This chapter is brewing in me a dislike for my Math book, sort of. It seems the reasoning applied to solve a particular problem is different from that used to solve another one. I've come with my issues to this site twice before (regarding conditional statements, logic, and proofs) and the answers provided by members on this site have shown me that it's not just me.

Anyways, here are some exercises I'd like help with.

I'm asking only about the conditional statement of each one, because I can thread with the converse, contrapositive, and inverse by myself.

  1. If an angle measures 167 degrees, then it is obtuse.

    • True, because an obtuse angle, by definition, is an angle whose measure is comprised between 90 and 180 degrees.
  2. If you are in a Physics class, then you will always have homework.

    • True, because the original statement has communicated it. There is no "outside logic" that makes it true, just the fact that the sentence says "You are in a Physics class, so you always have homework."
  3. If I take my driving test, I will get my driver's license.

    • Following the template of n_2 (regarding "outside logic"), I'm inclined to say true; but my gut disagrees, since you need to actually pass the driving test to earn your license.

I can understand your dislike for the book, because 2 and 3 are inferences, rather than if ... then statements.

But, the if... then statements you write in your post are most likely the ones that the book is looking for.

  • $\begingroup$ Hi, I want to ask how to tell/distinguish 1 v.s. 2&3 is in different sense? Traditional logic books seems always make it blurred. On the other hand, is the if-then-else syntax in programming language, for example, $\textsf{if(i<3) i++;}$ in C, corresponding to $\to$ in logic? Or $\vDash$ in logic? Or it should not be think of being related in logic? $\endgroup$ – Eric Jul 17 '17 at 15:14
  • $\begingroup$ @Eric If you think about it in logic at all, then definitely $\rightarrow$ instead of $\vDash$, because $\vDash$ says that $i++$ is somehow a logical consequence of $i < 3$, which of course it isn't ... you could have done anything in that case. $\endgroup$ – Bram28 Jul 17 '17 at 15:23
  • $\begingroup$ @Eric You can use logic to describe the meaning of imperative programs, but you make claims about states of the program. In a common approach, you convert the program to Single Static Assignment Form and then make claims about these single-assignment variables. Simplifying, $(i_0 < 3) \wedge (i_1 = i_0+1) \vee (i_0 \geq 3) \wedge (i_1 = i_0)$. (Those are equality signs, not assignments.) [EDIT: Didn't notice how old this question was...] $\endgroup$ – Fabio Somenzi Jan 5 '18 at 17:27
  • $\begingroup$ @FabioSomenzi Thanks for your informations! I'm now preparing an entrance examination for math graduate. So I have not studying logic for a long time. However I hope I can study logic deeply in my graduate times, by the time I'll come back to this topic. :) $\endgroup$ – Eric Jan 6 '18 at 14:49

In order for a conditional statement to be false, it must be impossible for the antecedent to be true and consequent false. Since it is possible to be in a Physics class and have no homework, then the second is false. Admittedly, this is only my interpretation, as both words in the phrase "always have" are somewhat ambiguous. Also, the third statement is false, for precisely the reason that you suspected, and that is non-ambiguous.

  • $\begingroup$ Your interpretation of 2 (and 3), makes me believe, then, that we can resort to "outside logic" to answer these questions – namely say, in this instance, that "it is possible to be in a Physics class and have no homework"; but but what do you think about the "so you always have homework" of n_2? Isn't it suggestive of anything to you? $\endgroup$ – Swift Jul 16 '17 at 20:43
  • $\begingroup$ My interpetation is that it means: "after every session of the Physics class, the teacher assigns homework." This need not be true. After all, near the time of the final exam, many classes do in-class review during the last few sessions, and assign no homework. Again, though, it isn't clear that my interpretation is the one intended, since it's put in colloquial (rather than logically precise) terms. $\endgroup$ – Cameron Buie Jul 16 '17 at 20:58
  • $\begingroup$ Another potential issue is the word "you." One could argue that this is intended to be you, in particular, so if you are not in a Physics class, the antecedent becomes false, and so the conditional statement becomes true, regardless of the consequent. $\endgroup$ – Cameron Buie Jul 16 '17 at 21:02
  • $\begingroup$ I see. The problem, then, becomes about interpretation. I'll get in touch with a teacher whenever I can for a definitive answer. (for better or worse, I have to suffice myself with my wits and those of the people here.) $\endgroup$ – Swift Jul 16 '17 at 21:36

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