J Marcos
Reputation
429
Top tag
Next privilege 500 Rep.
Access review queues
 16h comment How many ternary functionally complete connectives are there? You might want to have a look at this. Dec 5 comment Give a good reason to define a function from A to B as a triple (F, A, B) rather than a functional set of pairs with domain A and image included in B. The example may be generalized by noticing the coincidence between the graphs (ordered pairs) of the identity function $Id:C\to C$ and of the inclusion function $Inc:C\to D$, where $C\subseteq D$, both agreeing to map each element of the domain into itself. In contrast, when you consider the alternative definition of a function as a triple, there is no mistake about the codomain. In addition, if you admit non-total functions, information about the full domains cannot be extracted from their graphs. On the definition as a triple, there is also no mistake about the intended domain. Oct 19 comment Understanding Structural Induction Structural induction is just another name for well-founded induction (a.k.a. Noetherian induction), that is, induction on well-founded structures. Given dependent choice, a well-founded relation amounts precisely to a relation that does not allow for countable infinite descending chains, if you proceed top-down. Contrapositively, for this very reason, every object constructed bottom-up by well-founded (structural) recursion must be constructed in finitely many steps. Oct 9 awarded Yearling Oct 9 answered How to proofs work in three-valued Kleene logic? Apr 27 comment role of definitions in proofs This entry will also shed some light on the topic. Jan 26 comment Principle of explosion: Other arguments? As a historical complement to the comment by @RoryDaulton, the argument quoted from Wikipedia, using disjunction introduction and disjunctive syllogism, is essentially the one presented by Lewis & Langford in their Symbolic Logic [1932]. The second (quite distinct) argument for explosion to be found at the same Wikipedia entry, using contraposition (and double negation elimination), was offered in the 60s by Popper, in his Conjectures and Refutations. Jan 13 comment The definition of term in Enderton's Logic book @MauroALLEGRANZA Three things. First: terms are built-up from constant symbols, variables "and" other terms, via the use of functional symbols (you can't scape formulating Enderton's definition recursively). Second: your set TERM satisfies 1 "and" 2 "and" 3 ("and" the condition that it should be the smallest set simultaneously satisfying these three conditions). Third: Enderton's bottom-up definition of TERM via closure is perfectly formal; your top-down definition is in fact equivalent to it. Jan 13 answered The definition of term in Enderton's Logic book Jan 13 comment The definition of term in Enderton's Logic book Enderton is in fact describing a recursive construction. So, a term might also be 'built up' from other terms, as in $f(g(y),f(x,2))$. But it all starts (base step) with terms that are either constant symbols or variables. Jan 11 comment Structural Induction vs Normal (Mathematical) Induction Sect. 4.7 of Makinson's "Sets, Logic and Maths for Computing" provides a very accessible introduction to recursion and induction over well-founded sets (a.k.a. Noetherian induction). While mathematicians are happy with defining a structure as the least set closed under a certain set of constructing rules, a computer scientist is often happier to have data structures defined bottom up by the same set of rules, as this will allow her more naturally to implement operations by (structural) recursion and to perform verifications by (structural) induction. Oct 23 comment what are the Rosser Turquette axioms of Lukasiewicz 3 valued propositional logic? @DougSpoonwood I guess it all depends on what is your intention in axiomatizing a logic. In my earlier response I have pointed out that this can now be done for any finite-valued logic, using a uniform classic-like setting not unlike the one used by Rosser and Turquette, and giving rise to tableau systems with associated proof strategies and corresponding decision procedures. These systems would seem almost incomparable, in terms of usefulness, if you place them side by side traditional axiomatic systems. Oct 15 answered What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) Sep 30 awarded Explainer Sep 21 answered A knights and knave problem involving a native with a speech disorder Sep 21 revised A knights and knave problem involving a native with a speech disorder corrected spelling, corrected comment about the statement done by C Sep 21 suggested approved edit on A knights and knave problem involving a native with a speech disorder Sep 21 comment A knights and knave problem involving a native with a speech disorder Further, there is obviously a number of things that C could have meant to say, but some of them would not be enough for the kind of D to be determined. So, for deterministic solutions to be available to your problem, I understand you should postulate that the statement of the problem conveys enough information for one to uncover what kind of native D is (you might add to it the information that there was some logician present at the scene who overheard what C said and was then able to determine the kind of D). Do you accept that? If you do, the problem will turn out to be indeed solvable! Sep 21 comment A knights and knave problem involving a native with a speech disorder When you write about C that "You can't tell if he is identifying B or C as knights or knaves", just after having written that C "is saying something about B and D", I assume you meant to write about C that "You can't tell if he is identifying B or D as knights or knaves". Right? Sep 2 answered Having hard time understanding proofs by contradiction.