by Corry Shores
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[The following is summary. Boldface (except for metavariables) and bracketed commentary are my own. Please forgive my typos, as proofreading is incomplete. I highly recommend Agler’s excellent book. It is one of the best introductions to logic I have come across.]
Summary of
David W. Agler
Symbolic Logic: Syntax, Semantics, and Proof
Ch.6: Predicate Language, Syntax, and Semantics
6.5 Translation from English to Predicate Logic
Brief summary:
To translate between the language of predicate logic RL and colloquial English, we need to make a translation key to assign symbols to text, and we would follow the pattern given in this table:
(Agler 276)
Summary
6.5 Translation from English to Predicate Logic
Agler will explain how to translate English sentences into the language of predicate logic (RL) and vice versa.
6.5.1 Translation Keys
In order to make the translations between the verbiage of colloquial English and the symbolic language of RL, we will need a translation key that shows the equivalences from one system to the other. He shows some example keys. Here is one, along with his explanation for making translations on its basis:
D: human beings (alive or undead)
j: John
f: Frank the zombie
Txy: x is taller than y
Fxy: x wants to eats the brains of y
Hx: x is hungry
Lx: x is living
Using the above translation key, we can translate the following sentences:
(1) John is taller than Frank the zombie.
(2) Frank the zombie wants to eat the brains of John.
(3) Frank the zombie is hungry, and John is living. |
(1) to (3) can be translated using the key and the conventions for translating propositions involving predicates and singular terms: (1*) Tjf
(2*) Ffj
(3*) Hf∧Lj
(Agler 270-271)
But now we will need to see how to translate English sentences that contain quantifiers.
6.5.2 The Universal Quantifier (∀)
Agler will explain the translation of quantifiers by first looking at what are called bridge translations for wffs. He shows this key:
D: human beings (living or dead)
Hx: x is happy
Zx: x is a zombie
Mx: x is mortal
Rx: x is a murderer
Wx: x is wrong
(Agler 271)
Agler then will use this key to make bridge translations for some example wffs.
(1) (∀x)Hx
(Agler 271)
He writes that, “(∀x) is translated as for every x, for all x’s, or for each x. The second part of (1) says that x is H or x is happy” (271). So on the basis of our key, we might make this bridge translation for sentence 1: “For every x, x is happy” (271). Then, on the basis of this bridge translation, we can further translate it into more colloquial English as “Everyone is happy” (272). Here is another example that goes from wff, to bridge translation, to colloquial English.
(2) (∀x)¬Zx
(2B) For every x, x is not a zombie.
(2E) Everyone is a not a zombie.
(Agler 271, 272)
In the following example, we will expand the bridge translation in order to facilitate the conversion to colloquial English.
(3) (∀x)(Zx→Hx)
(3B) For every x, if x is a zombie, then x is happy.
(3B*) Choose any object you please in the domain of discourse; if that object is a zombie, then it will be also be happy.
(3E) Every zombie is happy.
(Agler 271, 272)
Agler gives another instance with an additional bridge translation.
(4) (∀x)(Zx→¬Hx)
(4B) For every x, if x is a zombie, then x is not happy.
(4B*) Choose any object you please in the domain of discourse consisting of human beings (living or dead); if that object is a zombie, then it will not be happy.
(4E) No zombies are happy.
(Agler 271, 272)
Agler will now show us how to translate a negated universal quantifier.
(5) ¬(∀x)(Zx→Hx)
Recall from sentence 3 how we translated it in the unnegated form.
(∀x)(Zx→Hx)
Every zombie is happy.
What we do not is place the negation at the beginning.
(5E) Not every zombie is happy.
(Agler 273)
Agler then deals with the operators. Here is conjunction.
(6) (∀x)(Zx∧Hx)
(6B) For all x in the universe of discourse, x is a zombie and happy.
(6E) Everyone is a happy zombie.
Here is disjunction. [I will add my own bridge translation, but it might be incorrect.]
(7) (∀x)(Zx∨Hx)
(7B) For all x in the universe of discourse, either x is a zombie or x is happy.
(7E) Everyone is either a zombie or happy.
