Universal/Existential Instantiation in Truth Trees (Confused)
 Universal/Existential Instantiation in Truth Trees (Confused) | |
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wlcgeek
Resident Usergroup: Members Joined: May 08, 2011 Total Topics: 48 Total Posts: 106 |
 #1
Subject: Universal/Existential Instantiation in Truth Trees (Confused) Hey guys - Wasn't sure if this belonged in the HW section, but please do feel free to relocate if need be. Mine, though, is more of a general question rather than a specific HW problem. Essentially, I'm confused about how to use the Universal Instantiation (UI) and Existential Instantiation (EI) rules for truth trees. Specifically, here's what I'm not sure of: (UI) PROBLEM: ...When you are doing UI and are told that you must instantiate for EVERY name/constant in a particular open path of a tree, is that for every name/constant that occurs BELOW where the universally quantified sentence is located, ABOVE, or BOTH? E.g., suppose you have: Line 1: a Line 2: b Line 3: (Vx) Fx Line 4: c Line 5: d If we want to decompose Line 3 using UI, then do we have to instantiate using ALL the name/constants (a, b, c, d) that occur in that same open path, irregardless of whether those name/constants occur above and/or below Line 3. Or, do we only instantiate with those above (a and b) or sole those below (c and d)? (EI) PROBLEM: ...Here, a similar/parallel question occurs for me. Do we have to instantiate with a NEW name/constant in our open path based on the names/constants in existence ABOVE, BELOW, or ABOVE AND BELOW where the existentially quantified sentence occurs? If using the same 5 lines from the UI example above and replacing Line 3 with (Ex) Fx .....Do we instantiate the existential here with something other than a,b,c and d (meaning something different from above AND below the line) ....OR is it just something that has to be different from ABOVE..or BELOW the line? Does that make sense? If not, please let me know and I'd be happy to try to specify/elaborate more guys. Thanks a great deal for your help. Love and always appreciate these forums! |
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the.yangist
(P |- P) = (P ⇒ P)  Usergroup: Members Joined: Sep 30, 2007 Location: 三重, 台北市 Total Topics: 19 Total Posts: 448 |
#2
Look for the contradictory one and close your tree early if you can. Otherwise, instantiate all of them, above and below. |
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wlcgeek
Resident Usergroup: Members Joined: May 08, 2011 Total Topics: 48 Total Posts: 106 |
#3
the.yangist wrote: Look for the contradictory one and close your tree early if you can. Otherwise, instantiate all of them, above and below. Thanks, Yangist (second or third time you've helped answer a q for me here). One quick follow-up is that I'm not sure about the Existential Instantiation issue I wrote about above. Your response was (presumably) for Universal Instantiation. ...But for Existential, when we're trying to choose a NEW name/constant, does the same also apply: i.e., choosing one that is different from those ABOVE and BELOW the existentially quantified sentence? OR, must we only have to worry about a different name/constant that is either ABOVE or BELOW that existential sentence? Thanks again! |
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the.yangist
(P |- P) = (P ⇒ P) Usergroup: Members Joined: Sep 30, 2007 Location: 三重, 台北市 Total Topics: 19 Total Posts: 448 |
#4
In EI, you don't want to accidentally insantiate with a constant that is not unique, so pick one that is unique above and below. For further interest, Analytic tableau/truth tree proofs have been automated, so most questions can be answered by plugging in some sample propositions and watching the program do its work: http://www.umsu.de/logik/trees/ |
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wlcgeek
Resident Usergroup: Members Joined: May 08, 2011 Total Topics: 48 Total Posts: 106 |
#5
One follow-up question guys. My book totally sucks and is unclear.
When we apply some Vx (universal) or Ex (existential) rule for instantiation...do we apply that sucker to EVERY branch of the trip. EXAMPLE: Suppose you have: Line 1.....Vx and there are THREE open branches stemming from top of tree. When we instantiate Vx, do we have to apply that instantiation to ALL THREE open branches? Why or why not? Similarly when instantiating an Ex, do we instantiate the sentence for ALL open branches? Or does it also matter WHERE the Vx and Ex sentence is? Like does it matter if they are at top of tree or at a lower branch of the tree, etc.?
This final question should clear up all my misunderstanding!!! THANK you all SO VERY MUCH!!! |
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Willemien
Intermittently sane Usergroup: Sponsors Joined: Apr 05, 2011 Location: London (UK) Total Topics: 36 Total Posts: 572 |
#6
You better give a complete example  but in principle yes yes (there were two questions) |
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