7 Jun 2016

Agler (6.4) Symbolic Logic: Syntax, Semantics, and Proof, "Predicate Semantics", summary

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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.]

[Note for this entry: I had some difficulty understanding the last part of this section. I found it useful to consult one of Agler’s class handouts which includes this material but gives a little more explanation:
Summary of

David W. Agler 

Symbolic Logic: Syntax, Semantics, and Proof
 
Ch.6: Predicate Language, Syntax, and Semantics

6.4 Predicate Semantics



Brief summary:
In the language of predicate logic (RL) we will want to give interpretations for constants and for predicate formulations and also to give truth evaluations for well-formed formulas (wffs), including when they involve quantifiers. To do these things, we construct models, which specify the domain as well as the interpretation functions that assign objects in the domain to names, and n-tuples of objects to n-place predicates. On their basis, we valuate truth and falsity for formulas using a function that assigns T for when the named objects are among the tuples in the predicate’s interpretation, and F otherwise. When truth-evaluating quantified wffs in RL, we, intuitively speaking, give a universally quantified formula the value T if all substitutions for the variables make the formula true, and it is F otherwise; and for existentially quantified formulas, if there is at least one substitution that makes the formula true, it is evaluated as T, and F otherwise. For certain technical reasons, the actual procedure cannot involve substituting every possible name or object into the variables. We instead need to test for every object in the domain by thinking of each one being a potential interpretation for a constant, then seeing if these variant interpretations make the formula true or not. Despite the differences in notation, however, the more proper procedure conducts basically the same operation as the intuitive one. The evaluation rules for wffs in RL are the following:
1    v(Rai) = T if and only if the interpretation of ‘ai’ is in ‘R.’
2    vP) = T iff v(P) = F
       vP) = F iff v(P) = T
3    v(PQ) = T iff v(P) = T and v(Q) = T
       v(PQ) = F iff v(P) = F or v(Q) = F
4    v(PQ) = T iff either v(P) = T or v(Q) = T
       v(PQ) = F iff v(P) = F and v(Q) = F
5    v(PQ) = T iff either v(P) = F or v(Q) = T
       v(PQ) = F iff v(P) = T and v(Q) = F
6    v(PQ) = T iff either v(P) = T and v(Q) = T or v(P) = F and v(Q) = F
       v(PQ) = F iff either v(P) = T and v(Q) = F or v(P) = F and v(Q) = T
7    v(∀x)P = T iff for every name ‘a’ not in ‘P’ and every a-variant interpretation ‘P(a/x) = T.’
       v(∀x)P = F iff for at least one ‘a’ not in ‘P’ and at least one a-variant interpretation ‘P(a/x) = F.’
8    v(∃x)P = T iff for at least one name ‘a’ not in ‘P’ and at least one a-variant interpretation ‘P(a/x) = T’.
      v(∃x)P = F iff for every name ‘a’ not in ‘P’ and every a-variant interpretation ‘P(a/x) = F.


Summary

6.4 Predicate Semantics

Agler will now cover the semantics of the language of predicate logic (RL). First he will explain sets and set membership, and secondly he will discuss what makes a formula in RL true (263).


6.4.1 A Little Set Theory

Agler first notes the "naive" sense of the meaning of set: it is “a collection of objects considered without concern for order or repetition” (Agler 263). These objects making up the set are called the set’s members or elements. Suppose set ‘M’ is the set of odd integers from 1 to 5. We would write:
M = {1, 3, 5}
(Agler 263)
To say that 5 is a member of set M, we write:
5∈M’
And to say that 2 is not a member of M, we write:
2∉M
(Agler 263)
“The members of a set are considered without regard to ordering or repetition”. Thus set T, containing the members Mary, John, and Sally can be written
T = {Mary, John, Sally}
T = {Mary, Sally, John}
|
T = {Sally, Mary, John}
T = {Sally, John, Mary, Mary, Mary, John}
(263-264)
These sets are all identical.
It could also be that the there are so many members in a set that it is impractical to list them all. We can use predication to specify members of sets that are large or even infinite. So
P = {x | x is a politician}
would be the set of all politicians (264).
The set of all positive even integers is infinite, so there is no way practically speaking to list them all between brackets. But using predication we can specify that set in the following way.
E = {x | x is a positive even integer}
(Agler 264)


