I am going to post an article tomorrow that I worked on a few years ago on the differences between the traditional Aristotelian system of logic that I use in my logic texts (and which is still used at many Catholic schools) and the modern system that is emphasized in most college logic courses today.
In the meantime, here is a response I wrote to William Randolf Brafford, who wrote an article at First Thoughts as a response to Peter Kreeft who has an article in the newest issue of Touchstone Magazine called, "Clashing Symbols: The Loss of Aristotelian Logic; the Social, Moral & Sexual Consequences." I haven't read the Kreeft article because it is available only to subscribers and I haven't gotten my magazine yet.
It would probably help if Mr. Brafford read some of the literature from the Aristotelian perspective on this, particularly the work of Henry Veatch. I recommend Two Logics, Intentional Logic, and Logic as a Human Instrument.
The most obvious difference is that symbolic logic assumes without warrant that existential import for particular (but not universal) propositions. The result of just these two assumptions is that five of the traditional 19 valid categorical syllogism forms are rendered invalid, and the "square of opposition" becomes the "cross of opposition," since three of the four kinds of opposition are eliminated.
This assumption is embedded in Venn Diagrams commonly used in logic courses.
Another big difference is that, in symbolic logic, all statements are considered truth conditional when they clearly are not. This was one of the reasons why Wittgenstein, who devised the system of truth tables, later repudiated them.
In fact, two of the principals involved in the Principia Mathematica (from which virtually all modern symbolic logic derives) later repudiated the project--Wittgenstein and Alfred North Whitehead. Russell stayed with it, and was not nearly so sanguine as Brafford about the two systems being consistent.
"I conclude that the Aristotelian doctrines with which we have been concerned in this chapter are wholly false," he says, in his history of Western philosophy, "with the exception of the formal theory of the syllogism, which is unimportant."
On the other side, Jacques Maritain said, "Logistics [which is what he calls modern symbolic logic] and logic remain separate disciplines, entirely foreign to one another."
Brafford says that it must have been possible for nominalists to use Aristotelian logic since nominalism goes back to the 1300s and modern logic does come along until about the turn of the 20th century. That ignores the fact that there was quite a bit of discomfort with a system of logic that philosophers knew was based on Aristotelian metaphysics. The problem was there simply wasn't any alternative until Frege and Boole began to develop the rudiments of the modern system, a system that was brought to fruition by Bertrand Russell, Whitehead, and Wittgenstein. This was one of the reasons that the then mostly logical positivist (and by implication nominalist) philosophical establishment immediately seized upon it.
To borrow a phrase from Richard Dawkins, the modern system made it possible to be an intellectually satisfied positivist.
William Barrett has perhaps the best popular account of how all this went down in his Illusion of Technique.
Veatch challenged the academic establishment on its almost exclusive emphasis on symbolic logic and as far as I can find in the journals, no one ever responded to him. I asked Kreeft about this one time and he said that, to his knowledge, no one ever did.