Quantified (General) Statements
Quantified (General) Statements With Only One Predicate Letter
Quantified (General) Statements With Two Predicate Letters
Quantified (General) Statements With More Than Two Predicate Letters
Statements in predicate logic will either be a singular statement ( or a truth functional combination of singular statements) or a quantified statement (or, at worst, a truth functional statement that contains a quantified statement as a component). If the statement is quantified (i.e., general), it will either contain one predicate, two predicates, or more than two predicates. We will consider each of theses cases in turn.
About all that is involved in translating singular statements is to identify the predicate letter and individual constant to use. For example, the singular statement 'Jane is married' might use the predicate letter 'M' for the attribute of being married and the individual constant 'j' for Jane. So, the translation would be Mj.
A general statement that contains only one predicate letter will either say that everything has the property, or that nothing does, or that something does, or that something does not have the property. For example:
Everything is made out of matter.
Nothing is made out of matter.
Something is made out of matter.
Something is not made out of matter.
The translation of these kinds of statements is rather straightforward:
(x) ( Mx )
(x) ( ~ Mx )
(Ex) ( Mx )
(Ex) ( ~ Mx )
where (Ex) is the existential quantifier.
General statements with two predicates will be one of the four traditional categorical statements or a variation on one of them. The four traditional categorical statements are:
Type English Pattern Translation
A All F are G (x) ( Fx -> Gx )
E No F are G (x) ( Fx -> ~ Gx )
I Some F are G (Ex) ( Fx & Gx )
O Some F are not G (Ex) ( Fx & ~ Gx )
General statements with more than two predicate letters are almost always a variation on one of the four traditional categorical statements. If we can identify which of the four basic patterns is involved, we then have most of the translation done because we then know:
a. Which quantifier to use
b. The main connective for the propositional function
c. The major grouping of the parts of the statement
d. Whether there are curls and where to put them.
Medications should be taken immediately or thrown away.
This is basically an A type. It says that all medications are blah...blah...blah. So, the quantifier is universal, the main connective is a horseshoe, the antecedent is the subject word 'medication,' and the consequent is the predicate phrase 'should be taken immediately or thrown away.' So, we have:
(x) ( x is a medication -> x should be taken immediately or thrown away )
All that's left is the either/or in the consequent:
(x) [ Mx -> ( Ix v Ax ) ]
Reading back into English:
For all x, if x is a medication then x should be taken immediately or x should be thrown away.
In better English:
All medications should be taken immediately or thrown away.
Some medications are dangerous only if taken with alcohol.
This is clearly an I type. So, the quantifier is existential, the propositional function is a conjunction, the subject term 'medications' is the left hand conjunct, and the predicate phrase 'are dangerous only if taken with alcohol' is the right hand conjunct. So, we have:
(Ex) ( x is a medication & x is dangerous only if taken with alcohol )
All that is left is the 'only if' construction in the right hand conjunct:
(Ex) [ Mx & ( Dx -> Ax ) ]
In English:
There is at least one medication and if it is dangerous then it is taken with alcohol.
In better English:
Some medications are dangerous only if taken with alcohol.
No fireman is competent unless he has been well trained.
This is clearly an E type. It is basically saying that no firemen are competent with a qualification. So, the quantifier is universal, the propositional function is conditional, the antecedent is the subject phrase 'fireman,' and the consequent is the predicate phrase 'not competent unless well trained.' So, we have:
(x) ( x is a fireman -> x is not competent unless well trained )
All that is left is the 'unless' construction:
(x) [ Fx -> ( ~ Cx v Wx ) ]
Some talented actors are not famous.
This is an O type. So, the quantifier is existential, the propositional function is a conjunction, the subject term 'talented actors' is the left hand conjunct, and the predicate phrase 'are not famous' is the right hand conjunct. So, we have:
(Ex) ( x is a talented actor & x is not famous )
Evidently, the phrase 'x is a talented actor' is referring to things which are both talented and actors. So, we get:
(Ex) [ ( Tx & Ax ) & ~ Fx]
Reading back into English:
There is at least one talented actor who is not famous.
Or in better English:
Some talented actors are not famous.
Only F's are G's.
None but F's are G's
These are clearly A types. The only question is the order of the predicates. Both put the predicate phrase in the consequent:
(x) ( Gx -> Fx )
All F's and G's are H.
For example, all doctors and lawyers are professionals. It is tempting to translate this pattern as:
(x) [ ( Fx & Gx ) -> Hx ]
but this is a mistake. In the example, we are not referring to things which are both doctors and lawyers, but rather things which are either one are the other. So, the correct translation for the pattern is:
(x) [ ( Fx v Gx ) -> Hx ].