(Agler 273, except for 7B)
And this is the biconditional [again with my own possibly mistaken bridge translation].
(8) (∀x)(Zx↔Hx)
(8B) For all x in the universe of discourse, x is a zombie if and only if x is happy.
(8E) Everyone is a zombie if and only if he or she is happy.
(Agler 273, except for 8B)
6.5.3 The Existential Quantifier (∃)
To show how to translate wffs with the existential quantifier, Agler will give examples using the following key:
D: human beings (living or dead)
Hx: x is happy
Zx: x is a zombie
Mx: x is mortal
Rx: x is a murderer
Wx: x is wrong
(Agler 273)
Agler explains that “(∃x) is translated as for some x, there exists an x, or there is at least one x.” So let us look at some wffs that are translated to colloquial English using bridge translations [I will add my own extra bridge sentence on the basis of Agler’s text description, but it is probably flawed.]:
(1) (∃x)Hx
(1B) For some x, x is happy.
(1B*) In the universe of discourse, there is at least one object x that is happy.
(1E) Someone is happy.
(Agler 274, except 1B*)
(2) (∃x)¬Zx
(2B) For some x, x is not a zombie.
(2B*) In the universe of discourse, there is at least one object x that is not a zombie.
(2E) Someone is not a zombie.
(Agler 274, except 2B*)
In the next example, we have the existential quantification being negated. We will do this by placing “it is not the case that” at the beginning of the non-negated translation.
(3) ¬(∃x)Zx
(3′) (∃x)Zx
(3′B) For some x, x is a zombie.
(3′B*) In the universe of discourse, there is at least one object x that is a zombie.
(3′E) Someone is a zombie.
(3E) It is not the case that someone is a zombie.
(Agler 274, except primes)
Agler then compares 2 and 3.
(2) (∃x)¬Zx
(3) ¬(∃x)Zx
(2E) Someone is not a zombie.
(3E) It is not the case that someone is a zombie.
(2) says that something exists that is not a zombie, while
(3) says that zombies do not exist.
(Agler 274)
Agler gives some more examples [I add some parts that might be flawed]:
(4) (∃x)(Zx∧Hx)
(4B) For some x, x is a zombie, and x is happy.
(4B*) In the universe of discourse, there is at least one object x that is a zombie and that is happy.
(4E) Someone is a happy zombie.
(Agler 274, I added 4B* and 4E)
(5) (∃x)Zx∧(∃x)Hx
(5B) For some x, x is a zombie, and for some x, x is happy.
(5B*) In the universe of discourse, there is at least one object x that is a zombie, and in the domain of discourse, there is at least one object x that is happy.
(5E) Someone is a zombie, and someone is happy.
(Agler 274, I added 5B* and 5E)
Agler then compares these two, as they might be confused with one another.
(4) (∃x)(Zx∧Hx)
(5) (∃x)Zx∧(∃x)Hx
(4E) Someone is a happy zombie.
(5E) Someone is a zombie, and someone is happy.
(4) asserts that there is something that is both a zombie and happy, while
(5) asserts that there is a zombie, and there is someone who is happy.
(Agler 274, the English translations are mine, from before)
Agler then shows how to translate wffs with other operators. [I am not sure why English expressions have a B subscript, in my version.]
(6) (∃x)(Zx→Hx)
(6B) For some x, if x is a zombie, then x is happy.
(6B) There exists something such that if it is a zombie, then it is happy.
(7) (∃x)(Zx∨Hx)
(7B) For some x, x is a zombie, or x is happy.
(7B) There exists something that is either a zombie or happy.
(8) (∃x)(Zx↔Hx)
(8B) For some x, x is a zombie if and only if x is happy.
(8B) There exists something that is a zombie if and only if it is happy.
(Agler 274-275)
6.5.4 Translation Walk Through
Agler now will walk us through the process of translating from a colloquial English sentence to a wff in RL. He begins with this one.
(1) Some rich people are not miserly, and some miserly people are not rich.
(Agler 275)
[At this point, we need a way to turn this into symbols. So we identify the important structural components of the English sentence that would need to find expression in RL, then we designate symbols for those parts.]
