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So the question is,

Show that the premises “It is not sunny this afternoon and it is colder than yesterday,”

“We will go swimming only if it is sunny,”

“If we do not go swimming, then we will take a canoe trip,” and

“If we take a canoe trip, then we will be home by sunset”

lead to the conclusion

“We will be home by sunset.”

The first part of given solution states,

Let p be the proposition “It is sunny this afternoon,”

q the proposition “It is colder than yesterday,”

r the proposition “We will go swimming,”

s the proposition “We will take a canoe trip,” and

t the proposition “We will be home by sunset.” Then the premises become ¬p∧q,r →p,¬r →s, and s →t. The conclusion is simply t.

I am unable to figure out how the premise of statement,

We will go swimming only if it is sunny

became,

$r \Rightarrow p$

Shouldn't it be,

$p \Rightarrow r$

The way I am thinking is,

The statement "We will go swimming only if it is sunny" can also be written as,

If it is sunny, then we will go swimming. .'. $p \Rightarrow r$

Please help me figure out what should be correct approach of turning the statement "We will go swimming only if it sunny" into proposition.

Thanks.

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2 Answers 2

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You are confusing "Swimming only if Sunny" with "Swimming if Sunny".

The former is meant by the problem poser to mean "If it is Sunny, we might or might not go Swimming, by we won't go Swimming if it is not Sunny".

The latter, which is how you interpreted it, means "If it is Sunny, we certainly go Swimming; I am saying nothing about what happens if it is not Sunny."

You therefore had the implicatoin in the opposite direction of what was intended.

This matter is further complicated by the fact that $A \mbox{ only if } B$ is often used in math texts to mean $A$ if and only if $B$, as in

"An even number is perfect only if it is of the form $2^{n-1}(2^n-1)$ with $(2^n-1)$ prime."

If you take "only if" to have that meaning, then your confusion is at least half justified.

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"We will go swimming only if it's sunny" can be rephrased in a few ways:

  1. it's not the case that [we will go swimming and it won't be sunny], that won't happen; equivalently,
  2. we won't go swimming if it isn't sunny; in other words,
  3. if it isn't sunny then we won't go swimming. This is equivalent to:
  4. if we go swimming then it is (or, will be) sunny.

Another way to see that this is the correct rendering is to apply De Morgan's law and then the equivalence $\neg A\lor B\equiv A\Rightarrow B$ to arrive at the same "if-then' form:

  • By De Morgan's law, 1. above is equivalent to either we won't go swimming or it will be sunny. This is of the form $\neg A\lor B$, equivalent to $A\Rightarrow B$:
  • if we (will) go swimming then it will be sunny.

"if" and "only if" are the converses of each other. "$p$ if $q$" is the same as "if $q$ then $p$", which is rendered symbolically as $q\to p$". "$p$ only if $q$" is the converse, and the implication is the reverse of "if": "$p$ only if $q$" is rendered as $p\to q$, the same left-to-right order of propositions. Thus:

  • "$p$ only if $q$" means "if $p$ then $q$", whereas
  • "$p$ if $q$" means "if $q$ then $p$".

In a sense, then, it's 'backwards' when "if" is in the middle of the two propositions and there's no "then".

This is why the *bi-implication' $\leftrightarrow$ is "iff" — if and only if. $p\leftrightarrow q$ is equivalent to $(p\to q)\land (q\to p)$. In English with variables, "$q$ if $p$ [left to right], and $q$ only if $p$ [right to left]. (Equivalently/by symmetry: $p$ if $q$, and $p$ only if $q$.)

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  • $\begingroup$ I understand now, so if the statement is like "We will go swimming if only it is sunny", positions of 'if' and 'only' interchanged, then it can be rephrased as "If it is sunny, then we will go swimming." am I correct? $\endgroup$ Commented Feb 26, 2016 at 18:26
  • $\begingroup$ Your example in the comment is backwards :) "We will go swimming only if it's sunny" is the non-reversed direction: it means "if we go swimming then it's sunny". $\endgroup$
    – BrianO
    Commented Feb 26, 2016 at 18:47
  • $\begingroup$ You're welcome. Got it now? $\endgroup$
    – BrianO
    Commented Feb 26, 2016 at 18:49
  • $\begingroup$ Yes absolutely! :) :) $\endgroup$ Commented Feb 26, 2016 at 18:50
  • $\begingroup$ So you should upvote both answers :) $\endgroup$
    – BrianO
    Commented Feb 26, 2016 at 18:51

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