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May 5, 2016

Beyond “Unless”: Advanced Conditional Reasoning

Beyond "Unless": Advanced Conditional Reasoning

It’s fair to say that conditional reasoning is either the bane of your test prep, or a welcome escape from the uncertainty that plagues causal reasoning.  In the first few months of test prep, you will likely see conditional reasoning everywhere: understanding conditional reasoning can easily turn into an obsession, prompting you to diagram whenever you come across any of the indicators of conditionality. The costs of this approach ultimately outweigh the benefits. As you progress through your studies, you will hopefully develop a more careful, judicious approach to diagramming. As we’ve said in the past, you should diagram only when you think it will help you better understand what the author is saying.

Now, look at the last sentence in the previous paragraph. Conditional reasoning is front and center in that sentence, thanks to the fairly uncommon necessary condition indicator only when. In recent years, test makers have made a concerted effort to convey conditionality using relatively unexpected phrasing, in part because they know that most test-takers have learned the basic approach to the more common conditional constructions. Here’s how you want to handle the less common ones:

Not Until 

Not until is primarily a temporal preposition, which literally means “not before a stated time or event.” Not every sentence using this construction will convey conditional dependence. When it does, the grammatical structure of these sentences is shown in the diagram below:

Not until [1st clause: subject + auxiliary verb] [2nd clause: auxiliary verb + subject]

The conditional relationship between the two clauses is as follows:

S                                 N

2nd clause    →    1st clause

For example:

Not until the car came to a complete stop did we see the damage to the front bumper.

S                                 N

See damage    →    Complete stop

Not until the stars aligned did I win the lottery

S                                 N

Win lottery   →    Stars aligned

Not until I got into Yale did I stop worrying about my future

S                                 N

Stop worrying   →    Got into Yale

In all of these examples, the clause that follows immediately after the preposition not until is the necessary condition. The remainder of the sentence functions as a sufficient condition. The same would be true with not unless and only when. All of these are necessary condition indicators.

None But, None Except, No… Except

The language immediately following these indicators is the necessary condition; the remainder is the sufficient. Think of all three of these indicators as synonymous with the word only – a classic necessary condition indicator. Let’s take a look at a few examples:

None but the brave die young.

S                                 N

Die young    →    Brave

None of the students finished the test, except for those who had prepared for it.

S                                 N

Finish test   →    Prepared

No cars except SUV’s  can safely travel this road.

S                                 N

Safely travel   →    SUV

Only vs. The Only

The difference between only and the only is small, but critical. Whereas only is a classic necessary condition indicator, the language immediately following the phrase the only is actually the sufficient condition. For instance, the following pairs of statements are identical in meaning, even though the first statement uses the word only, whereas the second uses the phrase the only:

Invitations were extended only to wealthy donors

Wealthy donors are the only ones who were invited

S                                 N

    Invite    →  Wealthy donors

Only sports cars can drive this fast

The only cars that can drive this fast are sports cars

S                                 N

Drive fast  →    Sports cars

All Except, All But

Unlike none except and none but, which are synonymous with only and function as simple necessary condition indicators, all except and all but are a bit more complicated: they must be translated not as single conditional statements, but as pairs of conjoined conditional statements. Such statements are also known as exceptive propositions. Let’s take a look at an example:

All but 1st year associates received a raise.

The sentence above contains a pair of conditional relationships: 1) no 1st year associate received a raise; and 2) everyone who is not a 1st year associate did:

S                                 N

1) 1st year associate   →    Receive a raise

2) 1st year associate   →   Receive a raise

Now, let’s consider the contrapositives of each conditional relationship:

S                                 N

1) Receive a raise   →    1st year associate

2) Receive a raise   →   1st year associate

When each of the two original relationships is combined with the contrapositive of the other, the two resulting propositions amount to the following bi-conditional relationship:

S                                 N

1st year associate    ←→    Receive a raise

Contrapositive:        Receive a raise    ←→   1st year associate

In other words, only one of two outcomes are possible: either you are a 1st year associate who didn’t receive a raise, or else you are not a 1st year associate who did receive a raise.

Keep in mind the above is not an exhaustive list of sufficient and necessary condition indicators: the more common ones have been discussed extensively on our Forum, Blog, and, of course, the Logical Reasoning Bible and LSAT course materials. If your goal is a top-1% score (and why would you want anything less?), it is important to prepare for each and every eventuality. As you’ve probably heard, curve balls on the LSAT are the new normal.

