Earlier, we became acquainted with the logical operators not, and, and or. Here we treat two more important operators.
The implication of two propositions is a proposition that is precisely true if the first proposition is false or the second proposition is true. In all other cases, the implication is false.
Implication is denoted by the if...then-operator #\rightarrow#. So, if #\blue p# and #\blue q# stand for propositions, then #\blue{ p}\rightarrow \blue{q}# indicates that #\blue p# implies #\blue q#.
Example
The proposition \(\blue{\textit{"If}}\) #\blue{\textrm{it rains}}# \(\blue{\textit{then}}\) #\blue{\textrm{the ground is wet"}}# is true if #\blue{\textrm{"It rains"}}# and #\blue{\textrm{"The ground is wet"}}# are both true statements.
The implication is also true whenever the proposition #\blue{\textrm{"It rains"}}# is false.
Operator notation: #\blue{\textrm{"It rains"}} \rightarrow \blue{\textrm{"The ground is wet"}}#
Implications express causality. They indicate that the truth of the first part of the proposition entails the truth of the second part. In a causal relation the second part might still be true or false if the first part is false. This is why #\blue p \rightarrow \blue q# is defined to be true when #\blue p# is false. More on this in Normal language.
In the definition we use the terms 'first proposition' and 'second proposition'. This is not completely accurate. This way of looking at implications can only be used if a proposition is formulated in the form "if...then...". If a proposition is formulated in the form "...if....", the second proposition causes the first.
In short, "if #\blue p# then #\blue q#" and "#\blue q# if #\blue p#" are equivalent statements (which means that they always take the same values; see below).
We need to be really careful when using implications, because we need to make sure we identify the correct proposition as the cause, and the correct proposition as the effect. We always write "cause #\rightarrow# effect".
Example
The propositions #\blue{\textrm{"If it rains then the ground is wet"}}# and #\blue{\textrm{"The ground is wet if it rains"}}# can both be written as
#\blue{\textrm{"It rains" }} \rightarrow\blue{\textrm{ "The ground is wet"}}#.
The propositions #\blue{\textit{"If } \textrm{the ground is wet } \textit{then } \textrm{it rains"}}#" and #\blue{\textrm{"It rains}\textit{ if} \textrm{ the ground is wet"}}# might both be written as
#\blue{ \textrm{"The ground is wet" } \rightarrow \textrm{ "It rains"}}#.
The propositions #\blue{\textrm{"It rains" }} \rightarrow\blue{\textrm{ "The ground is wet"}}# and #\blue{ \textrm{"The ground is wet" } \rightarrow \textrm{ "It rains"}}# have a fundamentally different meaning.
In daily language, the if...then statement is often interpreted slightly differently from the use in mathematics. If we say #\blue{\textit{"if } p\textit{ then }q\textrm{"}}#, we assume #\blue p# and #\blue q# to have some causal relationship where #\blue q# is true because #\blue p# is true.
In mathematics, this does not have to be the case: we only require if #\blue q# is not true, that #\blue p# be false.
To stress the mathematical meaning, cautious people often use expressions like "If #\blue p# is true, and I am not saying that it is true, but suppose that is true, then #\blue q# is true as well".
Example
Consider the proposition #\blue{ \textit{"If}\textrm{ all apples have purple dots}\textit{ then}}# #\blue{\textrm{ all bananas have red stripes"}}#.
Both parts of the proposition are clearly false.
In mathematics, we consider this implication to be true, since the proposition #\blue{\textrm{"All apples have purple dots"}}# is false.
So cautious persons might say "If all apples have purple dots, and I am not saying that they do, but suppose that they do, then all bananas have red stripes" in order to avoid being accused of purporting that all apples have purple dots.
In logic different kinds of arrows are being used for different levels of formal language. Here, we work with the single line arrow #\rightarrow# to represent ``if...then" statements in logic. Often, we use the symbol #\Rightarrow# in logic to talk about and analyse implications that concern the formal language in logic.
