Truth in Logic

 

Given the premises in a logical argument, 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. 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 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.

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) that employ logical constants like not and or, and and and if - then.

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. 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))).

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. 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.

How could we know these principles? 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: 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.

There is therefore no simple opposition of fact (science, mathematics and logic) to imagination (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.

A much fuller article can be found on TextEtc.

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Truth in Logic

 

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