Tag: ViHart
Logic on Math StackExchange
- Can propositional functions be categorized as tautologies, contradictions or contingencies by themselves? October 4, 2023I've read that compound propositions are classified as either tautologies, contradictions or contingencies, depending on whether they are always true, always false or dependent on the variable. I am wondering whether it is possible to apply the same concept to individual propositional functions rather than compound propositions such that if a given propositional function of […]jacob78
- $n$-ary relations vs subsets and binary relations October 4, 2023A relation between sets is a subset of the Cartesian product of those sets. A subset of any set is a binary relation between the subset(correction: superset) and the set $X = \{True, False\}$, that maps to $True$, i.e. a subset is a subset of the Cartesian product of the subset(correction: superset) and the set […]setblack7
- Is there an existential version for $ \forall x (\alpha \to \beta) \to (\forall x \alpha \to \forall x \beta) $? October 4, 2023Enderton's An Mathematical Introduction to Logic presents a logical axiom on p112: $$ \forall x (\alpha \to \beta) \to (\forall x \alpha \to \forall x \beta) $$ Why is it? Is it assumed to be true, or is it proved from something else? Since it is a logical axiom, is it assumed to be true? […]Tim
- Prove ((p→q)∧q) and q are equivalent using logic laws October 4, 2023So far I have ((p→q)∧q) ≡ ((~p∨q)∧q) ≡ (~p∧q)∨(q∧q) $\; \; \; \; \;$(Distributive law) ≡ (~p∧q)∨q But then I got stuck. And I'm not sure if what I've got so far is correct. Any help is greatly appreciated!Biki
- Do these two $\forall$-introduction rules agree with each other? October 4, 2023Enderton's An Mathematical Introduction to Logic says on p112: $\alpha \to \forall x \alpha$ if $x$ does not occur free in $\alpha$. Why is the requirement of $x$ not free in $\alpha$? Ebbinghaus' Mathematical Logic says on p69: 5.5 Exercises (b4) $ \frac{\Gamma \quad \vdash \quad \phi}{\Gamma \quad \vdash \quad \forall x \phi} $ if […]Tim
- Book request: I need a book on logic that tackles truth trees in a rigorous way. October 4, 2023So I recently finished reading the Book "Introducción a la lógica moderna" by Andrés Páez which you can see here. I thought the book was good but it wasn't rigorous at all, it made many claims that it never proved. What I found really interesting about this book was the truth trees, as I've never […]zlaaemi
- Can Substitution Rule for Equality be rewritten this way? October 4, 2023p68 of Ebbinghaus' Mathematical Logic says 4.4 Substitution Rule for Equality $$\frac{\Gamma \quad \vdash \quad \phi\frac{t}{x}}{\Gamma \quad t \equiv t' \quad \vdash \quad \phi\frac{t'}{x}}$$ Can it be rewritten as: $$\begin{align*} \Gamma \quad \vdash \quad \phi\frac{t}{x} \\ \Gamma \quad \vdash \quad t \equiv t'\\ \hline \\ \Gamma \quad \vdash \quad \phi\frac{t'}{x} \\ \end{align*}$$ ? Thanks.Tim
- Is this a typo in Enderton's Logic book? October 4, 2023On p114 of Enderton's An Mathematical Introduction to Logic, Is $\forall {v_3}$ a typo? Is it to be $\forall v_2$ instead?Tim
- "Some students in this class grew up in the same town as at least two other students in this class.” October 4, 2023Some students in this class grew up in the same town as at least two other students in this class. I'm thinking that I'm required to use the following information: $T(x,y):$ student $x$ grew up in town $y$ $C(x):$ student $x$ is in this class. I think "some students" means $∃x C(x).$ But my translation […]Rose Pink
- Gödel’s Incompleteness Theorems Simple Wikipedia October 4, 2023I recently wrote the outline of the proof of Gödel’s incompleteness theorems for Simple Wikipedia. I would like to get feedback on its clarity and logical validity so that I can make further improvements. I’ve reproduced relevant sections from the Simple Wikipedia article below. Gödel’s Theorems Gödel said that every non-trivial formal system (consistent and […]Caleb
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