Timeline for What are free variables and what does it mean for a statement to contain one?
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| Aug 11, 2013 at 16:38 | history | edited | Mozibur Ullah | CC BY-SA 3.0 |
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| Aug 11, 2013 at 16:24 | comment | added | Thomas Klimpel | In a certain sense, universal algebra is a subset of first order logic. The term algebra also occurs in first order logic in the proof of Henkin's theorem, but it is called term-structure/term-interpretation there. I just wanted to give a nice example where free variables are useful by themselves, and the information of the wikipedia page at least wasn't wrong (even so many useful details are missing), so I used it as reference. You are right that 'freely generated' & 'free variables' are different, but free variable still recalls/implies this "generic element" feeling. | |
| Aug 11, 2013 at 15:48 | history | edited | Mozibur Ullah | CC BY-SA 3.0 |
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| Aug 11, 2013 at 15:37 | history | edited | Mozibur Ullah | CC BY-SA 3.0 |
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| Aug 11, 2013 at 15:18 | comment | added | Mozibur Ullah | It seems to me that the use of free in 'freely generated' & 'free variables' are somewhat different, or have I misconstrued the article? | |
| Aug 11, 2013 at 15:14 | comment | added | Mozibur Ullah | @klimpel:Well, I'm certainly not correct about free variables in the propositional calculus - they don't have variables, never mind free ones! Now, according to your linked article, Term algebras are freely generated structures for a given signature, and the nearest equivalent in logic is a Herbrand universe - but this consists simply of all ground terms, that is all terms without any free variables in them. It doesn't say, but it looks likely that any quotient of the Herbrand universe doesn't have free variables either. | |
| Aug 11, 2013 at 10:04 | comment | added | Thomas Klimpel | You simply seem not to be at ease with free variables (in first-order logic), which puts you in an unfortunate position to answer this question. The term algebra "T(X)" and the related factor algebra "T(X)/Gl_X(K)" are examples from universal algebra where free variables are useful. Both the term algebra "T(X)" and "T(X)/Gl_X(K)" (the free algebra of the class of algebras "K" with the generating set "X" also noted as "F_K(X)") are free algebras. | |
| Aug 10, 2013 at 18:47 | comment | added | Mozibur Ullah | Well, the OP asked 'A good answer would explain the difference between (4) and (5)', with (4) being free, and (5) being bound. But you're right I've placed more weight on bound variables, and nor have I discussed assigning meaning to free variables, I only mentioned in passing that they're used in the propositional calculus. To discuss meaning for free variables there would mean discussing their model theory or truth tables. Are you suggesting that free variables are also useful in first-order logic? | |
| Aug 10, 2013 at 17:07 | comment | added | Thomas Klimpel | -1 Even after reading your answer multiple times, it's still unclear to me how you assign meaning to free variables. You make it pretty clear that bound variables are relatively unproblematic to you, but you fail to make it clear why free variables are useful at all, not even talking about how to assign meaning to them. | |
| Aug 10, 2013 at 0:55 | history | answered | Mozibur Ullah | CC BY-SA 3.0 |