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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.
Sep
2
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".
Feb
9
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.
Feb
9
answered What is the “correct” reading of $\bot$?
Feb
9
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.
Feb
8
awarded  Yearling