Is there any more info I can provide...

My understanding is this: A predicate is a functiont hat when invoked, returns a truth table.

For #1 we have the expression: "The police suspect that [1] is a murderer."

We can put whatever designator we want in to stand as [1], and in my estimation gramatically the predicate would be something like "[1] is a murderer" but logically, I don't think we could get a truth table out of this, becuase the truth value of the "predicate" will have no bearing on whether the police suspect who or whatever "1" is...

As a result, though I first though 1 would be a predicate, I now lean towards it not being. In contrast 10 would be a predicate...

I might not be right, but in my mind its simplified for me...

But I could use some affirmation or correection...

All these expressions contain English predicates, a predicate is simply the part of a sentence that assigns properties to its subjects, the question is how they can be formalised in logic.

You can't formalise it like that because "suspect" doesn't express a relation. If you've studied a bit of the set theory behind logic, this will be easier to understand. One place predicates express unary relations, which are simply sets e.g. {John, Roger, Tim}, and because first order logic (and set theory) is an extensional language we should be able to replace any designator in a sentence with another that refers to the same object without changing the truth value of the sentence. For example, 'George Orwell wrote 1984' and 'Eric Blair wrote 1984' both have the same truth value when we formalise them, but English does not always behave this way (it does in the Orwell case, but not in the case of your example). In your example, the Police may suspect Betrand Russell murdered Wittgenstein, but let's say that Bertrand Russell has changed his name to Peter Strawson has moved to Brazil. The sentence: 'The police suspect Peter Strawson is a murderer' would be false because the police may have no idea that Russell is living under the alias, Strawson, and may never have heard of such a person, let alone suspect him of murder. The best formalisation is: [1] suspect that Mr. Whoever is a murderer.


Best expressed as two place predicates in the form: It is necessary that [1] = [2]


The first two of these have the same problem as A. F is the way in which all such sentences must be formalised. Correct formalisations:
D. [1] expects to find something.
E. [1] believes something< 7.



Same problem as above. Formalisation:

([1] believes that something < 7) -> ([1]+[2]=[3])


This seems correct.


These two aren't incorrect so to speak, but they are better formalised as two place predicates as this more deeply reveals the logical structure of the sentences:
[1] has never met [2].

Ok. I think I am comfortable with these five...

A. The police suspect that [1] is the murderer.
I think the same principle at work in the "Inge knows that [1] wrote a novel" is at work here. Inge may know that Mark Twain or George Orwell wrote a novel, but not necessarily that Samuel Clemens or Eric Blair did. Even though Twain and Clemens, and Orwell and Blair are the same person, they render different truth values. Similarly, while the police might suspect John Doe as the murderer they might not suspect that Jim Doe (his alias) is the murderer. As a result, this is not a predicate. Were we to know the identity of [1] (let's suppose it is John Doe) we would have the expression "The police suspect that John Doe is the murderer," and were we to delete "the police" we would be left with the expression "x suspects that John Doe is the murderer," rendering, I think, "suspects that John Doe is the murderer" as the predicate.

F. [1] believes that 4 < 7.
This statement is true of anyone who believes that 4<7, and is false of anyone who does not believe that 4<7. As Mark Twain or Samuel Clemens can be inserted into the expression "[1] wrote a novel" as it is the individual named that is crucial and not how s/he is named, similarly here it can be as well. This is a predicate.


H. It is contingent that [1] is the murderer.
Where in expression A, we agree that the police might suspect that John Doe but not necessarily his alias Jim Doe, in this question it is not how the person is identified but simply that they are, and as a result, whether the murderer is known as Jim Doe or his alias John Doe, it is the person who such various names represent that is seen as the murderer. As a result, this is a predicate.

I. Tom has never met [1].
J. The Pope gave his blessing to [1].
These are predicates. In both of these examples, I think the subjects ("Tom" and "The Pope") might justifiably be deleted rendering our expressions "Tom has never met [1]" as "x has never met y" and "The Pope gave his blessing to [1]" as "x gave his blessing to y."

As for the others, I am not so sure yet...

Correct, I'll try and go into a bit more detail on the others.


A predicate is something that ascribes an object a property, or establishes a relation between two or more objects. The relation being established by these predicates is an identity holding between two objects (the two things on either side of the equals sign). If you where to fully formalise B into predicate logic (bearing in mind that it's a rather troublesome modal claim for predicate calculus to be dealing with) you would get:
∃xPxa
P: It is necessary that [1] = [2]
a: 2

Which reads "There exists an x such that it is necessary that x = 2". We've split the sentence into three parts: 'P' is the predicate, 'a' is a constant, and x is a variable.

If you want to go beyond the logical structure, and into meaning, the predicate, P, is satisfied by an infinite amount of values, any ordered pair <e,f> such that 'e' is identical to 'f' will give us a true sentence when we plug it in to the predicate. Therefore, we say that the semantic value of the predicate P is the set of ordered pairs {<e, f> : e is identical to f}. This is why the definition of a predicate you have been given, as something which yields a truth table, works; although, it's not really correct. Because relations are set theoretic objects, in order to be a relation it must be possible to replace any of the elements in the relation with another thing that is identical to it. This is why {<e, f>: e believes f} is not a relation, because sometimes we can't replace Eric Blair with George Orwell. A good rule for deciding whether or not an English predicate can be formalised into a logical predicate is to ask yourself, "does it express a relation?"


Consider the two sentences:
Oedipus expects to find Jocasta in his bedroom.
Oedipus expects to find his mother in his bedroom.

In the Greek myth, Jocasta is Oedipus' mother, but Oedipus doesn't know this. He may expect to find Jocasta in his bedroom, but not his mother. To put it more formally {<e, f>: e expects to find f} is not a relation.


'Belief' doesn't express a relation. Consider what happens when you ask most people, "Do you believe Mount Everest is the highest mountain in the world, or do you believe that Qomolangma is the highest mountain in the world?" Most people will say Everest, not knowing that Qomolangma is the Tibetan name for the same mountain.

I your example, you could plug in pi, or you could plug in "the ratio of a circle's circumference to its diameter". If Prince Charles does not know that pi is defined as the ratio of a circle's circumference to its diameter, then the truth value of the sentence is changed.


The If ... then ... bit of the sentence isn't formalised as a predicate, but it expresses a relation between two predicates: "If predicate X is true then predicate Y must be true". The two predicates it expresses a relation ship between are:
Prince Charles believes that [1] < 7 (which we have just shown is not a correct formalisation)

and

2+2=4
This does express a pretty clear cut relation between three objects, "The sum of one object and another object is identical to a third object", and there is no problem in formalising it is a three place predicate letter.

The sentence G as a whole, cannot be formalised as a single predicate letter, it should be formalised into two predicate letters, joined by the logical 'If... then... " symbol (the material conditional).