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OfflineFrenchSocialist
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Re: why are there so many loopholes, blanks, and contradictions in logic? [Re: shroomydan]
    #6587913 - 02/19/07 11:22 PM (17 years, 3 months ago)

Quote:

shroomydanMy understanding is that free logic cannot express that something exists, so it is limited to domains of non-existing things.




Wait, I thought the wiki said that free logic can express existence by using an existence predicate:

Quote:

In free logic, (1) is replaced with

1b. \forall xA \land E!t \rightarrow \exists xA, where E! is an existence predicate (in some but not all formulations of free logic, E!t can be defined as ∃y(y=t)).

Similar modifications are made to other theorems with existential import (e.g. the Rule of Particularization becomes (Ar → (E!r → ∃xAx)).



http://en.wikipedia.org/wiki/Free_logic

BTW I need to be registered in order to read that article and it says that free trials "are available for institutions only". Should I register, by pretending to be from an institution (I am just asking in case there are legal ramifications or I will be asked to prove my membership in some way)? Or would you PM me the article/post it via copy/paste?


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"Both liberty and equality are among the primary goals pursued by human beings through many centuries; but total liberty for wolves is death to the lambs" -- Isaiah Berlin

Edited by FrenchSocialist (02/19/07 11:37 PM)

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Invisibleshroomydan
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Re: why are there so many loopholes, blanks, and contradictions in logic? [Re: FrenchSocialist]
    #6589472 - 02/20/07 01:49 PM (17 years, 3 months ago)

Here's that article.

I think you are right about existence predicates, but it's not as simple as it might seem.

I'm not an expert in logic; I'm just a graduate student of philosophy suffering through an advanced course in first order logic. Next year I might take a more advanced logic course that deals with free logic, modal logic, tense logic, multivalued logic, and intuitionist logics.

So far my experience has been that no matter how hard you look into logic, the rabbit hole just keeps getting deeper and deeper.


Quote:

Free logics
We often need to reason about things that do not – or may not – exist. We might, for example, want to prove that there is no highest prime number by assuming its existence and deriving a contradiction. Our ordinary formal logic, however (that is, anything including standard quantification theory), automatically assumes that every singular term used has a denotation: if you can use the term ‘God’ – if that term is part of your language – automatically there is a denotation for it, that is, God exists. Some logicians have thought that this assumption prejudges too many important issues, and that it is best to get rid of it. So they have constructed logics free of this assumption, called ‘free logics’.

A ‘free logic’ is a system of quantification theory, with or without identity, that allows for non-denoting singular terms. In other words, some of the expressions that may be considered singular terms – individual constants, free variables, definite or indefinite descriptions – are not assigned any object in some of the models of the system. In a system of standard quantification theory, schemata such as ∃x(x = a) are logical truths; not so in free logics. Free logics reject the ‘principle of particularization’, A(t/x)⊃∃xA, and all equivalent principles or rules, among them the ‘principle of specification’, ∀xA ⊃A(t/x), and the rules of existential generalization and universal instantiation.

Historically, contemporary logicians since Frege have tended to dispense with non-denoting singular terms, adopting either Frege’s device of assigning them an arbitrary denotation or Russell’s of denying them singular term status (via his theory of descriptions; see Descriptions). It was Henry Leonard (1956) who first proposed that standard quantification theory be revised to allow for non-denoting singular terms. Subsequently, Hailperin, Leblanc, Hintikka, Lambert and Smiley proposed various formal systems to this effect (between 1959 and 1967 – see Lambert 1991). But, until the mid-1960s, no semantic interpretation of these systems was provided, for good reason.

Standard semantics is based on the ‘correspondence theory of truth’: a statement such as ‘Socrates is wise’ is true if and only if the object corresponding to ‘Socrates’ has the property corresponding to ‘wise’. But this scheme has no natural application to statements such as ‘Pegasus is white’, relating to nonexistent objects. So a semantics for a free logic (a ‘free semantics’) should provide for some alternative (or extension) to the correspondence theory, to account for those cases in which a singular term ‘corresponds to’ nothing – a problem for which it is difficult to find an uncontroversial solution.

