In propositional logic, we have worked with meaning at the level of simple and compound statements. What does it mean to say we have worked with meaning? It means that what we’ve been doing is representing meaningful units, i..e, sentences (statements in the case of Logic, as opposed to numbers, for instance, in the case of Mathematics). The smallest entity or unit we could identify was a simple statement, and the only representations we could make were to use a capital letter to stand for a single assertion, and to use operators to stand for combinations of single assertions into compound ones (conjunctions, disjunctions, etc.).
Now we can use the tools and machinery we’ve learned in propositional logic to go deeper into the representation of meaning. We can go inside the statements themselves and represent what is going on in there.
What is going on in there? Something is being predicated of something. That is, something is being said about something. We focused on this when we studied categorical propositions a while ago.
We can show the variety of ways that something can be said about something using the symbols of propositional logic. This takes us back to Categorical Propositions, but with a means of representation that dovetails with the symbolization we’ve become accustomed to.
Consider these examples:
All lemons are citrus.
All carrots are vegetables.
All dentists are doctors.
All leprechauns are short.
If you hesitate to agree to any of these, it’s probably because you don’t know if you want to go on the record saying that any sentence about leprechauns is true, since, after all, they don’t exist. But you might be willing to agree that
If anything is a leprechaun, it is short.
You understand that since this is a conditional statement (what we call a material conditional in Logic), it is true in those cases where its antecendent is false. So there’s no harm done in saying “If anything is a leprechaun,” since no thing is.
So if anything is a dentist, it’s a doctor, and if anything is a carrot it is a vegetable, and if anything is a lemon it is a citrus. These claims all make sense. Perhaps you can also see that, although they are conditional statements, they are equivalent in their truth value to what we were calling “universal affirmative” or “A” statements earlier. That shows you the beginning of how we can use the symbolization we’ve become used to in propositional logic to go inside the proposition and represent the predication it makes. Any universal statement can obviously be recast or rethought as being a conditional statement: “If anything is a rabbit, it is not a bear” expresses “no rabbits are bears.” “If anything is a pillow, it is not a tree” expresses “no pillows are trees.” So E and A statements both can be represented as conditional statements. The difference is just that in the conditional version of the E, the consequent is negated.
“No Popes are Hindus” will be “For any x, if x is a Pope, then x is not a Hindu.”
Here’s how we’re going to represent it:
(x) (Px > ~Hx).
When we use the construction “(x),” this stands for “For any x.” Using this makes it clear that we are writing a single statement, because “(x)” is the main operator of the statement, and the “⊃” is within the statement, connecting the subject term to the predicate term. This expresses what “are” expressed in the traditional categorical propositional form.
Now, however, we can use this same “machinery” to treat this expression (No popes are Hindus) as a statement that we can put into relation with other statements, like this one: Francis is not a Hindu, or ~Hf in this new approach.
As you can figure out, if “~Hf” means Francis is not a Hindu, then the capital letter “H” is now representing the predicate or property of “being a Hindu,” and the lower-case letter “f” is representing the individual, Francis.
So we can now say
If no popes are Hindus, then Francis is no Hindu
(x) (Px ⊃ ~Hx) ⊃ ~Hf.
The main operator of this line is the second horseshoe; the first horseshoe is governed by the phrase “for any x,” and basically means “are.” So what we have here is a conditional statement linking two simple statements. What we are doing in predicate logic is representing the predications that constitute the simple statements –so they may not look as simple as they did before, but they do still each contain one predication.
So, reviewing up to this point, what we’ve seen is that upper case letters now stand for predicates. These can be simple, represented by single words like “delicious” and “red,” as in (x) (Dx ⊃ Rx), meaning “everything that’s delicious is red.” Or they can be complicated, but still represented by single letters, like “big fan of the Colbert Report, as in “Ff” standing for “Francis is a big fan of the Colbert Report.”
That accounts for how to represent universal statements, and for how to represent singular statements. This is progress over that awkward phrasing we had in Categorical Logic, “All people identical to Francis are big fans of the Colbert Report.” Now we will use the lowercase letters “a” – “w” as what we’ll call “constants,” to name individuals. These individuals might be people, or planets, or buildings, or nations, etc.; the point is that they are not classes of things, but individual referents.
But there is another kind of statement to represent, the particular ones, which are also called “existential statements.” For this we have another new symbol.
(∃x) is that new symbol. Yes, it is a backwards “E.”
I wasn’t consulted in the choice of this symbol, so I’m not going to defend it. But you get used to it, and it’s kind of fun. In more advanced logic contexts, it is even used with an exclamation mark, “(∃!)” and is called (by at least some logicians), “E shriek.” As I recall, “E shriek” is used for talking about “worlds” or “possible worlds,” whereas the more mundane “(∃x)” always means only “there is” or “some.”
