Symbolizing Monadic Predicate Expressions
Names and Predicate Expressions
Here are some very simple sentences:
Spot is a dog
Jones is taking logic
Clarke is a cat
is a cat
Andrew drinks beer
Each of these sentences has a common structure. Each consists of a name, such as 'Spot' or 'Clarke', combined with a property. In each sentence, the property is predicated of the thing named. As the examples show, two different sentences might combine the same phrase with different names, or the same name with different phrases. We can picture this structure as follows:
'Spot' + '___ is a dog' = 'Spot is a dog'
'Andrew' + '___drinks beer' = 'Andrew drinks beer'
The expression '___ is a dog' is an expression with a
variable space in it. Any expression with a blank space is predicate expressions.
The property in the expression is predicated of (ascribed to) the thing
named. Remember, a predicate that contains free occurrences of an
individual variable is an open sentence, and lacks a truth value. A predicate expression
becomes a true statement when its blank space (or spaces) is (or are) filled in with
Symbolizing Simple Monadic Predications
We have expanded our symbolic language to accommodate the structure of English statements that involve predicates and quantifiers. . We will need two new classes of symbols, corresponding to the two new types of expression we've introduced (names and predicate expressions).
Table 1: Tools for Symbolizing Variable, Names, Predicates and Quantifiers
|For English||for example||we use||Symbolized as. . .|
|Names||Bob, Carol, Alice||Individual Constants||a, b, c|
|Variables||anyone, someone||Individual Variables||x, y, z|
|Properties||is red, is tall, is fun||Monadic Predicates||Rx, Tx, Fx|
|Relations||is in love with, is richer than, is between||Relational Predicates||Lxy, Rxy, Bxyz|
|Universal Claims||all, any, every||Universal Quantifier||∀x|
|Particular Claims||some, many, most, a few||Existential Quantifier||∃x|
We translate names with lower-case letters from the beginning of the alphabet. Just to make sure we never run out of names, we can add subscripts to these letters as much as we need.
Variables are letters taken from the end of the alphabet, with or without subscripts (we allow ourselves to have as many of these as we need:
Don't confuse variables with names: variables are lower-case letters from the end of the alphabet (u, v, w, x, y, z), while names are lower case letters from the beginning of the alphabet (a, b, c, d).
Symbolizing Predicate Expressions
Paraphrase. Play with different paraphrases of the English until you have one that is (1) equivalent to the original and (2) easier to translate than the original. In paraphrasing, remember that we have only three quantities: all, some, and none. Try to restate the sentence using one of these words explicitly.
Easy Cases: Predicates and Names
The easiest predicate expressions to symbolize are those which utilize only predicates and names. Examples of sentences of this sort are:
Adams is a Democrat.
Earth and Jupiter are both planets.
Neither Baxter nor Douglas is taller than Collins, even though both are Democrats.
In order to symbolize these expressions, we first need a dictionary. For our names, let:
'a' = Adams, 'b' = Baxter, 'c' = Collins, 'd' = Douglas, 'e' = Earth, and 'j' = Jupiter
For our predicates, let:
Dx = 'x is a Democrat', 'Px' = 'x is a planet', 'Txy' = 'x is taller than y'
In our symbolizations, the predicate letters will remain constant, but the variables will be replaced by names.
'Adams is a Democrat' becomes 'Da', 'Earth and Jupiter are both planets' becomes '(Pe ● Pj)', and 'Neither Baxter nor Douglas is taller than Collins, even though both are Democrats' becomes '~(Tbc ▼ Tdc) ● (Db ● Dc)'.
Easy Cases: Predicates and Quantifiers
Which quantifier? In deciding between quantifiers, ask yourself whether the English sentence commits itself to the existence of something or whether it remains non-committal. In the former case, use an existential quantifier; in the latter, use a universal quantifier. Remember how we symbolized the four basic categorical propositions.
Quantified predicate expressions are only slightly more challenging than predicate expressions that rely on just predicates and names. Using the same dictionary as above, and adding
'Rx' = 'x is a person'
Some people are Democrats.
This sentence asserts that there exists at least one x such that x is both a person and a Democrat.
(∃x)(Rx ● Dx)
Suppose you wanted to say that some people are Democrats and some aren't. This statement is a conjunction of 2 existentially quantified propositions.
(∃x)(Rx ● Dx) ● (∃y)(Ry ● ~Dy)
Why did I switch to a 'y' for my second quantifier? To avoid confusion! You could use 'x' again, but I think it is a very good rule to always use a new variable whenever you introduce a new quantifier.
Note: It is incorrect to try to express our statement as (∃x)(Rx ● (Dx ● ~Dx)) because such a an expression actually says "There is at least one person who is both a Democrat and not a Democrat'. Even with the amount of flip-flopping that goes on in modern politics, it is difficult to imagine this formula ever coming out true.