There are currently three main kinds of free semantics. An ‘outer domain’ semantics adds to the ordinary domain of quantification a second (‘outer’) domain and allows singular terms to be interpreted on either domain. Intuitively, the outer domain is constituted by nonexistent objects and the reason why ‘Pegasus is white’, say, is true is that the nonexistent object corresponding to ‘Pegasus’ has the property corresponding to ‘white’. A ‘conventional’ free semantics assigns a single truth-value (usually False) to all atomic formulas containing non-denoting singular terms, and then evaluates complex formulas accordingly. Both conventional and outer domain semantics are bivalent, and strong soundness and completeness theorems can be proved for them; that is, not only the set of logically true formulas but also the set of valid arguments can be axiomatized. Not so with the third kind of free semantics, ‘supervaluational’ semantics, due principally to Bas van Fraassen. This assigns truth-values to formulas containing non-denoting singular terms (relative to a model M), either by convention or by extending the domain, and then defines a ‘supervaluation’ (on M) that makes a formula true if it is true under all such assignments, false if it is false under all such assignments, and truth-valueless otherwise.

For illustration, suppose a is non-denoting in the model M, and consider the following formulas:

(1) Pa

(2) ∼Pa

(3) Pa ∨∼Pa

(4) Pa &∼Pa.

There will be conventions on M (or extensions of its domain) that make (1) true, and others that make it false, and similarly for (2); so the supervaluation on M will leave both (1) and (2) truth-valueless. But all these conventions (or extensions) will make (3) true and (4) false; so the supervaluation on M will agree on assigning them these truth-values. Intuitively, supervaluational semantics makes a statement containing non-denoting singular terms true (false) if it would be true (false) were its atomic components containing non-denoting singular terms to have any combination of truth-values (or its non-denoting singular terms to have any combination of denotations) – a ‘counterfactual’ theory of truth, as it is sometimes called.

Free logics have been applied in other parts of logic and to the formalization of scientific theories. Most typical of an application of the first kind is modal quantification theory, where a free logic base is quite natural (for consider: in the presence of the rule of ‘necessitation’, the theorem ∃x(x = a) of standard quantification theory can be strengthened to □∃x(x = a) – that is, whatever we can name not only exists, but exists necessarily). Most typical of an application of the second kind is set theory, where a free logic base will let us use the name of, say, the Russellian set without being committed to its existence – indeed, while being able to prove that it does not exist (it cannot, by Russell’s paradox).

Finally, free logics must be distinguished from ‘inclusive’ logics, that is, logics that allow for an empty domain of quantification. Though it is natural to require inclusiveness of a free logic, and though indeed most free logics are also inclusive, the definitions of the two are distinct, and independent of one another.


Free logics, philosophical issues in
The expression ‘free logic’ is a contraction of the more cumbersome ‘logic free of existence assumptions with respect to both its general terms (predicates) and its singular terms’. Its most distinctive feature is the rejection of the principle of universal specification, a principle of classical predicate logic which licenses the logical truth of statements such as ‘If everything rotates then (the planet) Mars rotates’. If a free logic contains the general term ‘exists’, this principle is replaced by a restricted version, one which licenses the logical truth only of statements such as ‘If everything rotates then Mars rotates, provided that Mars exists’. If the free logic does not contain the general term ‘exists’, but contains the term ‘is the same as’, the principle is replaced by a version which licenses only statements such as ‘If everything rotates then Mars rotates, provided that there is an object the same as Mars’.

Most free logicians regard the restricted version of universal specification as simply making explicit an implicit assumption, namely, that Mars exists. Indeed, free logic is the culmination of a long historical trend to rid logic of existence assumptions with respect to its terms. Just as classical predicate logic purports to be free of the hidden existence assumptions which pervaded the medieval theory of inference with respect to its general terms, so free logic rids classical predicate logic of hidden existence assumptions with respect to its singular terms.

There are various kinds of free logic, with many interesting and novel philosophical applications. These cover a wide range of issues from the philosophy of mathematics to the philosophy of religion. In addition to the issue of how to analyse singular existence statements, of the form ‘3 + 7 exists’ and ‘That than which nothing greater can be conceived exists’, of special importance are issues in the theory of definite descriptions, set theory, the theory of reference, modal logic and the theory of complex general terms.

