Timeline for Why does soundness imply consistency?
Current License: CC BY-SA 3.0
11 events
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| Feb 19, 2018 at 22:46 | comment | added | Derek Elkins left SE | My personal recommendation is not to be too philosophical about it and logic in general. I'm not saying ignore philosophical concerns entirely, but you can just recognize semantics as a useful tool for figuring out whether some statements are or are not provable, or, even more practically, one of the applications of formal logic. Interpreting syntax into mathematical objects is a useful handle to have on those mathematical objects. Semantics doesn't have to be about defining what logic "means" or what "truth" "is". | |
| Feb 19, 2018 at 22:34 | comment | added | Derek Elkins left SE | That's where ones philosophy of math may come in. Platonists believe the truth of the set theoretic statements (say) are just knowable without needing recourse to logic. Arguably for them, the set theoretic expressions are the meaning of logical formulas. Formalists will use syntactic, rather than semantic approaches, i.e. "true"="provable". Constructivists have a different notion of "truth" and the more computation-oriented sub-school of them would witness the "truth" via a program. | |
| Feb 19, 2018 at 22:23 | comment | added | Charlie Parker | So what do they do to get around that issue? | |
| Feb 19, 2018 at 22:20 | comment | added | Derek Elkins left SE | This typical set theoretic semantics is not the most satisfying thing because ZFC set theory is based on classical first-order logic. $[\![\varphi]\!]\cap(D\setminus[\![\varphi]\!])=\varnothing$ unravels to $\forall x\in D.x\in[\![\varphi]\!]\land x\notin[\![\varphi]\!]\iff\bot$, which, if we were taking a philosophical perspective, is essentially the thing we were trying to figure out in the first place. | |
| Feb 19, 2018 at 22:20 | comment | added | Derek Elkins left SE | Presumably, in your example $D$ would be $\mathbb R$, the real numbers. $[\![x\text{ is rational}]\!]$ would presumably get mapped to the set $\{x\in D^1\mid \exists m,n\in\mathbb N.x=m/n\}$. $[\![2\text{ is rational}]\!]$ would become $\{x\in D^0\mid\exists m,n\in\mathbb N.[\![2]\!]=m/n\}$ where $D^0$ is the set of $0$-tuples of $D$, $[\![2]\!]$ is the real number corresponding to the numeral $2$. This last set expression will either be $D^0$ or $\varnothing$. | |
| Feb 19, 2018 at 22:15 | vote | accept | Charlie Parker | ||
| Feb 19, 2018 at 18:57 | comment | added | Charlie Parker | Derek, if you have time do you mind perhaps making a concrete example of the domain and how it indeed leads to the empty set? (I am also happy to make a new question if you prefer) I had in mind an example but didn't know how to complete it. The example was showing that 2 is rational AND 2 is irrational lead to the empty set (or with $\sqrt 2$). I had in mind D is tuple of integers. Then $[\![ 2 \text{ is rational} ]\!] $ mapped to $(2,1)$ but I wasn't sure what $[\![ 2 \text{ is irrational }]\!] $ mapped to. Do you know how to finish this example in a sensible way?Or point me to an example? | |
| Feb 19, 2018 at 18:42 | comment | added | Derek Elkins left SE | That's the typical semantics for classical propositional logic, which can be viewed as a special case of classical first-order logic where all predicates are nullary. The Boolean "truth" values do indeed map to the empty set and the singleton set in this view. One of the not-so-blatant points of my first paragraph was to suggest that different logics have different notions of semantics. Even for a fixed logic, there are multiple possible semantics that could be given for it. There's a reason I say "the typical semantics" and not just "the semantics". | |
| Feb 19, 2018 at 18:32 | comment | added | Charlie Parker | interesting, I always thought that "truth" meant that we mapped a statement to boolean values 0 and 1. But it seems that was incorrect. I guess we can sort of fix my wrong model by having empty set map to 0 and non-empty to 1. Otherwise, I'm not sure how one is able to re-write your proof in "my definition of truth as the function mapping to 1 or 0". | |
| Feb 19, 2018 at 14:33 | comment | added | Charlie Parker | feel free to recommend me a book on logic, I don't really know what a good reference is, especially for beginner in logic. The funny thing is that I have take algorithms and real analysis, so I've never actually thought rigorously about the logic itself. | |
| Feb 19, 2018 at 7:53 | history | answered | Derek Elkins left SE | CC BY-SA 3.0 |