TRUTH IN LOGIC

truth in logicOverview

A whistle-stop tour through modern logic and its philosophies. Read this section to discount any simple opposition of fact (science, mathematics and logic) to fiction (art and literature). Not only is logic difficult, fascinating and (in its further reaches) contentious, but it seems ultimately to depend on some human sense of rightness: perhaps not so different from poetry in the end.

Introduction: What is Logic?

Though literature is commonly thought to tell some kind of truth, it is surely not one comparable to logic. A time-honoured example: {1}

Socrates is a man

All men are mortal

Therefore Socrates is mortal.

Given the premises, we intuitively grasp the conclusion as true. How could it be otherwise? It offends some sense of rightness to deny it, just as we cannot assert that something is at once the case and not the case, p and not-p. But what is this intuitive sense?

Broadly, there are four views. {2} The first is that the laws of logic are generalized, empirical truths about how things in the world behave, like the laws of science, but more abstract. Few believe this. That "ravens are black" is not an inevitable truth in the way "all bachelors are unmarried" necessarily must be. We accept that ravens are indeed black, but could conceive of some being not so. But a married bachelor is a contradiction in terms, something we can't seriously entertain.

A second theory, is that the laws of logic are not given to us by experience, but are true in ways more fundamental than our sense impressions: they are true because that is how the world is. The mind's power of reason gives us insight into the inherent nature of things: truth is a property of the world rather than our reasoning processes. But what is this property? We don't derive our laws of logic through insight, and there are sufficient conundrums in the physical world (e.g. quantum mechanics where an electron is sometimes a particle and sometimes wave occupying a position with some percentage of probability) for us to doubt if logic can be safely grounded in the outside world.

A third view is that logic is isomorphous with mind functioning, that humans by their constitutions are unable to entertain contradictions once they become apparent. Our brains are simply constructed ("hardwired") so as to reject logical inconsistencies. But logic is not a branch of psychology or physiology; and we have as yet only a rudimentary understanding of brain functioning. A theory so dependent on unknowns is not one securely based.

The fourth view is simpler: the laws of logic are verbal conventions. We learn through social usage the meanings of and, and not, true and false. In one, trivial sense this is undeniably true. But if the terminology is arbitrary, we still cannot rationalize away our sense of truth and correctness is this manner. The sentence "p and not-p" remains a contradiction, whatever term we give it.

Sentential Logic

Let's move on, difficulties notwithstanding. Logics that aim to represent situations in simple, context-free sentences are called sentential (also propositional, or propositional calculus), after Gottlob Frege who founded modern mathematical logic. Sentential logic is built with propositions (simple assertions) {3} that employ logical constants like not and or, and and and if - then. Such logics cannot deal with expressions like he believed her (which appeal to the common understanding of the human heart) but are very powerful in their own field.

Symbolic Logic

Indeed, once connectives are used ( &, ~, &exist, &sup, ∀ and, not, some, supposing, all) very complex sentences can be built up where the truth value of the whole sentence is dependent only on the truth values of its components. We arrive not only at secure judgements, but see clearly how the individual propositions systematically play their part in the overall truth or falsity of the sentence.

Symbols are commonly used. Take a sentence like: John exists. We recast that as : There is something that is John, and that something is identical to John. Expressed symbolically that becomes: (&exist x) (x = John). Everything is green becomes: (&forall x) (Green (x)). Using the negative ~ we can express: everything is green as: it is not the case that everything is not green: ~ (&exist x) ( ~ green (x)).

Is this helpful? Immensely so. Numbers can be defined in this way. Perplexing sentences like: The King of France is bald can be re-expressed as a conjunction of three propositions: 1. there is a King of France, 2. there is not more than one King of France, and 3. everything that is a King of France is bald. Put another way, this becomes: there is an x, such that x is a King of France, x is bald, and for every y, y is a King of France only if y is identical with x. In symbols: (&exist x) (K(x) & b(x) &(y)(K(y) &sup (y =x))). {4}.

Metalanguages

But how do we handle logical paradoxes like the following: { What is written between these brackets is false.} If what is written between the brackets is false, it is also false that What is written between these brackets is false — i.e. the sentence must be true. But we have accepted it as false. How can we stop sentences referring to themselves? Alfred Tarski's solution {5} was to consider the primary sentence as written in an object language, and that commenting on the primary sentence is in another language altogether, a metalanguage. Both languages had to be logically formulated to avoid the tangles and vagueness of everyday speech, but only the metalanguage could refer to the object language, not the other way about.