That’s right, we are back to the basic ideas of categorical logic: “All,” “No,” or “Some,” or else “this individual.”
So try to imagine what this symbolization means: (∃x) (Sx ∙ Rx), where “S” stands for “is a senator” and “R” stands for “is a Republican.”
Give up? How about simply “Some Senators are Republicans”? Reading the symbols a bit more closely, we’d say it reads “There is an x such that x is a Senator and x is a Republican.” So it will be no surprise that (∃x) (Sx ∙ ~Rx) means “Some Senators are not Republicans.”
The first and major task in predicate logic is getting accustomed to representing these predications in this new way that combines the machinery of propositional logic with the focus of categorical logic. You are going to love it!
Here are examples to practice with:
1. All bears are dangerous
(x) (Bx ⊃ Dx)
2. No sloths are energetic
(x) (Sx ⊃ ~Ex)
3. Some trees are willows
(∃x) (Tx ∙ Wx)
4. Some misanthropes are not idiots
(∃x) (Mx ∙ ~Ix)
OK, easy enough, that’s just the four basic forms.They never change; these are the basic structures of the A, E, I and O statements of categorical logic, re-presented in the garb of symbolic logic.
But what if you complicated the predications, like this:
5. All bears are dangerous carnivores.
That requires two elements in the predicate:
(x) (Bx ⊃ (Dx ∙ Cx)):
If x is a bear, then x is dangerous and x is a carnivore.
6. Some belligerent jerks need counseling. This would have to be represented as:
(∃x) ((Jx ∙ Bx) ∙ Nx),
which says literally “there is an x such that x is a jerk and x is belligerent and x needs counseling.”
7. Oranges and apples are fruits.
You’ll probably want to say
(x) ((Ox ∙ Ax) ⊃ Fx),
but that won’t really be good enough. Why not? Because that says “for any x, if x is an orange and x is an apple then x is a fruit,” and as you well know, no oranges are apples, so that’s just not right. Instead you need to make it a disjunction (x) ((Ox v Ax) ⊃ Fx), saying for any x, if x is an orange or x is an apple, then x is a fruit.
Here are more you can practice on. The answers to them are on the next page. Try them yourself first. One major task is to determine how many predicate letters you should use; we could have disagreements, but most of the time we probably won’t. Try to show as much meaning, in general, as it makes sense to show. As in propositional logic, always show negations explicitly.
1. Paris is beautiful.
2. Tokyo is overcrowded.
3. If Paris is beautiful then it’s popular.
4. If Gonzales is tortured, then he’ll talk.
5. All lawyers are members of the Bar Association.
6. Some flowers are not pretty.
7. All laptop computers have batteries.
8. No students carry cellphones, but Mary is not a student.
9. Not all Senators are communists.
10. Obama is running for President and so is Romney.
11. Obama’s not a Muslim and neither is Santorum.
12. Either Clinton will be the candidate or there will be no woman candidate.
13. Horses exist, but not unicorns.
14. Sea lions are mammals.
15. Squirrels live on this campus.
16. Only snakes and lizards thrive in the desert.
17. Peaches are delicious unless they are rotten.
18. Dogs bite if they are frightened or harassed.
19. Bears and eagles are talked about alot on The Colbert Report.
20. Sean Penn and Steven Colbert love metaphors.
21. Only arguments can be valid.
22. Arguments are sound if and only if they are deductive, valid, and have all true premises.
23. All sentences that are statements are either true or false.
24. If all sentences are statements, then they are all true or false.
25. Only inductive arguments are weak or uncogent.
26. A categorical syllogism is invalid if it has an undistributed middle term.
27. A good violin is rare and expensive.
28. Violins and violas are stringed instruments.
29. A room with a view is available.
30. A room with a view is expensive.
31. Hurricanes and tornados are violent and destructive.
32. Scooter is guilty if all the witnesses told the truth.
(Use the same predicate letters for 32 and 33.)
33. If any witnesses lied, then Scooter’s innocent.
34. If all journalists are interesting people, then Bob Novak is an interesting person.
35. Not all jazz fans like Monk.
36. Balcony seats are never chosen unless all the orchestra seats are taken.
37. Some employees will get raises if and only if some managers are overly generous.
38. Some local artists ask too much for their work, but not Bill Harris.
39. If the scientists and technicians are conscientious and exacting, then some of the mission directors will be either pleased or delighted.
40. The physicists and astronomers at the symposium are listed in the program if they either chair a session or read a paper.
41. Some words are predicates, but not “existence.”