For instance, the implication ``if #\blue p\to\blue p# is always true then #\blue{p\wedge p} \to\blue p# is always true" talks about two ``if...then" statements in logic. We can use the symbol #\Rightarrow# to represent these kinds of expressions: \[\blue p\to\blue p \text{ is always true } \Rightarrow \blue{p\wedge p} \to\blue p \text{ is always true}\]
Having a distinction between symbols in the language of logic and symbols to ``talk about" and to prove implication statements about logic is useful.
We sometimes also use the reverse arrow #\leftarrow#. The proposition #\blue{p}\leftarrow \blue{q}# means the same as #\blue{q}\rightarrow \blue{p}#.
For instance the propositions \[\begin{array}{c}
\blue{\textrm{"It rains "} \rightarrow \textrm{" The ground is wet"}} \\
\blue{\textrm{"The ground is wet"} \leftarrow \textrm{" It rains"}}
\end{array}\] represent the same proposition. In natural language, they correspond to, respectively, \[\begin{array}{c}\blue{\textrm{"If it rains then the ground is wet"}}\\
\blue{\textrm{"The ground is wet if it rains"}}\end{array}\]
According to the definition, the implication #\blue p\rightarrow \blue q# is equivalent to #\neg \blue{p}\lor\blue{q}#. This means that both propositions have the same value for each combination of values of #\blue{p}# and #\blue{q}#.
Use of implication makes it possible to express equivalence of propositions.
The bi-implication of two propositions is a proposition that is true exactly when both propositions have the same value. This means that the bi-implication is true if the two propositions are either both true or both false. If not, the bi-implication is false.
Bi-implication of two statements #\blue p# and #\blue q# expresses equivalence in the sense that #\blue p# and #\blue q# always have the same value (true/false).
Bi-implication is denoted by the if and only if-operator #\leftrightarrow#.
Example
The proposition #\blue{\textrm{"I am Janine's child"}}# #\blue{\textit{if and only if} }# #\blue{\textrm{"Janine is my parent"}}# is true since #\blue{\textrm{"I am Janine's child"}}# and #\blue{\textrm{"Janine is my parent"}}# both are either true or false statements at the same time.
Operator notation:
#\blue{\textrm{"I am Janine's child"} }# #\leftrightarrow# #\blue{\textrm{"Janine is my parent"}}#.
A bi-implication indicates that the truth of the first part of the proposition implies the truth of the second part, and the truth of the second part implies the truth of the first part.
As was the case with implication we only work with the single line arrow #\leftrightarrow# to indicate bi-implication and not with the double line version #\Leftrightarrow#.
A close look at the definition shows that for propositions #\blue p# and #\blue q#, the proposition #\blue p\leftrightarrow \blue q# is equivalent to \[ \left(\neg\blue{p}\land \neg\blue{q}\right)\lor \left(\blue{p}\land \blue{q}\right)\]Later we will see that this is also equivalent to the conjunction #\left(\blue{p} \rightarrow \blue{q}\right)\land \left(\blue{q} \rightarrow \blue{p}\right)# of two implications. Hence the name bi-implication.
Instead of ``#\blue p# is true if and only if #\blue q# is true", we also often say "#\blue p# is true precisely if #\blue q# is true". The word ``precisely" distinguishes the bi-implication from a unilateral implication.
| type |
example |
meaning |
| implication |
#\blue{p}\leftarrow\blue{q}# |
#\blue p# is true if #\blue q# is true. |
| bi-implication |
#\blue{p}\leftrightarrow\blue{q}# |
#\blue p# is true precisely if #\blue q# is true. |
Suppose that we know"If I study hard, then I will pass my exam."
Is the following a correct logical consequence?
"I did not pass my exam, so I did not study hard"
Yes, this is a correct logical consequence.
Given the proposition ''if I study hard, then I will pass my exam'', we know that studying hard implies passing. This means that if I did not pass my exam, I have not been studying hard. If I would have studied hard, I would have passed. Therefore, we can indeed conclude that I did not study hard.
Implication