Many-Valued Logics

Are these the only categories of logic? Not at all. There are three-valued systems that operate with true, false and possible/indeterminate. There are systems that use more than three values. And there is a large branch of logic (modal logic) that deals not with simple propositional assertions, but with concepts like possible, impossible, contingent, necessary and absurd.{6} And since what is true today may not have been so yesterday, some have argued that time should come into logic, {7} either by changing our understanding of logical operators, or by extending standard logic. Why have these alternatives been developed, and how do they modify a search for an ideal, logically transparent language?

A two-value system of logic is unsatisfactory in some areas of mathematics on two counts. Certain propositions cannot be declared true or false because that truth or falsity hasn't been demonstrated. Secondly, the adoption of either true and false values for a proposition may lead to contradictions in the mathematical treatment of quantum mechanics and relativity. As an alternative to standard logic, Jan Lukasiewicz developed a three-valued system of logic: true, false and possible, usually denoted as 1, 0 and 1/2 — where the possible was defined by his pupil Tarski in 1921. {8} Once the possible was denoted as 1/2 the way was open to many-valued logics, and such logics are indeed used to solve problems associated with independence, non-contradiction and completeness of axioms.

Much richer than these is the practical or deontic (normative) logic developed by G.H. von Wright, {9} who recognized two aspects of knowledge: theoretical and practical. The last he divided into logics of values, names and imperatives. Four modal categories applied to each of these three logics — truth (necessary, possible or contingent), knowledge (verified, falsified or undecided), obligation (compulsory, permitted, forbidden) and existence (relation of modal logic to quantification). The matter is technical, of course, and contested, but has been applied to legal issues.

If nonstandard logics like modal and the many-valued escape the restrictions of standard logic, are they more widely applicable? Surprisingly, the answer is no. They apply more cogently in certain specific areas (in quantum mechanics, in computer circuitry, or the problems of relay and switching circuits in electrical theory) but lose their universality because the two-valued tautologies no longer apply. {10} Many workers regard them as degraded systems, no more than interesting novelties.

Role of Logic and Its Limitations

So where does logic fit into philosophy? Mostly as a means to an end, i.e. to thinking clearly, and expressing that thought succinctly. The psychologist Jean Piaget certainly regarded thinking as secondary to the actions of the intelligence. For him (as it was for Cassirer) logic was a science of pure forms, structures simply representing the processes of thought. Logic was too narrow, arid and mechanical to properly represent human thought processes. René Poirier argued for an organic logic where modalities operated on various levels: symbolism, experience and mental certitude. Symbolic logic was only the syntactical manipulation of signs empty of content. Logic should start further down, thought Petre Botezatu, by studying the structures of thoughts themselves. Above all, thought Anton Dumitru, we must know directly the fundamental ideas and principles of logic. {11}

Reference and Naming

How could we know these principles? Moreover, to take something more straightforward, how do we even make reference to objects that form the subjects of simple two-value propositions? There is more to it than pointing and uttering a name. Many words denote things abstract, or never seen, or possibly not even existing. And the matter is crucial. However logically transparent the sentences, they will not make sense unless they hook up to the world beyond. How is this done? What is the answer to literary critics like Derrida who claim that words point not to objects, but to other words, and these to yet more, and so on endlessly?

At their simplest, prior to their use in propositions and sentences, words refer to things. But do they need to have meanings, or can they simply denote things — i.e. do they describe or simply point? Russell opted for both {12}: his theory of descriptions combined sense and reference: F denotes x iff F applies to x. (Additionally, there was a special category of logically proper names that denoted simple objects, these simple objects being the results of direct acquaintance, i.e. of sense impressions.) But in general Russell's ordinary proper names were identified with description, even though different speakers might carry around different descriptions in their heads. And where the simple object denoted did not exist (the King of France) then matters could be arranged so that one at least of the propositions was false.

