2.2.1 Definitions

In this chapter, I'll define the semantics of our language. Since the formal logical connectives will keep the same meaning as our metalanguage's, I'll seize the opportunity to clarify our metalanguage by specifying what is meant by "and", "or" and other usual connectives.

Truth tables

The semantics of propositional logic revolves around the truth values of wffs. I'll denote the truth value "true" by "\(\mathcal{T}\)" and the truth value "false" by "\(\mathcal{F}\)". "\(\mathcal{T}\)" and "\(\mathcal{F}\)" are said to be opposites from one another. One defines the meaning of any logical connective by settling the truth value of the compound statement for each combination of truth values of the substastement/s the connective applies to. This is usually done via a truth table, which enumerates all possibilities for the truth/falsity of the substatement/s.

In the following definitions, the letters \(A\) and \(B\) are metavariables.

Negation is denoted by the symbol \(\neg\). When negation is applied to a wff, the resulting statement takes the opposite value of its substatement. Its truth table is

The and connective, also called conjunction, is denoted by the symbol \(\land\). Conjunction is true if and only if both substatements are true and is false otherwise. The order in which the substatements appear is irrelevant (for example, "he fell and he got up" doesn't necessarily mean that first he fell and then he got up). Its truth table is

The or connective, also called disjunction, is denoted by the symbol \(\lor\). Disjunction is false if and only if both substatements are false and is true otherwise. Notice that by or, I mean the inclusive or: \(A\) or \(B\) is also true when both \(A\) and \(B\) are true, whereas in English, by or we usually mean the exclusive one, in which \(A\) or \(B\) is instead false when both \(A\) and \(B\) are true. In Mathematics in general (as well as in Math From Scratch), when or is used, the inclusive or is meant. Thus, in our context, "or" is the same as "and/or". Its truth table is

The exclusive or is denoted by the symbol \(\oplus\). It is true when the truth values of its substatements are different from one another. Its truth table is

Implication is an interesting one, as many people find its truth table unintuitive. Implies is denoted by the symbol \(\rightarrow\). In the expression \(A \rightarrow B\), \(A\) is called the hypothesis and \(B\) is called the conclusion. \(A \rightarrow B\) is false if and only if the hypothesis \(A\) is true and the conclusion \(B\) is false. Otherwise, it's true. Its truth table is

If you think that when \(A\) is false, then \(A \rightarrow B\) should be false, then you are likely confusing \(A \rightarrow B\) with \(A \land (A \rightarrow B)\). One is saying "if \(A\), then \(B\)" and the other "\(A\), so \(B\)", which is the same as just saying "\(A\) and \(B\)". Indeed, if you were to define \(A \rightarrow B\) as being false when \(A\) is false, you would get the same truth table as \(A \land B\).

The reason that \(A \rightarrow B\) is true whenever \(A\) is false is that one should interpret wffs as rules that are true whenever they are satisfied and false otherwise. Let's illustrate this with two examples.

Imagine you have four cards, each containing a letter on one side and a number on the other. Now suppose you can only see their front side and that they are labelled "A", "B", "5" and "6". Which card/s do you need to turn to check that the rule "If there is an A on one side of the card, then there is definitely a 5 on the other" is satisfied?

If your answer is A and 5, you are wrong! You do indeed need to turn card A to check whether there is a 5 on the other side, but you do not need to turn card 5 to check whether there is an A on the other, as the rule doesn't specify which letter should be on the other side for a 5. The answer to this question is instead card A and card 6, as if there were to be an A on the other side of card 6, then the rule wouldn't be satisfied.

In case you are still a bit confused about this, here's a second example: You are at a bar which has the rule that anyone under 18 cannot drink alcohol. There are four people at the bar, all drinking something: the first person is 16 years old, the second is 24 years old, the third is drinking water and the fourth is drinking a beer. Which people should you "investigate" to check that the bar's rule isn't being broken? Clearly you do not care about the person who's 24 nor the one who's drinking water. You do however need to check the 16 year old as well as the person drinking beer.

Notice that those two examples are equivalent, with the differences that in one we have letters/numbers whereas in the other we have the age/drink. The first person is like card A, the second like card B, the third like card 5 and the fourth like card 6.

I think that a source of many people's confusion comes from their thinking that there should be some causal link between the hypothesis and the conclusion. "If pigs can fly, then Lisbon is the capital of Portugal" is true, because the rule is always satisfied. Pigs can't fly (at least as far as I'm aware), so whichever the conclusion, the implication is true, even though there is no link between the flying ability of pigs and the city that's Portugal's capital. In the same way, "if the sky is blue, then the Rubik's Cube was invented by Ernő Rubik" is true because that rule is satisfied, but "if Darth Vader is Luke Skywalker's father, then I (the author, not necessarily you) am the fastest speedcuber ever" is false, because Luke is Vader's son, but I'm definitely not the best speedcuber ever.