1 Kinds of free logic
2 Philosophical semantics
3 Implications and applications
1 Kinds of free logic
Though there are various historical antecedents of free logic, it is a relatively recent development in the logic of terms – that is, in the development of predicate logic with or without identity. In fact, serious technical and philosophical study of this alternative to classical predicate logic originated only in the second half of the twentieth century with Henry Leonard’s 1956 paper, ‘Logic of Existence’. The technical rudiments of free logic were then worked out in the subsequent decade and a half, the expression ‘free logic’ being first coined in 1960 (by Karel Lambert). Since then many studies discussing or employing free logic in general philosophy, philosophical logic and in the philosophy of science have appeared.

‘Free logic’ is short for ‘logic free of existence assumptions with respect to both its general terms (predicates) and its singular terms’. It restricts the principle of universal specification (which licenses the logical truth of statements such as ‘If everything rotates then (the planet) Mars rotates’) to a version which licenses only statements such as ‘If everything rotates then Mars rotates, provided that Mars exists’ (if the logic contains the general term ‘exists’) or such as ‘If everything rotates then Mars rotates, provided that there is an object the same as Mars’ (if the free logic does not contain the general term ‘exists’, but contains the term ‘is the same as’).

When a free logic also entertains the empty universe of discourse, it is called a universally free logic. In such comprehensive free logics, statements (perhaps without singular terms) such as ‘There is an object which is tall or is not tall’ and ‘There is an object which is self-identical’ – statements which Russell (§§9, 11) regarded as impure logical truths because of the evidently factual implication that there exists at least one thing – are also excluded from the class of logical truths.

In any free logic, universal or otherwise, the quantifier context ‘There is an object…’ has existential force just as it does in classical predicate logic. So in any free logic containing the general term ‘is the same as’ (the identity symbol), a statement such as ‘Mars exists’ can be taken as shorthand for the statement ‘There is an object the same as Mars’. Indeed, statements of the latter form can be false in free logic with identity, and hence, in contrast to classical predicate logic with identity, are not logically true. An example is the false statement ‘There is an object the same as (the planet) Vulcan’.

Free logics may be divided into three classes depending on the treatment of atomic (or simple) statements containing at least one singular term t such that the statement ‘t exists’ is false. For example, the statement ‘Vulcan rotates’ is an atomic statement containing a singular term, ‘Vulcan’, such that ‘Vulcan exists’ is false. Those which count all such statements as false are called ‘negative’, those which regard some such statements as true are called ‘positive’, and those which assign no truth-value at all to such statements – except perhaps statements of the form ‘Mars exists’(or ‘Vulcan exists’) – are called ‘neutral’.

In virtue of these different views about the truth-value of atomic statements with singular terms that refer to no existent object there is a significant inferential difference between the different kinds of free logic. For only in negative free logic does a version of the classical principle of particularization hold. In a negative free logic, the principle of particularization licenses the logical truth of the conditional statement ‘If Vulcan rotates then there is an object which rotates’ because the antecedent, ‘Vulcan rotates’, is an atomic statement. This does not alter the falsity of the conditional ‘If Vulcan is not an object then there is an object which is not an object’ because its (true) antecedent, ‘Vulcan is not an object’, is not atomic.

The kind of free logic one adopts has epistemological significance. For instance, when Descartes’ declaration ‘I think, therefore I am [exist]’ is construed as an argument and formulated in a positive free logic it is invalid. For when the singular term ‘I’ is replaced by the singular term ‘Sherlock Holmes’, and the general term ‘think’ is replaced by the general term ‘is self-identical’, the resulting premise ‘Sherlock Holmes is self-identical’ is true in virtually any positive free logic but the resulting conclusion ‘Sherlock Holmes exists’ is false. On the other hand, if formulated in a negative or neutral free logic, Descartes’ argument is valid because the premise is either false or without truth-value when ‘I’ is replaced by a singular term such as ‘Sherlock Holmes’, no matter what general term is substituted for ‘thinks’.

