Phil 340: Logical Notation

Here’s a guide to some of the logical notation Lewis uses in his article.

First, we can abbreviate a long sentence like this:

Thor gave a speech and then Thor thanked Odin for helping to defeat Loki.

As just T or perhaps T[Thor]. The advantage of second abbreviation is then we can also talk about T[Freyja], which would be the same sentence, but with all occurrences of Thor replaced with Freyja, thusly:

Freyja gave a speech and then Freyja thanked Odin for helping to defeat Loki.

We can also work with abbreviations like T[Thor, Odin, Loki]. Then T[Freyja, Frigg, Baldr] would be:

Freyja gave a speech and then Freyja thanked Frigg for helping to defeat Baldr.

and T[Odin, Freyja, Frigg] would be:

Odin gave a speech and then Odin thanked Freyja for helping to defeat Frigg.

Sometimes instead of names, we use symbols like t1, t2, t3. When we want to talk about these three symbols as a combined triple, we can write it as t1, …, tn, and we call that an “n-tuple” (in this case, n = 3).

We sometimes write the n-tuple t1, …, tn as t (with an overbar) or as boldface t.

Thus if t1, t2, t3 = (Thor, Odin, Loki) and v1, v2, v3 = (Odin, Freyja, Frigg), then T[t] is the first of our sentences above, and T[v] is the fourth.

Another bit of logical notation that philosophers use is to write:

∀x (if x a Norse god, then x is famous).

to mean “Anything that is a Norse god is famous.” (This sentence doesn’t require that there actually be any Norse gods, just that if there are, they are famous.) Or to write:

∃x (x a Norse god, and x is famous).

to mean “There is at least one (possibly more) Norse god, and he/she/they are famous.”

We can also write:

∃x₁ ∃x₂ ∃x₃ (x₁, x₂, and x₃ are Norse gods, and T[x₁, x₂, x₃])

This sentence doesn’t reqire that x1, x2, and x3 be distinct people; thus one way for that sentence to be true would be if x1, x2, x3 = (Thor, Loki, Loki) and Thor was thanking Loki for helping to defeat Loki.

These symbols ∀ and ∃ are called “quantifiers.” The first one looks like it does because it has a meaning similar to the English “all” and “any”; the second has a meaning similar to the English “there exists”.

Sometimes we want to say not just that there is at least one (possibly more) things that are such-and-such; instead we want to say there is exactly one thing (or n-tuple of things) that are such-and-such. In English, for examnple, we might want to say:

There is a unique triple of things (x₁, x₂, x₃) such that T[x₁, x₂, x₃]

which could also be written as:

There is a unique triple of things x such that T[x]

Lewis writes this as:

∃₁ (x₁, x₂, x₃) such that T[x₁, x₂, x₃]

or, abbreviating the n-tuple, as:

∃₁x such that T[x]

Instead of ∃₁, you can also see this written as ∃!. (Not as E!, that has a different meaning.) Or you can see it written like this:

∃x ∀y (T[y] ↔︎ x = y).

The notation p ↔︎ q means that “If p is the case, then q is the case; and also if q is the case, then p is the case.” Or as philosophers abbreviate it, p iff q. Iff stands for for “if and only if”. So expanding that last sentence, it’s equivalent to:

∃x ∀y (x = y → T[y] and T[y] → x = y).

which is equivalent to:

∃x ∀y (T[x] and T[y] → x = y).

Sometimes we think there is a unique thing (or n-tuple) that satisfies a certain condition, and we want to talk about that thing, without using its specific name. (We may not know its specific name.) In such cases, we can use what philosophers, linguists, and logicians call a “definite description.” In English that’s written using the word the. So if there is a unique Norse god who defeated Loki, then we can talk about “the Norse god who defeated Loki”, or, more formally, as:

the x: x is a Norse god who defeated Loki

If Thor defeated Loki and nobody else also defeated Loki, then this description will designate Thor. If nobody defeated Loki, or multiple people did, then that would be called an “improper description.” Philosophers disagree about what happens with them. Russell talked about the description “the present king of France.” He was writing long after France stopped having a king. Russell’s view was that sentences like “The present king of France is bald” should be false, since there is no present king of France. Other philosophers argued that this sentence is neither true nor false.

Instead of using the word the, philosophers sometimes write definite descriptions using an upside-down Greek letter iota, so it looks like this:

℩x: x is a Norse god who defeated Loki

Lewis does this at one point in his article, where he writes:

t = ℩x T[x]

What he means by that is that t1, …, tn are identical to the unique x1, …, xn such that T[x1, …, xn].