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Embedded Bioconditionals

Mr.lopezMr.lopez Legacy Member
in General 172 karma
Hello everyone,

how do you draw out the logic for the following statement, and take the contrapositive.

Statement: The only way to stop the baby from crying is to give her some cold medication or a nip of wiskey, but not both.

Comments

  • steve-10steve-10 Alum Member
    192 karma
    crying ---> (cold med <----> nip of whiskey)

    Negation of the necessary condition: (cold med <---> nip of whiskey) or (cold med <---> nip of whiskey); i.e., both or neither, but not just one. Let's abbreviate this as NotNC.

    Contrapositive of the whole thing: NotNC --> crying
  • Rigid DesignatorRigid Designator Alum Member
    1091 karma
    I'd do it this way. Using propositional logic notation...

    1. StopCrying --> ((ColdMed or NipWhisk) & ~(ColdMed & NipWhisk))

    I'd then read this as. If 'Baby stops crying' then 'Cold Medication' or 'Nip of Whiskey' and not both 'Cold Medication' and 'Nip of Whiskey'.

    The contrapositive of this requires the negation of the consequent of conditional 1. The consequent of conditional 1 is "((ColdMed or NipWhisk) & ~(ColdMed & NipWhisk))". There are 2 ways this can be made false.

    A. You give baby neither ColdMed nor NipWhisk. This would make 1 false because it would make (ColdMed or NipWhisk) false, since you've given baby neither.

    B. You give baby both ColdMed and NipWhisk. This would make 1 false because it would make ~(ColdMed & NipWhisk) false. All ~(ColdMed & NipWhisk) is saying is not both ColdMed and NipWhisk. Thus, giving baby both would negate this, thus negating 1.

    This would make me interpret the contrapositive of this sentence as just 'If we give baby both the medication and the whiskey, or neither of the medication and the whiskey, the baby will continue crying'.

    In a real test I wouldn't use such a detailed notation unless I had the time. But for our purposes that is an accurate representation of it in terms of logic.
  • Q.E.DQ.E.D Alum Member
    556 karma
    @"Rigid Designator"
    are you a Phil major or is it just a hobby?
  • Rigid DesignatorRigid Designator Alum Member
    1091 karma
    @"Q.E.D" I have a BA and an MA in Philosophy!
  • Mr.lopezMr.lopez Legacy Member
    172 karma
    Thanks everyone.
  • Q.E.DQ.E.D Alum Member
    556 karma
    @"Rigid Designator"
    Cheers! And welcome.

    I suppose this isn't the place to start a fight about PWS. You're lucky bc I would designate your a** to the next possible world, sonnn.

    Srsly though, good to see you here. You will enjoy the LSAT.
  • Rigid DesignatorRigid Designator Alum Member
    1091 karma
    @"Q.E.D" A year ago I would've taken on the challenge to defend my boy, Kripke. But having just spent 20,000 words arguing that A Puzzle about Belief is not really a puzzle, I'm less inclined to die on the hill of Kripke-fandom, haha.

    And I'm sitting it this December, wish me luck!
  • quinnxzhangquinnxzhang Legacy Member
    edited November 2016 611 karma
    @"Rigid Designator" said:
    A year ago I would've taken on the challenge to defend my boy, Kripke. But having just spent 20,000 words arguing that A Puzzle about Belief is not really a puzzle, I'm less inclined to die on the hill of Kripke-fandom, haha.
    Funnily enough, I feel the same way. As controversial as Naming and Necessity is, I think it gets a lot of things right, and I think the attack on descriptivism is brilliant. However, then I find out the same person who wrote Naming and Necessity also wrote Wittgenstein on Rules and Private Language, and I wonder how he could have gone so wrong.
  • wildernesswilderness Alum Member
    edited November 2016 133 karma
    @Mr.lopez :

    Method 1

    CM = Cold Medicine
    NW = Nip Whiskey
    SC = Stop Crying

    We have: SC --> (( CM v NW ) & ~ (CM & NW) )
    Negating it we do

    ~ (CM v NW) v (CM & NW)

    Which turns into:

    (~CM & ~NW) v (CM & NW) --> ~SC

    ---
    Method 2

    Alternatively, the statement Cold Medication or a Nip of Whiskey But not both could be translated as

    ~CM <-> NW (Remembering our biconditional rules)

    This is because we're really combining two things. Cold Medicine or a Nip of Whiskey and not both. Which would be:

    (1) CM v NW
    (2) ~ (CM & NW)

    But in 7sage, we learn this as:

    (1) ~CM --> NW
    (1b) ~ NW --> CM

    (2) CM --> ~NW
    (2b) NW --> ~CM

    Combining (1) and (2b), we get: ~CM <-> NW
    Combining (1b) and (2), we get: ~NW <-> CM

    Note that both of these are equivalent to each other bc they're just contrapositives. Now, the statement we have reads as:

    SC --> ~CM <-> NW

    The contrapositive would be:

    ~ (~CM <-> NW) --> ~SC

    When we say that it is not the case that a biconditional exists, we're really stating that:

    ~ ( (~CM --> NW) & (NW --> ~CM) )

    Which of course turns into

    ~ (~CM --> NW) v ~ (NW --> ~CM)

    Using Demorgan's Law, we translate to:

    (~CM & ~NW) v (NW & CM)

    Remembering the necessary condition, we add to get:

    (~CM & ~NW) v (NW & CM) --> ~SC

    Note that this is exactly the same result as step 1 above. This is an instance in which you could save a shit ton of time if you translate the thing into lawgic not using conditionals/biconditionals... but just as simple or and and statements and remembering your rules.
  • Q.E.DQ.E.D Alum Member
    edited November 2016 556 karma
    And I'm sitting it this December, wish me luck!
    Done!
    As controversial as Naming and Necessity is, I think it gets a lot of things right
    I can't not take a stance. I'm partial to Searle's cluster descriptivism, as briefly laid out in Proper Names, and I think it's significant that we can replace names with descriptions in an ideal language, as remarked by Quine. Kripke's work obviously deserves its acclaim, but if we're to admit alethic modalities into our purest language, we should define them syntactically, in classical terms. Carnap's definition of necessity as "L-truth" was right, just failed in the execution.

    We can get a pretty muscular (Quinean) account of logical truth with quantification alone, but L-truth will end up reducing to truth simpliciter without a PWS-like structure, so I buy it on the grounds that we need it to capture the irreducibly modal nature of L-truth. But in that case, it needs to account for all logically possible object-predicate combos. That would place me squarely in the combinatorialist camp with folks like Cresswell. I say our mainstream use of PWS is in error bc we countenance such things as '□p', where 'p' is an atomic wff, which is senseless on its own bc it wouldn't be true in a combinatorialist semantics aimed solely at logical consequence. Carnap's modal system made '◇~p' a theorem for that reason, but that obviously has disastrous consequences when you treat 'p' as a schematic variable for any wff, including L-true wffs. Combinatorialist PWS gets around both absurdities, but you can see how easily we can abuse PWS.

    I think I've hijacked this thread long enough. Cool to see ya'll around. I'll spin up a metalogic thread at some point when ppl have developed a tolerance for me.
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