2 Philosophical semantics
It is a common misunderstanding that in free logics some singular terms need not refer. There do exist, however, semantic developments of free logics which entertain disjoint (and possibly empty) universes of discourse. Usually (though not always) these are philosophically interpreted as sets of existent objects and sets of nonexistent objects, after Alexius Meinong. In such developments, a singular term always refers, but not always to an existent object – the singular term, ‘Sherlock Holmes’, for example, may be taken to refer to a nonexistent (fictional) object. On the other hand, there also exist semantic treatments of free logic with a single (and possibly empty) universe of discourse. This is virtually always interpreted philosophically as the set of existent objects, after Russell. In such developments, a singular term can fail to refer to an existent object, and hence to any thing at all. Since free logics can be based on either Meinongian or Russellian ontologies, it is only correct to say that in such logics a singular term may fail to refer to an existent object. In free logics, successful or failed reference is explained as follows. Let ‘X’ be a place-holder for the name of a singular term, and let ‘——’ be a place-holder for the singular term named. Then a statement of the form ‘X refers (to an existent object)’ is true if and only if a statement of the form ‘—— exists’ is true – hence, when the language contains identity, if and only if a statement of the form ‘There exists an object the same as ——’ is true. For example, the statement ‘The satellite of the Earth refers (to an existent object)’ is true if and only if the statement ‘The satellite of the Earth exists’ is true (or, if the language contains identity, if and only if the statement ‘There is an object the same as the satellite of the Earth’ is true).

The quantifiers of free logic can be interpreted either objectually or substitutionally (see Quantifiers, substitutional and objectual); if interpreted in the latter way, then it is necessary to say in the clause for a quantifier context of the form ‘There is an object such that it ——’ that it is true just in case a sentence of the form ‘it ——’ becomes true when ‘it’ is replaced by some name ‘n’ such that ‘n exists’ is true. This condition is needed to give the quantifier context in question the requisite existential force. In current parlance, quantifier contexts of the form ‘There is an object such that it ——’ in free logic are ‘actualist’. Treatments seeking to preserve classical predicate logic by interpreting the quantifier context in question substitutionally while nevertheless admitting singular terms that refer to no existent object are not free logics because that quantifier context has no existential force; it is, in other words, ‘non-actualist’ in such treatments.
3 Implications and applications
Free logics are not committed to the philosophical doctrine that existence is a ‘predicate’ (see Existence) – indeed, there are free logics without identity or an existence symbol. Neither are they committed to the philosophical doctrine that proper names are truncated definite descriptions. They are not even committed to the doctrine that the reference of most grammatically proper names is determined by the reference of some description, a view often attributed to Frege and Russell (see Proper names §1). But an important result in the proof theory of positive free logic does bear on the traditional problem of defining (and, hence, explaining away) existence. Since it can be shown that, in the otherwise classical formal idiom minus identity, statements of the form ‘Mars exists’ (or ‘Vulcan exists’) are not definable in positive free logics, then, if one chooses to couch one’s logic of an existence predicate in positive free logic, that predicate cannot be eliminated save by means of the complex general term (predicate) ‘is the same as something’ or its equivalents.

Applications of free logic to important philosophical concerns are numerous. They range from the philosophy of mathematics to the philosophy of religion. Of special philosophical importance are the topics of definite descriptions, modality, the notion of reference, set theory and the theory of complex predicates.

In contrast to the Fregean tradition, which assigns an existent, sometimes arbitrarily, as referent to ‘unfulfilled’ definite descriptions such as ‘the planet Vulcan’, in contrast to the Russellian tradition which treats all definite descriptions as grammatical but not as genuine singular terms, and in contrast to the Hilbert–Bernays tradition which regards unfulfilled definite descriptions as formally ungrammatical (ill-formed), free theories of definite descriptions hold all definite descriptions to be genuine singular terms, but do not regard unfulfilled definite descriptions as referring to existents if they refer at all (see Descriptions). More precisely, in free theories of definite descriptions, if the noun or noun phrase following the word ‘the’ in a definite description is not true of exactly one existent thing, then it refers to no existent thing if it refers at all.

Free theories of definite descriptions are all based on the basic principle that

Everything is such that it is the same as the —— if and only if it and it only is ——.

This principle is known as Lambert’s Law in the literature. Free theories of definite descriptions form a continuum. At one end there is a Russell-like theory which counts all atomic statements having at least one unfulfilled definite description as false, and contains scope distinctions à la Russell. At the other end there is a Frege-like theory which counts all atomic identity statements having only unfulfilled definite descriptions as true, and collapses all scope distinctions held to be dependent on definite descriptions. There are also a multitude of intermediate cases in the literature which, in contrast to the other traditions, is dramatic evidence of the fecundity of free logic in the logical treatment of definite descriptions.

In many, if not most, treatments of the logic of singular and general terms, supplemented by the logical modalities ‘it is necessary that’ and ‘it is possible that’, the purely quantificational fragment is free. For in such developments, the conditional

C If every (existent) thing exists, then Mars exists,

an instance of the classical principle of universal specification, is rejected on the ground that it leads, via unimpeachable modal principles, to the conditional

C If necessarily every (existent) thing exists, then necessarily Mars exists.

