Do any LR questions have less than two terms which both appear twice?

attorneysomerville

It seems to me that it is impossible to form a complete logical argument in which the key terms do not appear at least twice, and it also seems that every logical argument but one necessarily uses at least two terms. (The only logical argument that only uses one term is the "double negative," which states: "A," therefore "~(~A)," and I have never seen this in an LSAT stimulus.)

As far as I can tell, a complete LSAT question (by which I mean the stimulus plus the correct answer) must necessarily include at least two terms, and each of these terms must appear at least twice.

I may be failing to consider certain unusual question types that don't actually consist of logical arguments as such. If so, that might be the exception that proves the rule.

If anybody can provide an example of an argument (other than the "double negative" above) that has less than two terms, or refers to a key term only once, please let me know.

quinnxzhang

I suspect that you mean something different by "term" than what logicians mean by "term". For the sake of understanding, can you give an example of an argument with at least two terms and point out what exactly the terms are?

attorneysomerville

A->B
A
Therefore B

attorneysomerville

I chose the term "term" after looking around hard for another word to express the concept. If you know a better word for the symbols we use in conditional statements, please let me know. I know that logicians routinely refer to the "terms" of a categorical statement as "terms," which is why I used the word in this context.

Cant Get Right

I think that’s probably safe to say for any question involving conditional reasoning. Of course some question types won’t necessarily apply, but if the answer involves conditionals, then you can’t link up a correct answer with the stimulus without connecting the relevant terms. They also plant traps here by substituting terms which are easily mistaken for the relevant terms in the stim. I think there’s one where they sub in “utensils” for “ironing pins” or something.

quinnxzhang

In propositional logic, the symbols are referred to as "sentence symbols". The abstract entities that the sentence symbols represent are called "propositions" or "sentences" (hence the name "propositional"/"sentential logic"). The word "term" is a technical term in predicate logics, used to refer to names of objects, not to propositions or sentences.

It seems like you're asking if there are arguments consisting of fewer than two atomic propositions. There's certainly nothing wrong with an argument consisting of one atomic proposition. For example, here's an argument: P. Therefore, P. And here's another argument: P or ~P. In fact, valid formulas (e.g. (P or ~P), ~(P & ~P)) are the limiting case of valid arguments -- they're just arguments with no premises.

Moving outside of formal logic, I'm not sure if there are LSAT questions which have arguments with fewer that two atomic propositions. It doesn't seem like the kind of thing an LSAT question writer would do. If it were to show up, it'd probably show up in necessary assumption questions or something similar, where we have to fill in the gaps with assumptions.

attorneysomerville

Thanks, Quinn. I took academic logic a long time ago... very likely before you were born, which means you're much more likely to be correct than I am, working from memory.

Insofar as the large majority of LR stimuli involve conditional or categorical reasoning, I'm happy to use the term "atomic proposition" if you think it would be clearer than "term." (I certainly don't find the term "term" to be unambiguously clear.)

So, partly echoing you and partly repeating myself from above, there are completely trivial arguments that only use one atomic proposition, but I haven't seen this in any actual LSAT stimuli. Until we discover one, I'm sticking with the observation that a VERY large majority of LR questions have at least two atomic propositions.

Moving on to the more controversial claim--I can't think of a way to write a conditional or categorical argument in which an atomic proposition only appears once. Can you?

quinnxzhang

@attorneysomerville said:
Moving on to the more controversial claim--I can't think of a way to write a conditional or categorical argument in which an atomic proposition only appears once. Can you?

I assume by "conditional argument", you mean an argument involving a wide-scope conditional sentence, usually of the form "If P, then Q". I assume by "categorical argument", you mean an argument involving a universal claim, usually of the form "All A's are B's".

The first thing to note is that these are **NOT** the same thing. I realize LSAT instructors almost universally treat them the same way, but insofar as you're asking a question about formal logic and not LSAT logic, "If P, then Q" and "All A's are B's" are very different. In fact, in propositional logic, "All A's are B's" would be represented by a single sentence symbol, such as "P", because the entire sentence would be treated as a single proposition.

Bracketing "categorical" arguments, I'm not sure if you're asking whether there are arguments in which *some* atomic propositions don't show up more than once, or if you're asking whether there are arguments in which *all* atomic propositions don't show up more than once.

For the former, you can have reductio type arguments where the consequent of a conditional is absurd and only shows up once. For example: Either the axioms of naive set theory are true or they're not true. If the axioms of naive set theory are true, then 1+1=3. Therefore, the axioms of naive set theory are not true. In this case, we see that the proposition "1+1=3" only shows up once.

For the latter, I don't think it's possible in propositional logics, but it is possible in predicate logics with trivial cases.

attorneysomerville

Your argument assumes that "1+1=3" is false without stating that it is false. If you spelled it out fully instead of making that assumption, the atomic proposition would appear twice.

It is certainly possible to make assumptions that are never spelled out in an argument. MOST LSAT questions either don't make such assumptions or else make the assumptions a part of the answer, but that is not always necessarily so as a matter of logic.

In trying to figure out the key parts of a stimulus, I want to be able to assure myself that there's something really there which one really can find if one really looks hard enough. (Really.) If I can count on there ALWAYS being two instances of every "key term" (in the stimulus+answer), then it's worth pushing them to keep hunting until they find it. If it could even theoretically be possible that a well-formed, complete, and non-trivial argument does not have at least two "key terms," both of which appear at least twice, then I don't want to go looking for them.

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quinnxzhang

@attorneysomerville said:
Your argument assumes that "1+1=3" is false without stating that it is false. If you spelled it out fully instead of making that assumption, the atomic proposition would appear twice.

You don't need to state that it's false for this argument to be valid, though. It's impossible for both the premises to be true and the conclusion to be false. This is because it's impossible for "1+1=3" to be true.

@attorneysomerville said:
In trying to figure out the key parts of a stimulus, I want to be able to assure myself that there's something really there which one really can find if one really looks hard enough.

Propositions will appear an infinite number of times if you want to make it such that every single assumption is neatly spelled out. For example, consider this argument: (P1) A. (P2) A→B. (C) B. You can always spell out more and more assumptions for this argument, such as (P3) (A & (A→B)) → B. Or (P4) (A & (A→B) & ((A & (A→B)) → )) → B. And so on.

At some point, you have to assume *some* things, e.g. modus ponens is a valid inference rule and 1+1 is not 3. If you don't, your claim is trivial -- we make an infinite number of assumptions in every valid argument, and each proposition will show up an infinite number of times. On the off chance you're interested, there's a very famous paper in philosophy of logic, written by Lewis Carroll, which discusses this: http://www.ditext.com/carroll/tortoise.html