6.4.2 Predicate Semantics


In PL, we determined whether a proposition was true or false. It is more complicated in RL, because we are also dealing with sentence parts that cannot easily be assigned truth values.
In PL, where the basic unit of representation is the proposition, we straightforwardly interpret the proposition relative to the world; as such, the sentences are interpreted as being simply true or false. In predicate logic, our basic units of representation are parts of sentences, and these parts cannot simply be assigned a truth value since names like John and predicates like is red are not things that can be true or false. While the goal of predicate logic is, much like in propositional logic, to assign sentences truth values, the manner in which this is done is more complicated since it takes into consideration the contribution the subsentential parts make to the truth or falsity of a sentence.
(Agler 264)
In RL, we will need to deal with models on the basis of which to determine the truth or falitiy of a wff. There are two main parts of a model: one part stipulates the domain and the other part interprets the formula in relation to that domain. This interpretation will be conducted by means of a function.
The truth or falsity of a wff in RL is determined relative to a model. A model is a two-part structure. There is the part that stipulates a domain, and there is the part that interprets the RL wff relative to the domain. In other words, a model consists of a domain (D) and an interpretation function (I).
Model: A model in RL is a structure consisting of a domain and an interpretation function.
An interpretation function assigns (1) objects in D to names, (2) a set of n-tuples in D to n-place predicates, and (3) truth values to closed formula.
Interpretation-function: An interpretation-function is an assignment of (1) objects in D to names, (2) a set of n-tuples in D to n-place predicates, and (3) truth values to sentences.
(264)
The interpretation function requires a specified domain of discourse [on which it operates]. This domain consists of all things a language is able to “meaningfully refer to or talk about” (264).
Domain: The domain of discourse (D) consists of all of the things that a language can meaningfully refer to or talk about.
(265)
[For some reason] we always assume that domains are non-empty. We can specify a domain in two ways. {1} We can simply list the members of D, as with:
D = {John, Sally, Mary}
(Agler 265)
Or {2} we can specify the class or classes of objects contained in the domain, as for example:
D = {x | x is a living human being}
(Agler 265)
We will look now at the workings of the interpretation function I. Our formula in RL will include wffs that may include something like ‘Tab’, for ‘a is taller than b’. We look first at how the interpretation function will interpret names such as ‘a’ and ‘b’ in our example. What the function will do is assign some item in the domain to some certain name. So suppose this is our domain:
D = {Alfred, Bill, Corinne}
Our interpretation function for constants might interpret the names a, b, and c in the following manner.
I (a) Alfred
I (b) Bill
I (c) Corinne
That is, Alfred in D is assigned to ‘a,’ Bill in D is assigned to ‘b,’ and Corinne in D is assigned to ‘c.’
(Agler 265)
We also have variables in RL, as for example Tx for ‘x is tall’, and Lxy for ‘x loves y’. Let us look at the first case where we have a one-place predicate. In these cases, the interpretation function would assign to the variable a subset of the domain which contains all the possible substitutions for that variable. So let us stick with this domain:
D = {Alfred, Bill, Corinne}
And suppose that Alfred is a bad short person, that Bill is also short, but not bad, and Corinne is bad, but not short. Let us make our predicates be the following:
Sx: x is short
Bx: x is bad
(Agler 265)
Our interpretation function would then do the following:
I (Sx): objects in D that are short (i.e.,{Alfred, Bill})
I (Bx): objects in D that are bad (i.e.,{Corinne, Alfred})
(Agler 266)
Agler then has us consider the two-place predicate Lxy for ‘x loves y’. Suppose we were to apply an interpretation function in this case which interprets only names. It would assign terms to x and terms to y. The interpretation function for the formula Lxy, however, should not just tell us about a collection of single objects but rather it should it should tell us about pairings, that is, of doubles. And for predicates with increasingly larger places, it should give us triples, quadruples, quintuples, sextuples etc., or in more general terminology: for an n-place predicate the interpretation function will assign n-tuples. We designate the groupings of members in a tuple using angle-brackets:
<Alfred, Corinne>
(Agler 266)
The interpretation function would then work in this way:
suppose that in D, Bill loves Corinne (and no one else), Corinne loves Alfred (and no one else), and Alfred loves no one. The interpretation of ‘Lxy’ in D would be represented by the following set of 2-tuples:
I (Lxy): {<Bill, Corinne>, <Corinne, Alfred>}
This says that the interpretation of the predicate x loves y relative to D consists of a set with two 2-tuples: one 2-tuple is <Bill, Corinne> and the other is <Corinne, Alfred>.
(Agler 266)
Now having seen how the interpretation function assigns objects to names and n-tuples to the variables of predicates, we will be able to see how we can determine whether a wff in RL is true or false. [To do this, we will use another interpretation function, it seems, namely, a truth-valuation function; but this function has a different domain of assignments than what is in the model’s domain, and those assignments are always limited to two items, true and false. It is also possible that it is wrong to think of the truth-valuation function in this way. I am not sure] [For the rest of the material in this text, my interpretations are less trustable. I took a lot of extra time trying to figure out the material, and I may have gotten it wrong. I recommend skipping to the quoted Agler text selections in the remainder of this summary, just to be safe.] [We will look at truth-valuations first for the relatively straight-foward cases, where there is a constant given with the predicate. The idea will be that we have a predicate, like S, and we have names for objects, a, b, c, etc. We then ask is Sa true? or is Sb true? The way we do that is by seeing if the object assigned to the name is also in the set of items assigned to the predicate. There will be a number of tricky things  in the following material that I had trouble grasping. One of them is the wording that speaks of an object being in a predicate, like the interpretation of ‘a’ being in the interpretation of S. For, whenever we give the interpretation function for S, we use a variable, as in Sx. Perhaps that is not something to worry about. Maybe we are supposed to understand “interpretation of ‘a’ in S” as also being “interpretation of ‘a’ in the interpretation of S(+variable)”. Maybe it makes no difference, but I find the wording a little confusing. Yet so far in this simple case of a one-term predicate, the evaluation is simple. Here is the relevant model data and the evaluation:
D = {Alfred, Bill, Corinne}
I (a) = Alfred
I (b) = Bill
I (c) = Corinne
I (Sx) = {Alfred, Bill}
v(Sa) = T
v(Sc) = F
Sa is true and Sc is false, because the interpretation of ‘a’ is Alfred, and it is in the interpretation of Sx; and the interpretation of ‘c’ is Corinne, and it is not in the interpretation of Sx.]
We will call an interpretation where a truth value is assigned to a wff a valuation (v). For this, let ‘R’ be an n-place predicate, let ‘a1,’ ... ‘an’ be a finite set of names in RL, and let ‘ai’ be a randomly selected name.
(1) v(Rai) = T if and only if the interpretation of ‘ai’ is in ‘R.’
This says that we can assign a value of true to ‘Rai’ if and only if our interpretation of ‘ai’ is in the interpretation of ‘R.’ To consider this concretely, we examine two examples. First, consider the following wff:
Sa
| Let’s say that ‘Sa’ is the predicate logic translation of Alfred is short. According to (1), ‘Sa’ is true if and only if ‘a’ is in ‘S,’ that is, if and only if an interpretation of the predicate ‘short’ includes an interpretation of the name Alfred. Earlier, we said that
I (Sx): objects in D that are short (i.e., {Alfred, Bill})
And so, ‘Sa’ is true in the model since Alfred belongs to the collection of objects that are short.
(Agler 266-267)
[Now, when we turn to multiple-place predicates, the notation situation gets complicated, and it will lead me to want to revise the above notation. So let us work with the predicate for ‘x loves y’. And let us evaluate whether Lca and Lcb are true.
D = {Alfred, Bill, Corinne}
I (a) = Alfred
I (b) = Bill
I (c) = Corinne
I (Lxy) = {<Bill, Corinne>, <Corinne, Alfred>}
v(Lca) = ?
