It’s understandable you’d find some of these a tad difficult; translating sentences from English to Predicate can sometimes be very challenging, as you need to try and preserve the semantics of each sentence as much as possible.
Let’s have a look at your first side problem, which has the form:
“All men are happy”.
When it comes to symbolising this sentence, we need to create what’s called an interpretation, which is just the kind of semantics we attach to the symbols (you’ll see what I mean as I go along). In this case, it matters what kind of symbols we use. So, an interpretation of the sentence above looks like this:
Domain:People
Mx:x is a man
Hx:x is happy
In order to come up with a symbolisation for this sentence, one useful way is to gradually introduce the logic into the English sentences, like this:
∀x(if x is a man then x is happy)
∀x(x is a man→ x is happy)
∀x(Mx→Hx)
Now, the symbolisation of the sentence is:
∀x(Mx→Hx)
Breaking a sentence up into its semantic components is a good way to understand what a sentence is actually saying. I think this might help explain why your symbolisation of this sentence as ∀x(Mx∧Hx) is incorrect. If we apply this process backwards, we get:
∀x (x is a man∧ x is happy)
∀x(x is a man and x is happy)
Everyone is a happy man
It’s clear that “Everyone is a happy man” is not semantically equivalent to “All men are happy”. In the former case, we need to specify the fact that the property of “happiness” only applies when the person is a man, not when they could be anybody. In the latter case, we are essentially saying that everything within the scope of the domain (namely people) possesses the properties of “happy” and “man”, which is clearly not what this sentence means. But we need to make an important observation. In almost all cases (and by that I mean cases where the English sentence actually makes sense), the main operator inside a universal quantifier is a conditional statement, whereas the main operator inside an existential quantifier is generally an and statement. The reason for this is that in the English language, we tend to refer to Some x is y to mean ‘there is at least one x and it has the property y’, whilst we refer to All x is y to mean ‘everything that is in x must also be in y’. The word ‘and’ in the former sentence pretty clearly denotes the logical hat symbol, but the phrase ‘must also be’ is actually equivalent, at least in meaning, to the conditional.
I will assume that you understand my reasoning on your first side example and proceed to do the same from now on, with the same logical analysis. Now let’s have a look at your second side example, which faces the same problem I’ve just indicated above (namely a conditional inside an existential):
“Some men are happy”.
The construction of the symbolisation looks like this:
∃x(x is a man and x is happy)
∃x(x is a man∧ x is happy)
∃x(Mx∧Hx)
Now, the symbolisation of the sentence is:
∃x(Mx∧Hx)
But a deconstruction of ∃x(Mx→Hx) looks like this:
∃x (x is a man→ x is happy)
∃x(if x is a man then x is happy)
There is a person who is happy if they are a man.
Clearly, “There is a person who is happy if they are a man” is not the same as our original sentence. In the former case, it states that some members of the set of men are happy; this sentence allows for the fact that some men may not be happy. In the latter, someone is a man
only when they are among the set of happy people; this sentence does not allow for the fact that some men may not be happy.
Now let’s tackle your so-called Actual Questions.
*QUICK NOTE: “All predicate statements are correct for their respective English statements”. You’ll actually see that this isn’t quite true, and that there are small errors in some of the symbolisations.
