, is that of absoluteness. Fordetails, see 6.
The Ehrenfeucht-Fraïssé game is a game-theoretictool for investigating to what extent two models are similar (seeentry on logic and games for a general introduction to Ehrenfeucht-Fraïssé games).Two isomorphic models would be very similar but normally we areinterested in similarity of models that are not isomorphic. TheEhrenfeucht-Fraïssé game of second-order logiccharacterizes \(\ma\equiv_L^2\mn\), i.e. the proposition thatexactly the same second-order sentences are true in \(\ma\) and\(\mb\), just as the Ehrenfeucht-Fraïssé game of firstorder logic characterizes the first order elementary equivalence\(\ma\equiv\mn\). See the entry on first-order model theory for more on elementary equivalence.
For simplicity we disallow function and constant symbols as well asfunction variables in this section. Suppose \(\ma\) and \(\mb\) aretwo models of the same finite relational vocabulary. In the game whichwe denote by \(G^2_n(\ma,\mb)\) two players I and II pick one at atime subsets (or elements) of A or subsets (or elements) ofB. During round i of the game player I can pick arelation \(A_i\) on A (or an element \(a_i\) of A) andthen player II has to pick a relation \(B_i\) on B of the samearity as \(A_i\) (or an element \(b_i\) of on B) and viceversa: Player I can instead pick a relation \(B_i\) on B (or anelement \(b_i\) of B) and then II picks a relation \(A_i\) onA of the same arity as \(B_i\) (or an element \(a_i\)) ofA. After n rounds the pairs of played elements\((a_i,b_i)\) form a binary relation R on \(A\times B\). Ifthis relation is a partial isomorphism of the structures \(\ma\) and\(\mb\) expanded by the played relations \(A_i\) and \(B_i\), i.e., itpreserves atomic formulas and their negations, we say that II has won.The models \(\ma\) and \(\mb\) satisfy the same second-order sentencesif and only if for each \(n\in \oN\) Player II has a winning strategyin \(G^2_n(\ma,\mb)\). A model class (i.e. a class of models of afixed vocabulary, closed under isomorphisms) K is definable insecond-order logic if and only if there is an n such that if\(\ma\in K\) and Player II has a winning strategy in the\(G^2_n(\ma,\mb)\), then \(\mb\in K\).
The first order version \(G^1_n(\ma,\mb)\), where players play onlyindividual elements, i.e., no relations are played, is very useful inworking with first order logic. Unfortunately the game\(G^2_n(\ma,\mb)\) is much more complex. It is very difficult toinvent winning strategies for Player II except in the trivial casewhen \(\ma\cong\mb\). If Player I plays a binary relation R onA, Player II should be able to play a binary relation \(R'\) onB such that whatever challenges Player I makes in the rest ofthe game, even involving new binary relations, the relations Rand \(R'\) look similar. If \(V=L\), then countable second-orderequivalent models (with a finite vocabulary) are actually isomorphic(Ajtai 1979). For details, see 10. Thus consistently in countable models there is no other winningstrategy for Player II than an isomorphism. Perhaps this explains whythe game is not as useful in second-order logic as it is in firstorder logic. However, if we restrict to monadic second-order logic,which in terms of the Ehrenfeucht-Fraïssé game meansrestricting the game to unary predicates, the situationchanges. A unary predicate just divides the model into two parts. IfPlayer I divides A into two parts, Player II should find asimilar division in B. This is more reasonable and thereactually are useful strategies for Player II. For examples in thefinite context see 16. 2ff7e9595c
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