Jon and Jerry, To specify a system of formal logic, there are many equivalent options for choosing the notation, the operators, the definitions, the axioms, and the rules of inference.
JA
One could hardly dispute the importance of implication relations like A ⇒ B. The set-theoretic analogues are subset relations like A ⊆ B...
Yes. And many logicians, such as Frege, chose IF and EVERY as the two primary operators because they correspond to his two primary rules of inference: modus ponens and universal instantiation. When you add NOT, you can define all the other operators. But other options may be preferable for other purposes. JA
implication in existential graphs is expressed in a compound form, as (A (B)), “not A without B”. For another, there is Peirce's own discovery of the amphecks, “not both” and “both not”, which appear to have a primary and and fundamental status all their own.
If you map natural languages to logic, consider the effect of various options. When you translate a text from any NL to any version of logic, the most common logical operators are AND and SOME. The third most common is NOT. EVERY, OR, and IF are less common; NOR and IFF (AKA if-and-only-if) are rare; and NAND (AKA not-both) is vary rare. But if you're designing computer hardware, NAND and NOR turn out to be very useful for reducing the total number of transistors on a chip. For his entitative graphs, Peirce started with EVERY, OR, and NOT as his basic operators. But when he gave examples of mapping English to and from the graphs, the readings were rather awkward. For example, if you want to say "A cat is on a mat" with the operators of entitative graphs, you need to say the equivalent of "It's false that for every x, for every y, either (x is not a cat) or (y is not a mat) or (x is not on y)." It took less than a year of such examples before Peirce switched to the operators SOME, AND, and NOT. That choice enabled him to say "A cat is on a mat" by a very simple existential graph: cat———on———mat, which is equivalent to "Some cat is on some mat." With Frege's choice of EVERY, IF, and NOT, "A cat is on a mat" becomes "It's false that for every x, for every y, if x is a cat, then if y is a mat, then x is not on y." Exercise for the reader: You can express everything with just one Boolean operator (NAND or NOR) plus one quantifier (SOME or EVERY). Pick one from each option and try to say "A cat is on a mat". JR
I thought ‘ergo’ was simply identical with ‘hence’, which is what follows ‘the’ and ‘but’ in the argument CP 5.189.
Natural languages have an abundance of vocabulary for describing a proof and emphasizing different aspects. When mathematicians are describing the steps of a proof, they might use the word 'therefore' when they emphasize the conclusion. But they might use the word 'hence' when they emphasize the previous step from which they derive the next one. For the final conclusion, they might say QED (Quod Erat Demonstrandum -- What was to be demonstrated). In any case, the words used to describe a proof are less important than the propositions at each step and the rules of inference that are applied to derive each step from a previous step (or steps). And by the way, Peirce's verb 'illation' is rarely, if ever, used today. The common term is 'inference'. Unfortunately, the Latin verb 'fero' (bear or carry) is a quirky verb with past participle 'latus'. In olden days, when every college graduate knew Latin, they understood that the English 'illation' was derived from Latin 'illatio' from 'illatus'. Today, logicians use the word 'inference', which is derived from the present participle 'inferens'. When quoting Peirce, it's essential to use his exact words. But when talking about illation, I recommend the modern word 'inference'. John
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