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I have read Wikipedia's term logic entry, and the quote by Gareth Evans in the Revival section that's supposed to argue for term logic's advantages over predicate logic:

"I come to semantic investigations with a preference for homophonic theories; theories which try to take serious account of the syntactic and semantic devices which actually exist in the language ...I would prefer [such] a theory ... over a theory which is only able to deal with [sentences of the form "all A's are B's"] by "discovering" hidden logical constants ... The objection would not be that such [Fregean] truth conditions are not correct, but that, in a sense which we would all dearly love to have more exactly explained, the syntactic shape of the sentence is treated as so much misleading surface structure" (Evans 1977)

However, I don't understand those arguments at all. Isn't term logic merely a different syntax for a very restricted subset of predicate logic?

Instead of writing

∀x philosopher(x) ⟶ mortal(x)

we would use a shorthand like

philosopher ⊂ mortal

(or whatever the actual syntax is), and similarly for the rest of the four kinds of propositions.

Is this just a question of shorter syntax?

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    Evans's passage is more of a hope/call for extending term logic to something as expressive as predicate calculus, but with more natural devices in the "homophonic" spirit of the syllogistic. Such extensions have eventually been offered, with plurally referring noun phrases and quantifiers modifying them rather than binding variables, see e.g. Ben-Yami, Logic & Natural Language, ch. 7. – Conifold Sep 10 at 4:46
  • In fact, Sommers was working on an extension of Aristotle's term logic to relational reasoning during 1970-s and published his system in Logic of Natural Language (1982). Englebretsen's book is a more accessible introduction, there is also online tutorial. Sommers's system is as expressive as predicate calculus, and "more elegant" according to Ben-Yami, although he questions that it is necessarily "more homophonic". – Conifold Sep 22 at 23:21
  • @Conifold Thanks for the links! I skimmed through the tutorial. As far as I can tell, it's a convenient notation for Aristotle's logic, that removes the need to memorize things, but I don't really see how it would make TL equivalent to FOL. Wouldn't you need something like variables for that? – MaxB Oct 2 at 22:49
  • Variables and functions of them is what predicate calculus uses, natural language does not. It uses plurally referring names instead and attaches quantifiers to them. The difference between Aristotle and Sommers is that the latter can represent relations (click on Newer Post at the bottom of the tutorial to see parts II-V). The calculus of +/- signs then allows applying monotonicity to make relational inferences. It can be shown that the resulting inferential system is intertranslatable with predicate calculus, see Englebretsen. – Conifold Oct 3 at 6:15
  • @Conifold thanks again! Does SETL distinguish between ∀boy ∃girl loves(boy, girl) and ∃girl ∀boy loves(boy, girl) ? Looking at the examples, it would seem SETL encodes both as -boy + (loves + girl). Are there distinct expressions for them? – MaxB Oct 3 at 22:13
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Gareth Evans is arguing that Aristotelean logic is closer to natural language usage and as such introduces fewer unfamiliar logical devices and has fewer counterintuitive features. This is true, but the vast majority of logicians consider this to be a price worth paying to have a much more powerful and expressive logic. Natural languages such as English have evolved to allow people to express common or garden thoughts, but they are not well equipped for precisely expressing propositions and the logical relationships between them. Logicians have invented stylized formal logics for much the same reason that computer scientists have invented computer languages. You wouldn't want to try to write a complex computer program without the benefit of a precisely defined syntax and semantics, and it is highly desirable to have comparable resources available in logic.

Some of the disadvantages of Aristotelean logic are:

  1. It does not readily express multi-place predicates.
  2. It limits propositions to one quantifier only.
  3. It provides little, if any, understanding of the logical terms 'and', 'or' and 'if'.
  4. It limits arguments to two premises.
  5. It provides only limited support for arguing by reductio.

Classical first order predicate logic (FOPL) overcomes all these limitations. Some of the counterintuitive features are:

  1. Statements of the form "all A's are B's" lack existential import.
  2. Arguments with tautological conclusions are valid no matter what the premises.
  3. Arguments with inconsistent premises are valid no matter what the conclusion.

Aristotlean logic can be mapped onto a fragment of typed FOPL. This was demonstrated by John Corcoran and Timothy Smiley back in the 1970s. So in that sense you could say that Aristotelean logic is a subset of predicate logic. However, because "all A's are B's" has existential import in Aristotelean logic, the statement is false if there are no A's. To render this correctly into FOPL we can type the variable and require the type to be non-empty.

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    With existential import, isn't philosopher ⊂ mortal equivalent to (∃p philosopher(p)) ∧ (∀x philosopher(x) ⟶ mortal(x))? Why the need for typed logic? – MaxB Sep 10 at 16:58
  • Corcoran proposed the typed variable approach, though as you say, you could express existential import using an additional existential term. Strawson showed in chapter 6 of Introduction to Logical Theory that you cannot retain all the features of aristotelean logic this way. – Bumble Sep 10 at 18:53
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    Predicate logic is semidecidable, while term logic is decidable. This has to be an advantage? – MaxB Sep 12 at 14:35
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    I wouldn't call that an advantage. All it amounts to is that Aristotelean logic is a decidable fragment of FOPL. There are many decidable fragments of FOPL, but it would be odd to describe them as having an advantage over FOPL as a whole. – Bumble Sep 13 at 0:28
  • Using a computer language as an analogy for expressing anything other than a neutered symbol system is quite curious. – Charles M Saunders Sep 16 at 17:39

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