Semantics

It's important to realize that the English does not in any way define these connectives; the English is at best an approximation of the meaning of the connectives. Their precise meaning is given via truth-tables, which tell us what the value of a statement is given the truth-values of its components. The simplest truth-table is that for negation, which informs us, unsurprisingly, that if $\phi$ is T then $\neg\phi$ is F (first row below double lines), and if $\phi$ is F then $\neg\phi$ is T (second row).

$$\begin{array}{c\vert\vert c} \phi & \neg \phi\\ \hline \hline T & F\\ F & T\\ \end{array}$$

Here are the remaining truth-tables.

$$\begin{array}{c\vert c\vert\vert c} \phi & \psi & \phi \wedg... ...T & T & T\\ T & F & F\\ F & T & F\\ F & F & F\\ \end{array}$$

$$\begin{array}{c\vert c\vert\vert c} \phi & \psi & \phi \vee ... ...T & T & T\\ T & F & T\\ F & T & T\\ F & F & F\\ \end{array}$$

$$\begin{array}{c\vert c\vert\vert c} \phi & \psi & \phi \righ... ...T & T & T\\ T & F & F\\ F & T & T\\ F & F & T\\ \end{array}$$

$$\begin{array}{c\vert c\vert\vert c} \phi & \psi & \phi \left... ...T & T & T\\ T & F & F\\ F & T & F\\ F & F & T\\ \end{array}$$

There are a few things you probably should take special note of here before we proceed. First, notice that the truth-table for disjunction says that when both disjuncts are true, the entire disjunction is true. This is called inclusive disjunction. In exclusive disjunction, it's one disjunct or another, but not both. This distinction becomes particularly important if one is attempting to symbolize parts of English (or any other natural language). It would not do to represent the sentence

$$\mbox{\lq\lq George will either win or lose.''}$$

as

$$P_w \vee P_l,$$

because under the English meaning there is no way both possibilities can be true, whereas by the meaning of $\vee$ it is perfectly admissible as a scenario that P_w and P_l are both true. We will use $\vee_x$ to denote exclusive disjunction, which we can define through the following truth-table.

$$\begin{array}{c\vert c\vert\vert c} \phi & \psi & \phi \vee_... ...T & T & F\\ T & F & T\\ F & T & T\\ F & F & F\\ \end{array}$$

Before conclusing this section, it is worth mentioning another issue involving the meaning of English sentences and their corresponding symbolizations in propositional languages: the issue of the ``oddity" of conditionals (formulas of the form $\phi \rightarrow \psi$). An example will bring this oddity to the surface. Consider the following English sentence.

$$\mbox{If the moon is made of green cheese, then Dan Quayle will be the next President of the United States.}$$

Is this sentence true? If we were to ask ``the man on the street," the answer would likely be ``Of course not!" -- or perhaps we'd hear: ``This isn't even a meaningful sentence; you're speaking nonsense." These responses are quite at odds with the undeniable fact that when represented in the propositional calculus, the sentence turns out true. Why? The sentence is symbolized as

$$P_g \rightarrow P_q.$$

Since P_g is false (the moon isn't made of green cheese), the truth-table for $\rightarrow$ classifies the conditional as true. Results such as these have encouraged some to search for other formal accounts of the ``if - then's" seen in natural languages. For now, however, we will not embark on such a search, and will be content with the conditional as defined by the customary truth-table for $\rightarrow$ presented above.