The reverse implication \(A \leftarrow B\) is an implication in which \(A\) is the conclusion and \(B\) is the hypothesis. \(A \leftarrow B\) is called the reciprocal of \(A \rightarrow B\). Thus, its truth table is

Equivalence is denoted by the symbol \(\leftrightarrow\). The meaning of \(A \leftrightarrow B\) is the same as \((A \rightarrow B) \land (A \leftarrow B)\). Thus, if you check the truth tables of \(A \rightarrow B\) and \(A \leftarrow B\) and apply the definition of and, you get

or more simply

Notice that \(A \leftrightarrow B\) is true if and only if \(A\) and \(B\) have the same truth values. An example of equivalent formulas are the implication \(A \rightarrow B\) and its contrapositive, defined as \(\neg B \rightarrow \neg A\):

Their truth values are all the same, so \((A \rightarrow B) \leftrightarrow (\neg B \rightarrow \neg A)\) is true for all rows.

The symbol \(\uparrow\) is the negation of conjunction, called nand or Sheffer stroke. As the negation of conjunction, its truth table is

or more simply

The symbol \(\downarrow\) is the negation of disjunction, called nor. Its truth table is (I let the details for you to deal with)

The last three connectives whose meaning remains to be defined are \(\not\rightarrow\), \(\not\leftrightarrow\) and \(\not\leftarrow\). These symbols are the implication and equivalence symbols with a stroke. They are the negations of \(\rightarrow\), \(\leftrightarrow\) and \(\leftarrow\) and thus their truth tables are as follows:

Notice that \(\not\leftrightarrow\) has the same meaning as \(\oplus\).

What is meant by taking the conjunction, the disjunction, the implication, or whatever other application of a binary logical connective \(\circ\) to \(n\) metavariables (at least 2) ranging from \(A_1\) to \(A_n\), is to create the wff \(\bigcomp\limits_{i=1}^{n}A_i\).

Evaluating a wff

With those definitions in place, we are now in position to evaluate the truth value of a wff given an assignment of either \(\mathcal{T}\) or \(\mathcal{F}\) to each individual metavariable:

1. Evaluate the subwffs for which the truth assignments have been made
2. Given evaluated subwffs, evaluate the compound wffs created from them by the binding of logical connectives by checking the corresponding row in the truth table for the respective connective

Some abbreviation notation for the metalanguage

Sometimes, instead of writing "If \(A\), then \(B\)" or "\(A\) implies \(B\)", I may write "\(A \Rightarrow B\)". Also, instead of writing "\(A\) is equivalent to \(B\)" or "\(A\) iff \(B\)", I may write "\(A\) iff \(B\)" or "\(A \Leftrightarrow B\)". For propositional logic, saying \(A \Leftrightarrow B\) means that \(A\) and \(B\) have the same truth table, i.e. that all values of \(A \leftrightarrow B\)'s truth table's final column are true.

Do not mistake the symbols \(\Rightarrow\) and \(\Leftrightarrow\) for \(\rightarrow\) and \(\leftrightarrow\). The first two belong to the metalanguage, even though they have the same meaning as the object language's!

I'll be making an exception though, where the meanings of \(\Rightarrow\) and \(\Leftrightarrow\) will be different than \(\rightarrow\) and \(\leftrightarrow\). When writing various successions of metalanguage's implications or equivalences \(\bigcomp\limits_{i=1}^{n}A_i\), where \(\circ\) stands for the symbol \(\Rightarrow\) or \(\Leftrightarrow\), possibly mixed, instead of having the same meaning as when \(\circ\) stands for \(\rightarrow\) or \(\leftrightarrow\), it will mean \(\left(\bigwedge\limits_{i=1}^{n-1}A_i\right) \circ A_n\).

Since the metalanguage's goal is to communicate ideas informally, I might be loose in the use of those metalinguistical symbols by adding dashes on top of them to indicate their negation or by writing things like

Notice that for that last statement, the commas in the sentence "\(A \Rightarrow B\), and \(A \not\Rightarrow C\), and \(B \Leftrightarrow C\)" are intended to indicate that the "and" has a lower precendence that the other connectives, as without them, the sentence would mean "\(A \Rightarrow \)(\(B\) and \(A\)) \(\not\Rightarrow \) (\(C\) and \(B\)) \(\Leftrightarrow C\)".

Source:

• Math that will make you think. Twice....maybeZach Star (Youtube channel)