And the correct translation for the example is:
(x) [ ( Dx v Lx ) -> Px ]
Not all F are G
Not every F are G
Both of these are denials of an A type. So, a correct translation is:
~ (x) ( Fx -> Gx )
A better translation is to use the opposed O type:
(Ex) ( Fx & ~ Gx )
Not any F is G
This pattern is actually a variation on an E type:
(x) ( Fx -> ~ Gx )
If anything is an F, then it is a G.
If there is an F, then it is a G.
This one is an A type (although in a more complex setting it could be something else). So:
(x) ( Fx -> Gx )
Propositions can contain more than one quantifier in at least two different ways. One is to simply have a truth functional combination of independent quantified statements, for example:
If all students are honest, then no dorm room needs to be locked.
This is a conditional statement with an A type statement as its antecedent and an E type statement as its consequent. It could be translated as follows:
(x) ( Sx -> Hx ) -> (x) ( Dx -> ~ Lx )
where the predicates are Sx - x is a student, Hx - x is honest, Dx - x is a dorm room, and Lx - x needs to be locked.
Since we will now be using the letters 'a - t' as individual constants and the letters 'u - z' as individual variables, the example can also be translated as:
(x) ( Sx -> Hx ) -> (y) ( Dy -> ~ Ly )
which is better since each quantifier has its own variable associated with it.
The other way to have two or more quantifiers in the same statement is to embed a quantified statement within a larger quantified statement, for example:
If anyone is a thief, then if no dorm rooms are locked, then they will rob us blind.
This looks like an E type embedded inside a larger quantified statement. The embedded statement is functioning as a qualification. If we remove it, we have:
If anyone is a thief, then they will rob us blind.
As we saw above, the 'If any . . ., then it . . .' kind of construction is actually an A type statement. So, it could be translated as:
(x) [ ( Px & Tx) -> Rx ]
where the predicate letters are: Px - x is person, Tx - x is a thief, and Rx - x will rob us blind.
Using the predicates Dx - x is a dorm room and Lx - x is locked, we can translate the E type as:
(x) ( Dx -> ~ Lx )
So, the original, more complex statement could be translated as:
(x) { ( Px & Tx) -> [ (x) ( Dx -> ~ Lx ) -> Rx ] }
But this translation is not very good since it is not clear whether the 3rd, 4th, and 5th occurrences of the variable 'x' belong to the first quantifier or to the second. A better translation, then, is:
(x) { ( Px & Tx) -> [ (y) ( Dy -> ~ Ly ) -> Rx ] }
since each quantifier as its own variable associated with it and it is very clear which variable belong to which quantifier.
So, one tip for multiply general propositions is to always use a different variable for different quantifiers. If we do this consistently, we will not have to worry too much about the scopes of quantifiers since we will be able to tell at a glance which variables belong with which quantifiers.
Another tip bears on how to tell (usually) whether you are dealing with a multiply general proposition that is a truth functional combination of independent quantified statements or is one that has quantified statements nested inside of a larger quantified statement. In general, the construction:
If any . . ., then . . .
could be either.
If there is a pronoun (he, she, it, they) in the consequent that refers back to whatever is being referenced in the antecedent, the construction is essentially an A type. So, the translation will need to begin with a quantifier whose scope is the entire statement in order to bind the variable replacing the pronoun to the variables in the antecedent. The example above about thieves robbing us blind is like this. If there is another quantified statement involved, it (almost always) will be embedded within. The example about thieves robbing us blind if dorm rooms are left unlocked is like this. It is a good idea to look for this pattern (pronoun referring back to the antecedent). If it is present, one way to translate is to get the 'if any . . .' part while leaving the embedded statement(s) in English. In the above example:
(x) { ( Px & Tx) -> [ if no dorm rooms are locked, then Rx ] }.
Then translate the remaining connectives and the embedded statement:
(x) { ( Px & Tx) -> [ (y) ( Dy -> ~ Ly ) -> Rx ] }.
If there is no pronoun referring back from the consequent to the antecedent, you (almost always) have a truth functional combination of independent statements. In this case, the 'if any . . .' (almost always) means 'if there is at least one . . .' and so you will use an existential quantifier whose scope is just the antecedent of the entire statement. For example:
If anyone is a thief, then all doors should be locked.
Apparently this means if there are any thieves at all (at least one), we should lock our doors. The antecedent will be an I type and the consequent will be an A type:
(Ex) ( Px & Tx ) -> (y) ( Dx -> Lx ).
Finally, the constructions
If any . . ., then . . .
If every . . ., then . . .
If some . . ., then . . .
etc., are usually synonymous and can be treated the same as the 'if any . . .' construction.