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Thesis Predicate logic was discovered, not invented

Tomasz Szewczak Fellow
· 5 replies
Logic & argumentAnalyticIndianMimamsaGottlob FregeKumārila BhaṭṭaL. E. J. BrouwerRudolf Carnap#conventionalism#logic-discovered-or-invented#realism-about-logic
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Where it stands

Is logic discovered or invented? I want to argue that predicate logic is the strongest evidence that it is discovered.

Consider the inference: every horse is an animal, so every horse's head is an animal's head. People knew for two thousand years that this was valid. Nobody could formalise it, because syllogistic logic had no way to handle relations or to put one generality inside another. When Frege introduced quantifiers and variables in 1879, he did not stipulate that the inference should count as valid. It was already valid, and everyone already knew it. What he supplied was a notation that could finally capture it.

That looks like discovery to me. A convention can be changed at will: we could drive on the left or the right. But no change of convention would make "every horse's head is an animal's head" fail to follow. The relational structure was there before anyone had words for it, much as a continent was there before it was mapped.

So: the notation was invented. What it captured was discovered. And the fact that the new system turned out to contain all the old valid syllogisms as special cases, with no loss once existential import is made explicit, is exactly what you would expect if both were partial maps of the same territory.

Objections, especially from constructivists and conventionalists, are welcome.

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5 replies

For the thesis 2

  1. Govinda Prasad Upadhyaya

    For Fellow

    The Mīmāṃsakas would side with Tomasz, by a road he might not expect. Kumārila and his school held that the relation between a word and its meaning is not created by convention at all. It is original and eternal, and human usage only reveals it. Extend that attitude from words to forms of inference and you have Tomasz's position: the structure is there to be found, not made. The schoolboys I tutor in Sanskrit find it obvious. Carnap would find it very strange.

    Helpful · 3
  2. Derek Hollis

    For Fellow

    I'm with Tomasz, and I'd compare the notation to a telescope. Galileo built his own instrument and discovered the moons of Jupiter. Nobody says the moons were invented because you needed a device to see them. Frege's notation is the telescope. Relational validity is the moons.

Against the thesis 2

  1. Ana Clara Ferreira

    Against Fellow

    As an intuitionist I'll take the bait. Brouwer thought mathematics is a construction of the mind and that logic is abstracted from mathematical practice, not prior to it. The classical predicate logic you're praising includes the law of excluded middle applied to infinite domains, which intuitionists reject. So your "same territory" is already contested: two careful traditions draw the map differently, and not because one is careless. That looks less like discovering a continent and more like choosing a coordinate system.

    Helpful · 1
    1. Tomasz Szewczak

      Fellow

      Ana Clara's point is the strongest, and I'll concede part of it: if competing logics are equally good maps, then "discovered" needs care. But I'd answer that intuitionists and classical logicians agree on the horse's head, and on a great deal of everyday reasoning besides. Their disagreement concentrates at the frontier, around the infinite. A disputed coastline does not show that the continent was invented. Carnap I still find hard to answer, mostly because his view rules out the question rather than answering it.

      Helpful · 1
  2. Oskar Nyberg

    Against Fellow

    Carnap is the classic opponent. In The Logical Syntax of Language (1934) he wrote: "In logic, there are no morals. Everyone is at liberty to build up his own logic, i.e. his own form of language, as he wishes." His principle of tolerance says we choose a logical framework for its usefulness, and the question "but which one is true?" has no content outside some framework. Your horse's head inference is valid in the framework we have chosen, which is a fine thing to be, but not evidence of a framework-independent fact.

    Helpful · 2