But C is false because though ‘Necessarily every (existent) thing exists’ is true, ‘Necessarily Mars exists’ is false. In contrast, the principle of restricted universal specification yields only the innocent and trivial

C+ If necessarily every (existent) thing exists then necessarily Mars exists, provided that necessarily Mars exists.

The various treatments of free logic can influence the relationship between the meta-logical notion of reference and the logical notion of identity, the relationship reflected in the traditional adequacy condition that, for example, the singular term ‘the planet causing the perturbations in the orbit of Mercury’ refers to Vulcan if and only if the planet causing the perturbations in the orbit of Mercury is the same as Vulcan. To clarify the point, suppose the singular term ‘the planet causing the perturbations in the orbit of Mercury’ is replaced throughout the adequacy condition by the singular term ‘Vulcan’. Then one obtains the special case that ‘Vulcan’ refers to Vulcan if and only if Vulcan is the same as Vulcan. Clearly the traditional adequacy condition is sustained in negative free logics because both the reference claim and the identity claim are false since there exists no such object as Vulcan. Similarly, if the free logic is positive, but treats all singular terms as referring, in the spirit of Meinong, then the traditional adequacy condition is also sustained because both the identity claim and the reference claim will be true. However, if the free logic is positive and it recognizes that some singular terms do not refer, in the spirit of Russell, the traditional adequacy condition may have to be altered; in such a version of free logic, it is false that ‘Vulcan’ refers to Vulcan if Vulcan does not exist, but usually it is true that Vulcan is identical with Vulcan. In such a treatment of positive free logic the relationship between reference and identity is expressed in the following way: ‘the planet causing the perturbations in the orbit of Mercury’ refers to Vulcan if and only if Vulcan exists and the planet causing the perturbations in the orbit of Mercury is the same as Vulcan.

Free logic also exerts an influence in set theory; in free set theory, for instance, the naïve axiom of set abstraction, from which Russell deduced his famous paradox, is assertible without condition (see Paradoxes of set and property §4). Because of the replacement of universal specification by the restricted version of that principle, the most that can be deduced from the statement ‘Everything is a member of the set of sets which are not members of themselves if and only if it is not a member of itself’, an instance of the naïve principle of set abstraction, is that the set of all sets which are not members of themselves does not exist.

A final application of free logic concerns the important topic of complex general terms. In certain developments of free logic, both positive and negative, devices exist for generating complex general terms out of ‘open’ sentences, which are expressions without truth-value of the form ‘it rotates’. For instance, the open sentence in question can be turned into the complex general term ‘object such that it rotates’ by prefixing to it the general term-forming operator ‘object such that’. Then applying that complex general term to the singular term ‘Vulcan’ yields an expression capable of having a truth-value, the statement ‘Vulcan is an object such that it rotates’. In some developments this statement is equivalent to the statement ‘Vulcan rotates’ if and only if it is true that Vulcan exists. The result is a free logic in which scope distinctions with respect to the connective ‘it is not the case that’ can be made over the whole class of singular terms and not just with respect to the class of definite descriptions. For instance, in the developments in question, ‘Vulcan is an object such that it does not rotate’ may be false, although ‘It is not the case that Vulcan is an object such that it rotates’ may be true. If, however, ‘Vulcan’ is replaced by ‘Mars’, the distinction between negation in the general term and negation over the whole statement collapses because the resulting statements have the same truth-value in virtue of the existence of Mars. Indeed, the means are available in these augmented free logics to make a distinction in general between de re and de dicto predicates or properties (see De re/de dicto). In such treatments, ‘t exists’ can now be defined even if identity is not present – as Arthur Prior once anticipated. The requisite definition is that ‘t exists’ means ‘For any general term G, it is not the case that t is an object such that it is G if and only if t is an object such that it is not the case that it is G’. More idiomatically, the definition says that an object exists just in case denying it is G is tantamount to asserting it is non-G, and vice versa, for any G.



How to cite this article:
LAMBERT, KAREL (1998). Free logics, philosophical issues in. In E. Craig (Ed.), Routledge Encyclopedia of Philosophy. London: Routledge. Retrieved February 20, 2007, from http://www.rep.routledge.com/article/X003SECT3



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