Formal Logic, Models, Reality

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y adding a new element g not occurring in M. Finally they expand the model M byadding interpretations g and G of the new symbols g and G. Then we get a modelM' = (M', g, G, …) = (M', g, {g}, …)While M is supposed to be a model of reality, M' has the character of a thought experimentwhich the atheist and the monotheist agree to make. They have agreed to letg represent God in the model, and they have agreed that if God exists, then Godshould have the properties G. Finally, they extend the language once more by addingthe one-place predicate EL'' = L' {E}with the intended interpretation (7-2), and they expand the model M' further by addingtheir interpretations of E. Now they do not agree any longer. The monotheist andthe atheist expand M' in different ways by giving distinct interpretations to E:(7-3) M atheist = (M', g, G, E, …) = ({g, a, b, …}, g, {g}, {a, b, …}, …)(7-4) M monotheist = (M', g, G, E, …) = ({g, a, b, …}, g, {g}, {g, a, b, …}, …)We have,(7-5) M monotheist |= E(g)from which follows by logic(7-6) M monotheist |= x E(x)Though this sentence is true in M monotheist , it is not logically true because it means:(7-7) There exists something in the model which represents an entity that existsin reality.This follows because existence-in-the-model, represented by the existential quantifier,is not the same as existence-in-reality, represented by the existence predicate.Similarly, the atheistic standpoint is true in the model M atheist :(7-8) M atheist |= E(g)By logic,(7-9) M atheist |= x E(x)In contrast to Quine's (2-3), this sentence is not a contradiction in terms because itmeans(7-10) There exists something in the model which does not represent an entitythat exists in reality.Again the crucial point is the difference between existence-in-the-model, representedby the existential quantifier, and existence-in-reality, represented by the existencepredicate.12
We can use constants to denote God without contradictions arising because formallogic is concerned with models and not directly with reality itself. A tiny bit of logicsolves a problem which philosophers like Russell, Wittgenstein, Carnap, Quine, andseveral others found mind-boggling.3. Ontological Commitment3.1 Introduction. For many years ontology was banned from analytical philosophy.The common attitude was expressed by Gilbert Ryle in the slogan: "Ontologizing isout." The situation changed with Quine's idea of ontological commitment.3.2 Ontological Commitment. Quine's ideas on ontology can be expressed in twoprinciples. The first principle is the criterion of existence. This is expressed in a slogan:(2-1) To exist is to be the value of a bound variable.The thought here is that whenever we quantify over a kind of entities, then this kindof entities is explicitly or implicitly assumed to exist. In the simplest case, consider asentence(2-2) x Q(x)If the sentence is true, then this is claimed to imply the existence of an entity or atype of entities with the property Q. We note that it is the quantification, independentof whether it is existential or universal quantification, which by Quine is claimed toimply the existence of the elements in the domain quantified over. Thus the example‘x Q(x)’ should have worked as well. The second principle is the principle of ontologicalcommitment. If a person P believes in a theory T and T involves quantificationover a certain kind of entities, then P is, by his belief in T, committed to believein the existence of this kind of entities.3.3 EXAMPLES. (I) In Peano arithmetic PA, we quantify over natural numbers. Thisimplies the existence of natural numbers in PA. If we believe in Peano arithmetic,then we are ontologically committed to believe also in the existence of the naturalnumbers.(II) In the set theory ZF, there is a theorem which asserts the existence of sets of arbitrarilylarge infinite cardinality. Believing in ZF commits us ontologically to a beliefin the existence of sets, and especially in the existence of sets of arbitrarily large infinitecardinality.13
Page 1 and 2: Formal Logic, Models, RealityKaj B
Page 3 and 4: Since the existential quantifier is
Page 5 and 6: (6-3) A, if A then B |= BThe meanin
Page 7 and 8: 'A B' in (6-7) and 'A B is true'
Page 9 and 10: He refers to God as an individual b
Page 11: logic, if there is a unique god, th
Page 15 and 16: says that there exists at least one
Page 17 and 18: elation between the formal language
Page 19 and 20: The statements (6-5) and (6-6) cont
Page 21 and 22: SOLUTION:We do think primarily in p
Page 23 and 24: 5.11 EXAMPLE. Consider the problem(

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