Step 1: identify all the RL-relevant parts of the sentence and make a translation key for them.
D: unrestricted
Rx: x is rich
Px: x is people
Mx: x is miserly
(Agler 275)
Step 2: find the main operator in the English sentence. [Recall the sentence: Some rich people are not miserly, and some miserly people are not rich. We have two independent clauses joined by the conjunction “and”. So we have two propositions combined with the conjunction operator.] The structure of this sentence would look like:
(1*) [Proposition]∧[Proposition]
Step 3: find the subject for each of the two propositions.
(1**) Px∧Px
[It seems at this point we first establish what we are talking about. In both cases, we are saying something about people. So we need in this early step to establish that most basic feature of what we are talking about.]
Step 4: figure out what is being said about the subject in each case. [Again, the sentence reads: Some rich people are not miserly, and some miserly people are not rich. In these cases, we are adding more qualifications to the people we are describing. So we will use conjunctions of other predications. In the first clause, we have things that are people and that are rich and that are not miserly. In the second clause, we have things that are people and that are miserly and that are not rich.]
(1***) [Px∧Rx∧¬Mx]∧[Px∧Mx∧¬Rx]
Step 5: Determine the proper quantifiers and their scopes. [Again, the sentence is: Some rich people are not miserly, and some miserly people are not rich. Because we are using “some” rather than “all”, that means we are using the existential quantifier. What about the scope? I am not sure how exactly we determine that. I will guess based on the example. We have two main options: one is a global existential quantifier for the whole conjunction and the other is to use some number of other ones. The guess I will make is that, at least in existential quantification, finding the scope might involve asking which predicates are thought to refer or apply to the same possible things? In our sentence, we seem to be talking about one possible thing or one possible group of things, and in the second case, we seem to be implyng that we are talking about another thing or group of things. Why? well, because we are saying that they have different determinations. If we were saying that all of them had all the listed determinations, then we are talking about the same thing or group of things. But surely this cannot be the case, because then we have things that are both miserly and not miserly, and both rich and not rich. So since we are taking about two and just two different possible candidates or just two different possible sets of candidates, we would use two quantifiers. Will this work for universal quantification? Later Agler gives this two examples:
(∀x)(Sx ∨Gx) / Everything is either sweet or gross.
(∀x)(Sx)∨(∀x)(Gx) / Either everything is sweet, or else everything is gross.
In the first case, we are speaking of all things in the domain, and we are saying that for any given one that we pick out within that domain, it has one of two properties (or maybe both too): the thing is either sweet or it is gross. In other words, there is nothing in the domain that is neither sweet nor gross. They all have at least one of those properties, but some might be sweet, some might be gross, (and maybe some might be both). In the second case, we are saying that there are two possibilities: either all the things in the domain are sweet, or all things in the domain are gross. The domain cannot be made partly of sweet things and partly of gross things. It is made of things that altogether are one or the other. So how does this help us go backwards, taking into account what we said with existential quantification? Suppose we start with “Everything is either sweet or gross.” We have two choices. Either we make the quantifier range over both predications or we make a different quantifier for each one. We ask, in this first case, do both propositions refer to the same possible totality, or do they refer to two different possible totalities? They refer to the same one, as both predications can apply to things in the same totality. Suppose instead we start with, “Either everything is sweet, or else everything is gross.” We again have the same options. We ask the same question, do both propositions refer to the same possible totality, or do they refer to two different possible totalities? They cannot refer to the same one, because the things in the totality cannot have both predications. Rather, they refer to one of two different possible totalities. For that reason, perhaps, we use a different universal quantifier for each predication.]
(1****) (∃x)[(Px∧Rx)∧¬Mx]∧(∃x)[(Px∧Mx)∧¬Rx]
(Agler 275)
Agler then provides the following translations so we can see how to translate all the important formations.
(Agler 276)
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.
Some changes to the book quotations may have been made, as based on: http://markdfisher.com/wp-content/uploads/2014/02/PHIL_012_ONLINE_SYLLABUS_SP14-3-1.pdf .
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