First-Order Necessity and Validity First-Order Necessity and - PowerPoint PPT Presentation

First-Order Necessity and Validity First-Order Necessity and Validity Mark Criley IWU “BackOf,” “Between,” “Small,” etc. predicates other than “=”. necessity does. But FO-necessity pays attention to a little bit less than Logical First Order Attention First-Order Necessity and Validity Mark Criley IWU Unlike the truth table. and variables. Unlike the truth table. TT-necessity. does. Mark Criley First Order Attention FO-NPEC, etc. pays attention to a little bit more than TT-NPEC Mark Criley IWU arguments rather than sentences . IWU 10/27/2017 Mark Criley IWU First-Order Necessity and Validity Intro We have added some new pieces to our language: Quantifjers and variables. These new pieces are going to add a new layer of NPEC : N ecessity, P ossibility, E quivalence, C onsequence. This is going to be called “First-Order” necessity, possibility, etc. That’s “FO” necessity, possibility, etc., for short. The book refers to FO-necessities as FO validities . Ugh. I avoid that, just because I want to reserve the term validity for First-Order Necessity and Validity • It pays attention to the meanings of the connectives, just like • FO-necessity doesn’t pay attention to the meaning of any • But it also pays attention to the meanings of the quantifjers • So FO-necessity doesn’t understand the meaning of “Cube,” • And it pays attention to one special predicate: Identity (=).

Euler Diagrammin’ TT-necessary? Mark Criley IWU any predicates other than ‘=’.) attention to but that FO necessity doesn’t. (It doesn’t have Euler Diagrammin’ First-Order Necessity and Validity Mark Criley IWU But we saw that it was logically necessary. (Why?) We saw that this wasn’t TT-necessary. (Why not?) Exercise 10.1.1: Remember this sentence from last class? First-Order Necessity and Validity Remember that necessity (and equivalence and consequence) will Mark Criley IWU Can you think of an example of an FO-necessity that isn’t than TT did, expand when we add new things to force sentences to be true or false. forcing when we move from TT to FO, sentence that is FO-necessary but not TT-necessary. First-Order Necessity and Validity Mark Criley IWU First-Order Necessity and Validity Not all FO-necessities are TT-necessary. For that to be true, there must be at least one example of a • Because all of the connectives are still around to do the • All TT-necessities are FO-necessary. • But because FO has some more stufg around to do forcing • Not all FO-necessities are TT-necessary. ∀ x (x=x) Yes, ∀ x (x=x) is FO-necessary. (Why?) • It doesn’t contain anything that Logical necessity pays • Is it FO-necessary?

Euler Diagrammin’ Mark Criley IWU Mark Criley IWU FO-necessity doesn’t pay attention to. Examples will have to exploit the meanings of some predicates that Hint First-Order Necessity and Validity Mark Criley IWU not FO-necessary? Can you think of an example of a logical necessity that is not FO-necessary. If that is true, then there must be some logical necessity that is Not all logical necessities are FO-necessary. First-Order Necessity and Validity Equivalence) What is the relationship between logical necessity and fjrst order So here’s the Euler Diagram for Necessity (and Consequence and Euler Diagram First-Order Necessity and Validity Mark Criley IWU meaning of predicates like “Cube,” “Large,” etc. ) attention to stufg that the FO level doesn’t (namely, the quantifjers, variables, names, identity), that forces sentences to be true at the FO level (connectives, necessity? First-Order Necessity and Validity • Since logical necessity pays attention to all of the same stufg • All FO-necessities are logically necessary. • But since there is some stufg that logical necessity pays • Not all logical necessities are FO-necessary.

Logically Necessary, but Not FO-Necessary First-Order Necessity and Validity we’re going to need a strategy to get ourselves to ignore the meanings of the predicates that FO thinking can’t understand. Strategy: We’ll just replace all of the predicates FO thinking can’t understand with nonsense predicates, ones that we can’t understand. Mark Criley IWU We just have to make sure that we replace the same sensible Here’s an example: predicate with the same nonsense predicate everywhere it appears. to replace it with that everywhere. And we can’t reuse the same nonsense predicate for a difgerent sensible predicate. we can’t turn around and say it also means “Large”. Mark Criley IWU In order to get ourselves to think in a “First Order” frame of mind, Strategy! First-Order Necessity and Validity Mark Criley IWU That sentence is logically necessary. (Why?) But it isn’t FO-necessary. (Why not?) Mark Criley IWU First-Order Necessity and Validity But not just any example that uses a predicate that FO-necessity doesn’t understand will do the trick. For instance, FO-necessity doesn’t pay any attention to the meaning of “Cube”. But the following sentence is FO-necessary: You don’t have to know anything about what “Cube” means to know that that sentence is true. Same with this one: First-Order Necessity and Validity • ∀ x ((Cube(x) ∧ Small(x)) → Cube(x)) • ∀ x (Cube(x) → ¬ Dodec(x)) • ∃ x Cube(x) → ¬∀ x ¬ Cube(x) • If we replace “Cube” with “Caburble” in one place, we have • If we have already declared that “Caburble” means “Cube”,

