There’s a branch of math known as

*propositional calculus*that treats arguments like mathematical propositions. Using propositional calculus, you can demonstrate the truth or falsity of certain arguments.

Take the statement “It is raining here now.” Depending on when you make the statement, it can be either true or false. In propositional calculus, you’d represent the statement as “P,” and the opposite, “It is not raining here now” as “¬P.” If P is true, then ¬P has to be false; if ¬P is true, then P has to be false.

You can link together statements with connectors. Common connectors are AND, OR NOT, ONLY IF, and

*IF AND ONLY IF. If we say “It is raining here now,*

*and*it is raining where you are now as well,” we can label the second statement as Q. Represent AND with the symbol ⋀, and we can write “It is raining here now,

*and*it is raining where you are now as well” as P⋀Q.

Of course, maybe it is raining here or it isn’t; maybe it’s raining at your house and maybe it isn’t. Because the individual statements can be true or false, we can prepare a

*truth table*.

__P Q P⋀Q__

True True True

True False False

False True False

False False False

With

*and*as a connector, the proposition P⋀Q is only true if

*both*statements are true.

The connector OR (represented as “⋁”), on the other hand, makes the proposition true as long as

*at least one*of the statements are true. “It is raining here now OR it is raining where you are now” results in the following truth table.

__P Q P⋁Q__

True True True

True False True

False True True

False False False

Notice that OR is used here

*inclusively*rather than

*exclusively*. That is, P doesn’t exclude Q from being true. If it’s raining at my house, that doesn’t mean it’s not raining at yours.

Given the idea of propositional logic, it's easy to conclude that there are fallacies to go with it. The first of these is known as

*affirming a disjunct.*

**Affirming a Disjunct**

Also known as the fallacy of the alternative disjunct, or the false exclusionary disjunct, this particular fallacy occurs when you change an

*inclusive*OR into an

*exclusive*one. “It is raining here now

*or*it is raining where you are now” gets interpreted as “If it is raining here now, then it isn’t raining where you are now.”

In our symbolic structure, that gets represented as the following argument (with “therefore” represented by ∴).

P⋁Q

P

∴¬Q

That’s a fallacy because it could be raining both places. One doesn’t preclude the other.

While OR in logic always means an inclusive “or,” that doesn’t mean you don’t sometimes want to be more concrete. The logical operator XOR is an exclusive or. When you use it, you’re saying “one or the other, but not both.” The symbol for that is ⊻.

*More next week.*

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