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Posted by PowerScore Test Prep / Conditional Reasoning, LSAT Prep / Conditional Reasoning, LSAT Prep 3 Comments

  • Fara Soubouti
    May 06, 2016 at 8:49pm

    Wow, the blog post is awesome!!! When you explained that “not until” can be a necessary indicator in the forum, that was helpful, but this blog post is that x 10! The “all except, all but” is totally new to me and I’m so happy that I can add that to my conditional reasoning indicator arsenal.

    On that note, I have a question about “all except, all but” category. It is essentially establishing that 1 and only 1 of the 2 terms must be in/ occur/ take place (loner variable), not allowing for the possibility that both are out/negated or the possibility that both are in/ occurring/ taking place, whereas “if but only if” says both must be in or out (inseparable variable pair), right? Also would it be appropriate to say “all except” and “if but only if” are opposites, since they’re both double arrows with a single negated variable (for the “all except” statements) or with both variables positive or negative (for “if but only if”)? Drawing connections between the types of relationships these indicators represent helps me understand.

    Thanks!!

  • Nicolay Siclunov
    May 06, 2016 at 9:05pm

    Hi Fara,

    You are absolutely correct.

    Phrases such as “all except A are B” and “all but A are B” are exceptive propositions, and consequently yield pairs of conjoined categorial propositions. The inferences resulting from these propositions are as follows:

    A→NOT B (contrapositive: B→NOT A)
    NOT A→B (contrapositive: NOT B→A)

    Thus:

    A ←→ NOT B (contrapositive: B ←→ NOT A)

    The relationship is bi-conditional and can be represented with the same Double-Arrow that we use to diagram “if and only if” relationships. As far as truth value is concerned, only two outcomes are possible in each case:

    “All except A are” produces the following two outcomes: either A or B, but not both, must occur. In other words, you are either an A or a B: no one is neither, and no one is both.

    “A if and only if B” produces the following two outcomes: either A and B must both occur, or else neither A nor B can occur. In other words, you are either both an A and a B, or else you are neither A nor B.

    Once you compare these propositions side by side, you realize that they are logical opposites of each other. You came to this realization yourself, which is totally awesome 🙂

  • Tom
    December 03, 2017 at 4:35pm

    Can someone explain if x then y unless z statements?
    I have discovered that the proper way to diagram this statement is if X and ~Z –> Y.

    However I am curious about the contrapositve. is it ~Y–> ~x or z

    The example that has me stumped is this:

    “If there are sentient being on planets outside our solar system, we will not be able to determine this anytime in the near future UNLESS some of these beings are at least as intelligent as humans”

    Thank you.

  • Dave Killoran
    December 06, 2017 at 4:56pm

    Hey Tom,

    Thanks for the question! The relationship you are describing is known as a Nested Conditional, and we have a great post on that by one of our instructors on that topic, so let’s start there. It’s at: https://forum.powerscore.com/lsat/viewtopic.php?f=7&t=7992. I think that will help you out here, but if not, just let us know.

    Thanks!

Comments

  1. Jeebs says

    April 28, 2021 at 6:12 pm

    Since “all except” and “all but” are exclusive propositions, would that mean that they are mutually exclusive as well?

    Since this blog post was published a while back, would it be correct to notate the exceptive proposition example as:

    1st yr assoc. <--|--> raise
    [s]1st yr assoc.[/s] <--|--> [s]raise[/s]

    Reply
    • Jeebs says

      April 28, 2021 at 6:14 pm

      *double not arrow is between the two variables. Jeez.

      Reply
    • Jon Denning says

      May 3, 2021 at 7:13 pm

      Hi Jeebs – thanks so much for posting!

      Your take on this is exactly right! “All except” and “all but” are indeed mututally exclusive constructs, so in the example used in the article: being a 1st yr associate and receiving a raise are mutually exclusive, as are the propositions of *not* being a 1st yr associate and *not* getting a raise.

      The author above ultimately represented those with a double arrow and a single term crossed out (removed, negated, etc), but you’d achieve precisely the same meaning if you instead used a double-not arrow and left both terms in matching condition (both either with or without slashes through them). And that’s exactly what you’ve done in your question, so happily you’ve got this one cracked 🙂

      Nice work!

      Reply

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