We can therefore speak meaningfully of things that do not exist. The sentences are simply false, as would be those employing a fiction like Sherlock Holmes. But since there is a distinction between Sherlock Holmes was a detective, and Sherlock Holmes was a woman, subsequent philosophers have often preferred to use a formal language in a domain of fictional entities: the so-called free logics. Many things are not determinable in fiction, moreover (did Sherlock Holmes have an aunt in Leamington Spa?) so that these logics are often multi-value. {13}

Since a name might not be acceptable to everyone, or might conjure up very different descriptions in different minds, Strawson and Searle suggested that name and reference should be established by a cluster of descriptions, most though not all operating at any one time. {14} But what do we understand by: The man who murdered Sadat was insane? — insane because he murdered Sadat, or insane anyway? To distinguish, Donnellan {15} used the terms attributive and referential respectively.

Saul Kripke is critical of description. Descriptions may fix a reference, but do not give it sense. Some may even turn out later to be false. To preserve a reference from these mishaps what we should employ are rigid designators, entities which have the same reference in all possible worlds. {16} Remember, says Kripke, that references are often borrowed without being understood, and that we may have only the haziest notion of Cicero and the Cataline plot but still wish to refer to them. Let us therefore adopt a causal theory of reference. {17} A name is introduced by dubbing: ostension. People not present at the dubbing can pick up the name later, and in turn pass it on to others: the reference chains are called designating or d-chains. The name thus becomes independent of its first use or user, allows substitution by other words, and needs no elaborate descriptions. No doubt this mimics what actually happens in the world. But each speaker is now responsible for the reference: his meanings, and associations in using the name can all be referred back and checked. Words other than names are more difficult: they require reference fixing and theories for reference borrowing, which is where a good deal of contemporary work continues.

References

1. See any elementary text: e.g. Logic, traditional entry in Ted Honderich's The Oxford Companion to Philosophy (1995). Also chapter 8 of F.R. Palmer's Semantics (1976, 1981) for a non-technical introduction.
2. Chapter 8 in Stephen Barker's The Elements of Logic (1980) and Susan Haack's Philosophy of Logics (1980).
3. Any elementary philosophy text book: e.g. Chapter 2 of A.C. Grayling's An Introduction to Philosophical Logic (1982).
4. Any textbook. Particularly useful are the Logic, modern entry in Ted Honderich 1995, Chapters 4-6 of Susan Haack 1980, Barker 1980, and Chapters 5 and 6 of Roger Scruton's Modern Philosophy (1996). Also Chapters 4 and 7 of Anders Wedberg's A History of Philosophy: Volume 3: From Bolzano to Wittgenstein (1984) and Chapter 2 of Sybil Wolfram's Philosophical Logic: An Introduction (1989).
5. pp. 157-160 of Grayling 1982.
6. Chapter 53 in Anton Dumitru's History of Logic (1977).
7. pp. 156-62 in Haack 1980.
8. p.146 in Dumitru 1977.
9. pp. 171-3 in Dumitru 1977.
10. pp. 35-8 of Haack 1980, and Chapter 53 in Dumitru 1977.
11. Chapter 57 in Dumitru 1977.
12. Chapter 4 of Michael Devitt and Kim Sterelny's Language and Reality: An Introduction to the Philosophy of Language (1987), and Chapter 5 of Haack 1980 for this and references 13-15.
13. Chapter 5 of Devitt and Sterelny 1987, and Chapter 5 of Haack 1980.
14. Chapter 3 of Devitt and Sterelny 1987.
15. pp. 126-132 in Mark Platt's Ways of Meaning: An Introduction to the Philosophy of Language. (1979), Note 9.1 in Simon Blackburn's Spreading the Word: Groundings in Philosophy of Language (1984) and chapter 5 of Devitt and Sterelny 1987.
16. Chapter 3 of Devitt and Sterelny 1987.
17. Chapter 9 of Simon Blackburn's 1984.  