1) “A goal without a plan is just a dream”
Our interpretation is:
Domain:People
Gx:x is a goal
Px:x is a plan
Dx:x is a dream
Our symbolisation is:
∀x(if x is a goal and does not have a plan then x is just a dream)
∀x(if x is a goal ∧ does not have a plan then x is just a dream)
∀x(( x is a goal ∧ does not have a plan)→ x is just a dream)
∀x(( Gx ∧ ¬Px)→ ~Dx)
A couple of notes. The original English sentence is incredibly ambiguous. It is unclear whether ‘A goal” refers to a single goal, or if it implicitly refers to ‘Any goal’. For simplicity’s sake, I have assumed the latter, as an assumption of the former would produce a different formula. The second point I want to make is the use of brackets; it is also incorrect. The only instances when you can omit brackets are as follows:
*Successive application of ∨, i.e. ∨…∨
*Successive application of ∧, i.e. ∧…∧
*Any outer brackets
Whilst these conditions do not apply to Predicate in the strictest sense, the reason why it is okay to do so in a more relaxed sense is because none of these rules affect the truth-functional properties of a statement. Any other bracket omission does, and unfortunately, your ‘correct’ solution proves somewhat ambiguous. Does it mean
∀x(( Rx ∨ Mx)→ ~Gx) or ∀x(Rx ∨( Mx→ ~Gx))? So, the correct solution proves to be ∀x(( Rx ∧ Mx)→ ~Gx), and from now on I will take the problem of brackets as a given.
2) “Rainy Days and Mondays always get me down”
Our interpretation is:
Domain:Days and Emotions
Rx:x is a rainy day
Mx:x is monday(s)
Gx:x is gets me down
Our symbolisation is:
∀x(if x is a rainy day and x is monday then x gets me down)
∀x(if ( x is a rainy day ∧ x is monday) then x gets me down)
∀x(( x is a rainy day ∧ x is monday)→ x gets me down)
∀x(( Rx ∧ Mx)→ Gx)
Unfortunately, your supposedly ‘correct’ statement is in fact incorrect again, but for different reasons this time. The main issue here is the use of ∨ instead of ∧. The former refers to English statements that feature something like “Either…or…”; but this phrase is clearly absent from this sentence. The use of ‘and’ in the sentence is a very clear indication that we should apply ∧, as this is what this particular symbol denotes (at least in English).
3) “If every circuit works properly and all laptops are functioning then every test run will terminate”
Our interpretation is:
Domain:Technology
Cx:x is a circuit
Wx:x works properly
Lx:x is a laptop
Fx:x is functioning
Rx:x is a test run
Tx:x is will terminate
Our symbolisation is:
*if for every x, if x is a circuit then x works properly and every if y is a laptop then it is also functioning then if every test runs then it will terminate
(for every x,if x is a circuit then x works properly and for every y,if y is a laptop then it is
also functioning)→(if every test runs then it will terminate)
*(for every x(x is a circuit→x works properly) and for every y(y is a laptop→it is also functioning)→(if every test runs then it will terminate)
*(∀x(x is a circuit→x works properly)∧∀y(y is a laptop→it is also functioning)→∀z(z is a test runs→it will terminate)
*(∀x(Cx→Wx)∧∀y(Ly→Fy)→∀z(Rz→Tz)
∀x∀y∀z(((Cx ∧ Wx ∧ Ly ∧Fy)→(Rz→Tz))) has one main problem; the main operator of the English sentence is very evidently the conditional arrow and not the universal quantifier, which is nested within the conditional. So, barring bracket errors, (∀x(Cx→Wx)∧∀y(Ly→Fy))→∀z(Rz→Tz) is the correct symbolisation for this sentence. I should just say that unlike the symbolisation that you’ve given, I have assumed that the variables x, y, z are distinct. This is because nowhere in the sentence does it explicitly specify that x, y, z are not distinct, so I think it would be a disservice to the semantics to assume this (other than that, and the brackets of course, my formula is the same as yours).
I sincerely hope some of this helped, even if just a little. Feel free to ask if not, especially since I used more notation and introduced some new ideas.
Final note: If you’re looking for more examples of English to Predicate, I’d recommend looking at forallx; it’s an open-source textbook, which is handy in understanding Propositional and Predicate logic in a less technical format.
Best Answer
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Your translations are correct. Here's a simple rule for translating phrases like "some clear explanations" and "no excuses":
So, "Some clear explanations are satisfactory" translates as $\exists x (P(x) \land Q(x))$.
How about "No excuses are clear explanations"? Well, we can rephrase that as "All excuses are things that are not clear explanations". So this is $\forall x (R(x) \implies \neg P(x))$.