Skip to main content
Humanities LibreTexts

11.3.5: The Logic of Only, Only-If, and Unless

  • Page ID
    36228
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    The word only is an important one for logical purposes. To explore its intricacies, suppose that to get an A grade in Math 101 you need to do two things: get an A- or better average on the homework, and get an A average on the tests. It would then be correct to say, "You get an A grade in Math 101 only if you get an A- or better average on all the homework." Now drop only from the sentence. Does it make a difference? Yes, because now you are left with the false statement "You get an A grade in Math 101 if you get an A- or better average on the homework." Speaking more generally, dropping the only from only if usually makes a significant difference to the logic of what is said. Unfortunately, many people are careless about using these terms. Let's display the logical forms of the two phrases in sentential logic, using these abbreviations:

    A = You get an A in Math 101.
    B = You get an A- or better average on all the homework.

    Now, “A only if B” is true but “A if B” is false. So "A only if B" and "A if B" are not equivalent; they must be saying something different. They have a different "logic."

    Here is a summary of the different logical behavior of if as opposed to only if. The following three statement patterns are logically equivalent:

    (1) P only if Q
    (2) P implies Q
    (3) If P, then Q

    but none of the above three is equivalent to any of the following three:

    (4) P if Q.
    (5) Q implies P.
    (6) If Q then P.

    Yet (4), (5), and (6) are all logically equivalent to each other.

    The phrase if and only if is a combination of if plus only if. For example, saying, "You're getting in here if and only if you get the manager's OK" means "You're getting in here if you get the manager's OK, and you're getting in here only if you get the manager's OK."

    Exercise \(\PageIndex{1}\)

    Which of the following are true?

    For all x, x = 4 only if x is even.
    For all x, x is even only if x = 4.
    For all x, x is even if and only if x = 4.

    Answer

    Just the first one

    Figure \(\PageIndex{1}\)
    Exercise \(\PageIndex{1}\)

    Are all three of these sentences logically equivalent to each other? If not, which two are equivalent to each other? Watch out. This is tricky because your background knowledge about geography is useless here.

    a. If you're from the USA, then you're from North Dakota.
    b. You're from the USA only if you're from North Dakota.
    c. You're from North Dakota if you're from the USA.

    Answer

    All three are equivalent to each other, and all three are false

    The logical form of sentences containing the word unless is important to examine because so many errors and tricks occur with the word. It usually means the same as or, although many people, on first hearing this, swear it is wrong. You're going to go to jail unless you pay your taxes, right? So, either you pay your taxes or you go to jail.

    Consider a more complicated situation. Suppose you will not get an A in this course unless you are registered. Does it follow with certainty that if you are registered, you will get an A? No. Does it follow that you will not get an A? No, that doesn't follow either. Does it follow instead that if you do not get an A, you are not registered? No. What would, instead, be valid is this:

    You will not get an A in this course unless you are registered.
    So, if you get an A, then you are registered.

    The logical form of the reasoning is

    Not-A unless REG.
    So, if A, then REG.

    Does this really look valid? It is.

    What you may have noticed in this and previous chapters is in translating from English to symbolic form we often make use of our background knowledge of what is equivalent to what. That is what happens when we translate "She is here" as S but also "She's here" as S even though the two sentences have different letters; one has an apostrophe, and the other does not. When we translate conditions, we make considerable use of our background knowledge about equivalence. We make use of our knowledge that tense (past, present, future) and mood (indicative, subjunctive) is or is not important for the argument. For example, notice the subtle move from present tense to future tense in this valid argument:

    If Samantha takes a car, she'll get there faster.
    She will take a car.
    So, she will get there faster.

    "She takes" is present tensed, but "She will take" is past tensed. We ignore all this tense information when we translate the argument as

    If TAKES, then FASTER.
    TAKES.
    So, FASTER.

    It takes a good understanding of the language to know when you can ignore time information and when you cannot. You cannot ignore it in this argument:

    If she takes a car, she'll get there faster, but not if she waits another five minutes.
    She will take a car, but not for ten minutes.
    So, she will get there faster.

    No, she won't. This is invalid reasoning.

    Here is another example of making use of background information. Is this argument valid?

    If Samantha takes a care, she'll get there faster.
    She will take a car.
    So, she will get there faster.

    Maybe. We said it was valid a couple of paragraphs ago, but it is valid only if we are justified in ignoring the fact that the word "she" might be referring to Abraham Lincoln's wife, and not to Samantha. Our background knowledge tells us whether we need to pay attention to this possibility or not.


    This page titled 11.3.5: The Logic of Only, Only-If, and Unless is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Bradley H. Dowden.

    • Was this article helpful?