NOTE: Existential quantifiers typically take conjunctions. "Some humans are inhumane": "(∃x)(Hx ● Ix)". To see this, paraphrase the original thus: some things have two properties, namely, being human and being inhuman.
NOTE: Watch the scope of your quantifier. Be sure that all of the individual variables in an expression are bound by a quantifier lest your formula be an open sentence lacking a truth value.
Suppose you want to assert that 'All Democrats are people'. This statement says that 'for any object x, if x is a Democrat, then x is a person'.
(∀x)(Dx → Rx)
NOTE: Universal quantifiers typically take conditionals. All humans are mortal: "(∀x)(Hx → Mx)". To see this, note that statements about "all" things of a certain kind (e.g. all humans) contain an implicit "if, then" structure: For all things in the universe, if they are humans, then they are mortal.
To symbolize 'All frogs are green' letting 'Fx' = 'x is a frog'; 'Gx' = 'x is green'
(∀x)(Fx ● Gx) is WRONG; this says: for all x, x is a frog and x is green, i.e., everything is a green frog.
(∀x)(Fx → Gx) is RIGHT; this says for all x, if x is a frog, then it is green
Mildly Challenging Cases: Mixing Names with Quantifiers
Sometimes we need to mix variables bound by quantifiers and individual constants together in the same statement. Consider:
If anything is a planet, then Jupiter is.
This statement is a conditional whose antecedent is a quantified expression and whose consequent is a predicate statement using a name. Paraphrasing this statement, we get something like:
If there are any planets at all, then Jupiter is one of them.
(∃x)Px → Pj
NOTE: This formula is in a from that is called purified form-- there are no variables within the scope of any quantifier not bound by that quantifier. The formula is equivalent to (∀x)(Px →Pj), which is in prenex form--all the quantifiers to the left, the entire matrix within the scope of every quantifier-- but I prefer the former version. Limiting the scope of your quantifiers as much as possible helps avoid errors. Later we will learn how, and why, we expand their scope. As a general rule, symbolize in purified form, or as close as you can get to it.
Suppose we know that both Baxter and Douglas are dedicated Republicans. One might say, a bit facetiously:
If either Baxter or Douglas is a Democrat, then everybody is!
After paraphrasing, this becomes:
(Db ▼ Dd) → (∀x)Dx (We do NOT put parentheses around Dx because of the scope rule.)
Problem Cases: Quantifiers and Negations
When we add negations to quantified statements, symbolization can become tricky.
No F's are G's. We translate this as "(∀x)(Fx → ~Gx)". By transposition, this is equivalent to ∀(∀x)(Gx → ~FAx)∀; hence in translation, use either one. If no F's are G's, then no G's are F's.
Not all / all not. Logicians have little patience with English speakers who confuse these expressions. "Not all" means "some are not", not "none are". "All are not" means "none are". For example, if you want to warn the lady in Led Zeppelin's "Stairway to Heaven" that glittering things are sometimes gold, but sometimes not gold, then you should say "Not all that glitters is gold." Despite this, the proverb says "All that glitters is not gold," which is absurd; it says that no glittering things are gold, not even gold.
Suppose that you need to symbolize "Not all people are Democrats". An easy way to do this might be simply to symbolize "All people are Democrats" and then stick a negation onto the front prducing ∀~(∀x)(Px → Dx)∀. This formula, however, is neither in purified nor prenex form--it is the negation of a formula in prenex form. In order put this formula into prenex form (and also purified form, in this case they are identical), we need to drive the negation across the quantifier. The rules for moving a negation from one side of a quantifier to the other are called quantifier negation rules. Whenever a negation passes through a quantifier, the quantifier switches from universal to existnetial and vice versa.
Negation (QN) equivalences
(1) ~(∃x)Fx :: (∀x)~Fx
(2) ~(∀x)Fx :: (∃x)~Fx
rules can be mapped directly onto categorical propositions, giving us:
Categorical quantifier negation (CQN) equivalences
Categorical quantifier negation (CQN) equivalences
(1) ~(∀x)(Fx → Gx) :: (∃x)(Fx × ~Gx) ~A :: O
(2) ~(∃x)(Fx × Gx) :: (∀x)(Fx → ~Gx) ~I :: E
(3) ~(∀x)(Fx → ~Gx) :: (∃x)(Fx × Gx) ~E :: I
(4) ~(∃x)(Fx × ~Gx) :: (∀x)(Fx → Gx) ~O :: A
Above, we listed 'any' along with 'every' and 'all' as one of the words used in forming simple universal quantifications. In combination with negations, however, 'any' and 'every' don't behave alike. Compare these two sentences:
Clarke didn't eat every bird.
Clarke didn't eat any bird.
The first sentence leaves room for a lot of bird-eating on Clarke's part, but the second says that she's totally innocent. This problem becomes most pronounced once we introduce relational predicates and multiple quantifiers.