Causality An Implication is an utterance of causality. The truth of the left-hand side of the proposition \(\varphi\rightarrow \psi\) also means that the right-hand side is true. In a causal relationship, the right-hand side can still be true or false if the left-hand side is false. This is why thruth valuse of \(\varphi\rightarrow \psi\) is defined as true in case \(\varphi\) is false.

Natural language We have already noted a few times that in natural language the proposition "If \(p\), then \(q\) " is often interpreted as a causal relationship that makes \(q\) true because \(p\) is true. But in classical logic this need not be the case. We can only draw the following two conclusions:

1. (Modus ponens) \(q\) follows from \(p\) and \(p\rightarrow q\).
2. (Mode tollens) \(\neg p\) follows from \(\neg\,q\) and \(p\rightarrow q\).

Computer language The if-then construction in most programming languages differs from the if-then construction in propositional logic. In most programming languages, the code "if condition then statement" means "if the condition is met, then execute the code, otherwise don't". As soon as the condition is not met, the code in the then part is no longer executed.

Reverse arrow If you want to keep the propositional notation as close as possible to natural language, then a reverse arrow notation is obvious. The sentence "Marthe stays home when she is sick." is then transcribed to \(q\leftarrow p\) with \(p=\text{"She is ill."}\) and \(p=\text{"Marthe stays at home"}\)

Notation In practice you will also encounter the implication arrow \(\implies\) instead of \(\rightarrow\) to make an "if ... then ..." statement in logic. Although they are often used interchangeably, there is a subtle difference. With \(\varphi\rightarrow\psi\) the truth depends only on the truth values of \(\varphi\) and \(\psi\); the definition of the implication arrow \(\rightarrow\) is defined via a truth table. The symbol \(\implies\) is actually meant to be an implication symbol in the meta-language through which we talk about logic formulas. An example may make this subtle distinction clearer.

Suppose \(\varphi\) is a formula that contains only the connectives \(\neg\), \(\land\), and \(\lor\). Then the dual formula \(\varphi^{\mathrm{d}}\) is defined as the formula resulting from replacing every occurrence of \(\land\) in \(\varphi\) with \(\lor\) and replacing every occurrence of \(\lor\) in \(\varphi\) with \(\land\). Now suppose that \(\varphi\) and \(\psi\) are equivalent formulas containing only the connectives \(\neg\), \(\land\), and \(\lor\). From this we may conclude that \(\varphi^{\mathrm{d}}\) and \(psi^{\mathrm{d}}\) are also logically equivalent. This is a statement about logical formulas that does not quite fall within the system of propositional logic. We therefore write this down with another symbol: \[\varphi\leftrightarrow \psi \implies \varphi^{\mathrm{d}}\leftrightarrow \psi^{\mathrm{d}}\]

Truth table If \(\varphi\) and \(\psi\) are logical formulas, then the following truth table is applicable: \[\begin{array}{cc|c} \varphi & \psi & \varphi\rightarrow \psi \\ \hline 0 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 0\\ 1 & 1 & 1 \end{array}\]

Natural language In natural language, the distinction between implication and equivalence is often blurred. If a mother says to her child "If you finish your plate, you get dessert." she usually implicitly means that her child only gets a dessert if s/he finishes her/his plate. If the child gets dessert, then s/he has finished her/his plate, even though this is not explicitly stated in the mother's statement.

Notation In practice you will also come across the double implication arrow \(\iff\) instead of \(\leftrightarrow\) to make an "if and only if" statement in logic. Although they are often used interchangeably, there is a subtle difference. With \(\varphi\leftrightarrow\psi\) the truth depends only on the truth values of \(\varphi\) and \(\psi\); the definition of the equivalence arrow \(\leftrightarrow\) is defined via a truth table. The symbol \(\iff\) is actually meant to be an equivalence symbol in the meta-language where we talk about logic formulas. An example may make this subtle distinction clearer.

Suppose \(\varphi\) is a formula that contains only the connectives \(\neg\), \(\land\), and \(\lor\). Then the dual formula \(\varphi^{\mathrm{d}}\) is defined as the formula resulting from replacing every occurrence of \(\land\) in \(\varphi\) with \(\lor\) and replacing every occurrence of \(\lor\) in \(\varphi\) with \(\land\). Now suppose that \(\varphi\) and \(\psi\) are equivalent formulas containing only the connectives \(\neg\), \(\land\), and \(\lor\). From this we may conclude that \(\varphi^{\mathrm{d}}\) and \(\psi^{\mathrm{d}}\) are also logically equivalent, and vice versa. This is a statement about logical formulas that does not quite fall within the system of propositional logic. We therefore write this with another symbol: \[\varphi\leftrightarrow \psi \iff \varphi^{\mathrm{d}}\leftrightarrow \psi^{\mathrm{d}}\]

Truth table If \(\varphi\) and \(\psi\) are logical formulas, then the following truth table is applicable: \[\begin{array}{cc|c} \varphi & \psi & \varphi\leftrightarrow \psi \\ \hline 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\\ 1 & 1 & 1 \end{array}\]