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 Jan26 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. Jan13 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. Jan13 answered The definition of term in Enderton's Logic book Jan13 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. Jan11 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. Oct23 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. Oct15 answered What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) Sep30 awarded Explainer Sep21 answered A knights and knave problem involving a native with a speech disorder Sep21 revised A knights and knave problem involving a native with a speech disorder corrected spelling, corrected comment about the statement done by C Sep21 suggested approved edit on A knights and knave problem involving a native with a speech disorder Sep21 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! Sep21 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? Sep2 answered Having hard time understanding proofs by contradiction. Sep2 comment Proof by Contradiction with Multiple Axioms Your "second format" is a particular case of your "standard form". Note indeed that (i) $\neg Q$ is equivalent to $\top\land\neg Q$, and that (ii) $\top\Rightarrow Q$ is equivalent to $Q$. Thus, your "standard form" $((P\land\neg Q)\Rightarrow \bot)\Rightarrow(P\Rightarrow Q)$ instantiates into $((\top\land\neg Q)\Rightarrow \bot)\Rightarrow(\top\Rightarrow Q)$, and this is turn equivalent, given (i) and (ii), to $(\neg Q\Rightarrow \bot)\Rightarrow Q$, a slight variation of your "second format". Feb9 comment What is the “correct” reading of $\bot$? @MauroALLEGRANZA No, in both cases you could use it: rule $\lor$-intro allows for any formula to be introduced, be it a propositional parameter, or a nullary connective, or a complex well-formed formula with many propositional parameters and connectives of all kinds. The point in this case is that the natural deduction rule is formulated with schematic letters, and such strategy automatically embeds substitution into the formalism. Feb9 answered What is the “correct” reading of $\bot$? Feb9 comment Is Paraconsistent Negation Really Negation? @Willemien Start with $LP$, a weakly expressive well-known three-valued logic having $\{0,\frac{1}{2},1\}$ as truth-values, having $0$ as its sole undesignated value, and where $\neg x=1-x$ and $x\vee y=\max(x,y)$. Now upgrade $LP$ by adding an implication such that $x\supset y=0$ if $x\neq 0$ and $y=0$, and $x\supset y=1$ otherwise. Note now, in particular, that $A\supset A\models A\lor\neg A$ yet $\neg(A\lor\neg A)\not\models \neg(A\supset A)$. The phenomenon is not uncommon. Feb8 awarded Yearling Feb8 comment Is Paraconsistent Negation Really Negation? Saying this is 'the most basic definition of negation' is really arguable. The literature on paraconsistent logics and the literature on many-valued logics both abound with examples of negations failing such global version of contraposition, and with good reason. However, paraconsistent negations with modal semantics (such as the ones that Restall is interested upon), may well satisfy this property. In fact, any normal modal logic may be rewritten in a language that extends the classical language by the addition of a paraconsistent negation satisfying global contraposition.