Internet Resources

1. Truth. Jan. 2004. http://en.wikipedia.org/wiki/Truth. Wikipedia's introduction to four concepts of truth.
2. Origins of Analytical Philosophy. http://www.people.virginia.edu/~msg6m/ dummett.html. Review by Mitchell Green of Dummett's 1994 book.
3. Analytic Philosophy.http://en.wikipedia.org/wiki/Analytic_philosophy. Wikpedia's entry: brief but with links.
4. Classical Logic. Stewart Shapiro. Oct. 2000. http://plato.stanford.edu/entries/logic-classical/. Straightforward if technical account.
5. Gottlog Frege. Jan. 2004. http://en.wikipedia.org/wiki/Frege. Introduction, with in-text links.
6. Friedrich Ludwig Gottlob Frege. John J O'Connor and Edmund F Robertson. Jan. 2004. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Frege.html. Introduction on very extensive site devoted to mathematicians.
7. Gottlob Frege (1848-1925). Kevin C. Klement. 2001. http://www.utm.edu/research/iep/f/frege.htm. Simple but thorough introduction, with extensive references.
8. Gottlob Frege. Edward N. Zalta. Jan. 2002. http://plato.stanford.edu/entries/frege/ Extended entry in the Stanford Encyclopedia of Philosophy.
9. Bertrand Russell Gallery. Paul Barrette. Jul. 2002. http://www.humanities.mcmaster.ca/~bertrand/. Very full site, with much background material.
10. Bertrand Russell. May 2003. http://plato.stanford.edu/entries/russell/. Covers all Russell's activities and includes extensive bibliography and site listings.
11 . Modal Logic. James Garson. Oct. 2003. http://plato.stanford.edu/entries/logic-modal/. Good introduction, but technical.
12. Many-Valued Logic. Siegfried Gottwald. Apr. 2000. http://plato.stanford.edu/entries/
logic-manyvalued/
. Introduction to the many types: technical entry in the Stanford Encyclopedia of Philosophy.
13. Resources for Many-Valued Logic. Reiner Hähnle. Reiner Hähnle Aug. 200. http://www.cs.chalmers.se/~reiner/mvl-web/ NNA. Some site listings, but mostly books.
14. Georg Henrik von Wright (1916-2003). 2003. http://www.kirjasto.sci.fi/gwright.htm. Short article on Books and Writers site.
15. Georg Henrik von Wright. Dec. 2003. http://en.wikipedia.org/wiki/Georg_Henrik_von_Wright. Short article with in-ext links.
16. What is Analytical Philosophy? Cesare Cozzo. http://w3.uniroma1.it/cog-found/what%20is%20analytical%20philosophy.pdf. Short article touching on von Wright, Dummett and others.
17. René Poirier http://www.mises.org/rothbard/toward.pdf. Argues that mathematical logic is subsidiary to verbal logic.
18. Analysis of Ordinary language. Garth Kemeling. Oct. 2001. http://www.philosophypages.com/hy/6u.htm. Brief entries on Ryle, Austin and Strawson.
19. Present King of France. Nov. 2003. http://en.wikipedia.org/wiki/ Present_King_of_France. Interpretations by Russell, Strawson and Donnellan.
20. Conversation with John Searle. Harry Kreisler. 1999. http://globetrotter.berkeley.edu/people/Searle/searle-con0.html. Easy introduction to Searle and his outlook.
21. Searle, John. http://www.artsci.wustl.edu/~philos/MindDict/searle.html. Brief introductions to main ideas.
22. John Searle. Dec. 2003. http://en.wikipedia.org/wiki/John_Searle. Brief Wikipedia article.
23. Minds, brains and programs. John Searle. 1980. http://members.aol.com/NeoNoetics/ MindsBrainsPrograms.html . Introduction to Searle's Chinese Room argument.
24. The Chinese Room Argument. Larry Hauser 2001. http://www.utm.edu/research/iep/c/chineser.htm The Chinese Room argument in more detail.
25. John Searle. The Philosophical Journey: An Interactive Approach. William Lawford. 2001. http://www.mhhe.com/mayfieldpub/lawhead/
chapter3/john_searle.htm
. An extended series of articles, somewhat technical.
26. Grice, Herbert Paul. Christopher Gawker. http://www.artsci.wustl.edu/~philos/ MindDict/grice.html. Brief introduction to his ideas.
27. Paul Grice. Nov. 2003. http://en.wikipedia.org/wiki/Grice. Brief Wikipedia entry.
28. Grice, H. Paul. Kent Bach. http://online.sfsu.edu/~kbach/grice.htm. Entry in MIT Encyclopaedia of the Cognitive Sciences.
29. Saul Kripke. Jan 2004. http://en.wikipedia.org/wiki/Saul_Kripke. Wikipedia entry with in-text links.
30. Sorites. http://www.sorites.org . Electronic magazine of analytical philosophy: technical.
31. Electronic Journal of Analytical Philosophy. http://ejap.louisiana.edu/. Archived 1993-98 papers.


      C. John Holcombe   |  About the Author    | ©     2007 2012 2013.   Material can be freely used for non-commercial purposes if cited in the usual way.