# Logical Truth

## Overview:

 Some ways of defining "Logical Truth" are introduced and their merits discussed.
 Analytic We could define logical truths in terms of analyticity, and then look for a good definition of analyticity (which doesn't get us back where we started)... a priori or alternatively in terms of the a priori, which concerns to the kind of evidence relevant to establishing the truths... Necessary Truths or in terms of necessity, sometimes described as a metaphysical characteristic.
 Evaluation Criteria How do we know when we have a good definition? Here are some criteria which we can use to assess candidates. definition in terms of formal systems Logic may be defined as the study of formal logical systems. The results of logic would then be meta-theory about formal systems or formal theorems derived in such systems. This is not far off the mark for mathematical logic in academe.
 True in Virtue of Form Possibly the most popular way of attempting to characterise the truths of logic, though this risks circularity if its purpose is to provide a philosophical rationale to underpin the use of formal systems Topic Neutral Another popular one which seems inextricably intertwined with the "in virtue of form" definition, so that one may have to take them as two parts of a single definition.

## Evaluation Criteria:

 How do we know when we have a good definition? Here are some criteria which we can use to assess candidates.
 Fundamental, not ad. hoc. Definitions involving enumeration of a specific set of logical constants are a typical example of a definition which appears arbitrary and hence unsatisfactory, unless grounds were offered for the belief that the concept thus defined was of importance. Real Content Its easy to give definitions which have very little content at all. e.g. Quine's "Logic is the systematic study of the logical truths". No harm in this so long as you follow through and define logical truth. To get a good definition you need to work through several layers.
 Good Closure Properties Examples of desirable closure properties are closure under logical conjunction and closure under logical equivalence. So a definition (like that of Kant for analyticity) in terms of subject predicate sentences, which was not hedged to ensure closure would be unsatisfactory. Hedging with a phrase like "equivalent to" risks circularity! Well Definedness Settling for a form of words which doesn't really say anything is all too easy with these fundamental concepts. Meta-circular formal definitions can help if you can get that far.
 Relationship with Analytic and a priori Assuming you don't define logical truth directly in terms of analytic or a priori truth then the relationship with these is worth looking at. If there are substantial differences in extension of the three concepts then one is probably badly defined. If there are small difference its worth thinking about whether there really are two important but just slightly different concepts or whether one of them is not quite correctly formulated. Utility A sense of purpose may be helpful in sorting out good from bad definitions.

## Formal Systems:

 Logic may be defined as the study of formal logical systems. The results of logic would then be meta-theory about formal systems or formal theorems derived in such systems. This is not far off the mark for mathematical logic in academe.
 strict formalism Extreme formalists concern themselves exclusively with matters of syntax, regarding formal notations as devoid of meaning. From this point of view logic would be considered the study of formal "deductive" systems, and the question of logical truth may not arise. For a starting point in this kind of study try starting with What is Logic? (the un-philosophical version) and try to ignore the references to semantics. disadvantages: The disadvantages of this strict approach include some increased difficulty in accounting for the applicability of logic and mathematics, and some difficulty, in the light of Gödel's incompleteness results, in justifying interest in formal systems for arithmetic, which must fail in their purpose, without being able to mention that purpose.
 moderate formalism Mathematical logic as practised is not irrefrangibly hostile to semantics. Tarski showed how to render the semantics of first order logic, and a great deal of further work has been done since. It is therefore possible to define logic as the study of formal systems without abstaining from consideration of semantics. If this is done, detailed consideration of logical truth is brought into the subject rather than being part of the definition of the subject. In effect this amounts to the adoption of a definition of logical truth in terms of analyticity, where analyticity is defined in terms of meaning. This avoids both the problems mentioned in connection with strict formalist definitions of logic. A key merit of this approach is that it does not depend upon any arbitrary choice of formal system.

## True in Virtue of Form:

 Possibly the most popular way of attempting to characterise the truths of logic, though this risks circularity if its purpose is to provide a philosophical rationale to underpin the use of formal systems.
 Formal Systems Symbolic logic is normally presented as codified in formal logical systems. Such codifications are called formal because they define criteria for theoremhood which remove the necessity for any understanding of the meaning of the sentences or terms involved in the demonstration of a theorem, making use exclusively of criteria which relate to the form of the sentences rather than their content. Logic is therefore essentially concerned with formality and it is natural to expect that the subject matter of logic, the logical truths, should be defined formally. For a classic statement of the importance of this view of logic, together with an indication of some of the problems which it causes, see: Bertrand Russell on the definition of Logic and Defining Mathematics and Logic. See also: Alfred Ayer on the analyticity of mathematics.
 Quine adopts a similar position, holding that logical truths are to be explained through something which he starts out calling grammar and then calls logical grammar, and which turns out to be his own account of the structure of first order logic. Why that particular grammar is logical and not any other (like the grammar of modal or higher order logic or of serbo-croat, or even my account of the grammar of first order logic in which "A or not A" and "A and not A" have the same grammatical structure) is not so easy to establish from his writings. Unlike Russell he doesn't seem aware of any difficulty here. For a starter see his Philosophy of Logic, which is a great little book but just never gets close to justifying the position it adopts.

## Topic Neutral:

 A logical truth is distinguished by being perfectly general, not referring to particular individuals or properties.
 Proponents include Bertrand Russell, Alfred Ayer, Willard Quine. Russell thought that logic should be defined in terms of the universality or generality of its truths or as arising in some way from the form of the sentences which express logical truths. However he had some difficulty in seeing how this could be carried through. His concerns are described in a brief passage from the Introduction to the second edition of The Principles of Mathematics.
 Topic Neutrality It is traditional to take the view that logic is topic neutral. To the extent that necessary truths, being true in all possible worlds, can say nothing which is particular to any one of those possible worlds, a necessary truth can be considered to have no subject matter in the real world. However, some propositions widely accepted as necessary are most naturally supposed as having a subject matter. For example, the truths of arithmetic may be considered necessary propositions whose subject matter is the collection of abstract entities known as the natural numbers.
 Even if the logicist thesis that mathematics is logic is put aside, the claim of topic neutrality seems to me difficult to justify. Classical propositional logic is the theory of boolean operators, they are its subject matter. There are many non-classical variants of propositional logic most of which have some purpose and rationale. For these alternatives to have any validity we must surely accept that they differ in subject matter. Neither intuitionistic nor many-valued logics are about boolean operators, even though they may use exactly the same symbols as classical logic.

## Necessary Truths:

 We consider the notions of necessary and contingent partly through consideration of possible worlds, in order to explicate logical truth.
 introduction We compare three different definitions of necessity (and hence of contingency). some necessary propositions Four examplary propositions are presented, one for each kind of necessity considered, and one which is unequivocally contingent.
 defining necessity and contingency We then sketch three alternative kinds of necessity, between which the examples serve to discriminate. possible worlds Finally we consider the relationship between necessity and possible worlds, showing the conception of possible world which corresponds to each of our notions of necessity.