Apply this technique to the FO-necessary sentences from before: But even so, without knowing anything about schwenky stufg or Mark Criley IWU then the sentence is not FO-necessary. you have replaced its sensible predicates with nonsense ones, FO-necessary. replacing its sensible with nonsense ones , then the sentence is First-Order Necessity and Validity Mark Criley IWU That has to be true, right? All schwenky caburbles are caburbles. caburbles, I still know this: First-Order Necessity and Validity Mark Criley IWU I have no idea! I don’t know! I don’t want to know! What do “Caburble” and “Schwenky” mean? What’s A “Schwenky Caburble?” First-Order Necessity and Validity Mark Criley IWU Now the earlier sentence becomes “Schwenky”. replace the sensible predicate “Small” with the nonsense predicate Replace the sensible predicate “Cube” with “Caburble”. And First-Order Necessity and Validity • ∀ x ((Cube(x) ∧ Small(x)) → Cube(x)) • ∀ x ((Caburble(x) ∧ Schwenky(x)) → Caburble(x)) • ∀ x ((Caburble(x) ∧ Schwenky(x)) → Caburble(x)) • If you can determine that a sentence must be true even after • If you can’t determine whether a sentence must be true after

Another example Now let’s apply this technique to another sentence we encountered Mark Criley IWU And that means that it is not FO-necessary. depends on the meanings of its predicates. It depends on what “Curdiddle” and “Doodiddle” mean. I have no idea. Is it necessary? Does that have to be true? Can No Curdiddles Doodiddle? First-Order Necessity and Validity Mark Criley IWU Does that have to be true? tired of Caburbles, though.) Replace its sensible predicates with nonsense ones. (I’m getting First-Order Necessity and Validity First-Order Necessity and Validity This becomes: Now, what about this sentence: earlier: First-Order Necessity and Validity Mark Criley IWU So the original sentence is FO-necessary. That’s just got to be true, no matter what “caburble” means. Mark Criley IWU But I do know this: What’s a caburble? I still have no idea! • ∃ x Cube(x) → ¬∀ x ¬ Cube(x) • ∀ x (Cube(x) → ¬ Dodec(x)) • ∃ x Caburble(x) → ¬∀ x ¬ Caburble(x) • If something caburbles, then not everything fails to caburble. • ∀ x (Curdiddle(x) → ¬ Doodiddle(x)) • ∀ x (Cube(x) → ¬ Dodec(x)) • ∀ x (Curdiddle(x) → ¬ Doodiddle(x)) That means that the necessity of ∀ x (Cube(x) → ¬ Dodec(x))

FO-Validity sensible predicate everywhere with the same nonsense predicate. Mark Criley IWU At least one thing doesn’t scebby Bill. Nothing is a slolly. If everything scebbies Bill, then there is at least one slolly. First-Order Necessity and Validity Mark Criley IWU What do you think? FO-valid. (It might be logically valid, or TW-valid.) depend on the meanings of its predicates.) original sensible argument is FO-valid. (Its validity doesn’t First-Order Necessity and Validity We have just used the technique to see whether a sentence was Can we determine whether this argument is valid? Mark Criley IWU Do the nonsense replacement, making sure to replace the same First-Order Necessity and Validity FO-necessary. We can use the same technique to determine whether an argument is FO-valid or not. Consider the following argument: Is this argument FO-Valid? Mark Criley IWU First-Order Necessity and Validity ∀ x SameSize(x,b) → ∃ x Small(x) ∀ x ¬ Small(x) ∃ x ¬ SameSize(x,b) ∀ x Scebbies(x,b) → ∃ x Slolly(x) ∀ x SameSize(x,b) → ∃ x Small(x) ∀ x ¬ Slolly(x) ∃ x ¬ Scebbies(x,b) ∀ x ¬ Small(x) ∃ x ¬ SameSize(x,b) ∀ x Scebbies(x,b) → ∃ x Slolly(x) ∀ x ¬ Slolly(x) ∀ x Scebbies(x,b) → ∃ x Slolly(x) ∃ x ¬ Scebbies(x,b) ∀ x ¬ Slolly(x) ∃ x ¬ Scebbies(x,b) • If we can determine whether this argument is valid, then the • If we can’t , then the original sensible argument is not