v(Lcb) = ?
Let me stop here for a second. This presents my next confusion. Our evaluation rule indicates that we should look at the members of I(Lxy), which are <Bill, Corinne> and <Corinne, Alfred>, and for evaluating Lcb, we need to see if the objects assigned to ‘c’ and ‘b’ are among them. This is if we take the rule strictly. My confusion is that we have singular terms for the name assignments, but we have 2-tuples for the predicate assignments. Suppose we are allowed to ignore the angle brackets. In that case, the assigned objects for ‘c’ and ‘b’ are in the set. Thus “Corinne loves Bill” would be evaluated as true. But of course that is false. So we cannot ignore the brackets. But suppose then that we take each angle-bracketed 2-tuple as a singular item in the set. Then we cannot match the 2-tuple predicate assignments to the name-object assignments. In other words, we have ‘a’ being assigned to Alfred and ‘c’ being assigned to Corinne, but neither <Bill, Corinne> nor <Corinne, Alfred> is the same as Alfred or Corinne. Perhaps I am mistaken about something here. But if I am not, then would there not need to be some other step in the procedure that would allow us to properly compare the objects designated by the names with the tuples of objects assigned to the predications? One possibility for this that I can think of is to make another assignment which places the names into tupled relations. In other words, before we can evaluate v(Lca), perhaps we need to first evaluate I(Lca). So let me redraw this model and its evaluation, and I will change the notation a little:
D = {Alfred, Bill, Corinne}
I (a) = {Alfred}
I (b) = {Bill}
I (c) = {Corinne}
I (Lxy) = {<Bill, Corinne>, <Corinne, Alfred>}
I (Lca) = {<Corinne, Alfred>}
I (Lcb) = {<Corinne, Bill>}
v(Lca) = T
v(Lcb) = F
One problem with this is that the interpretation functions I(Lca) and I(Lcb) are not given literally in the model. I formulated them by placing the name assignments into the predicate structures, so to construct the 2-tuples. But by doing so, we can then explicitly show how Lca is true and Lcb is false. Probably there is some better way to go about this. In fact, the way Agler does it of course is sufficient, but I am just confused about how it all works exactly, for the reasons I gave above. Let me quote.]
Second, consider the following more complex wff:
Lca
‘Lca’ is a predicate logic translation of ‘Corinne loves Alfred.’ According to (1), ‘Lca’ is true if and only if the predicate ‘loves’ includes the ordered pair <Corinne, Alfred>. Since it does, the interpretation function assigns a value of true to ‘Lca.’ That is,
v(Lca) = T
(Agler 267-268)
[As I mentioned in my prior comments, I suggested that I might want to revise the notation for the single-place predicate situation. I might want to do that in order to make it resemble the notation I was using for the two-place predicate. So I might rewrite it as:
D = {Alfred, Bill, Corinne}
I (a) = {Alfred}
I (b) = {Bill}
I (c) = {Corinne}
I (Sx) = {<Alfred>, <Bill>}
I (Sa) = {<Alfred>}
I (Sc) = {<Corinne>}
v(Sa) = T
v(Sc) = F
In the exercises for this section, Agler has in one model:
I (H) = {<a>, <b>, <c>}
So perhaps it makes sense to use the angle brackets even when the predicate is only one-place.]
We will now give the rules for truth-evaluating wffs with logical operators. [It seems that these depend first on truth-evaluating the unoperated wff using rule 1, and then changing the value according to the operator. But I am not sure. Let me quote.]
The interpretation of wffs that involve truth-functional operators as their main operators is the same as in propositional logic and straightforward given that we know the truth values of their components. Given that ‘P’ and ‘Q’ in (2) to (6) are well-formed formulas, then relative to a model,
2     vP) = T iff v(P) = F
        vP) = F iff v(P) = T
3     v(PQ) = T iff v(P) = T and v(Q) = T
       v(PQ) = F iff v(P) = F or v(Q) = F
4     v(PQ) = T iff either v(P) = T or v(Q) = T
       v(PQ) = F iff v(P) = F and v(Q) = F
5     v(PQ) = T iff either v(P) = F or v(Q) = T
        v(PQ) = F iff v(P) = T and v(Q) = F
6     v(PQ) = T iff either v(P) = T and v(Q) = T or v(P) = F and v(Q) = F
        v(PQ) = F iff either v(P) = T and v(Q) = F or v(P) = F and v(Q) = T
(Agler 267)
Now we will need a way to evaluate wffs taking the form ‘(∀x)P’ and ‘(∃x)Q.’ [For some reason] we assume for such formulas that they “are closed; that is, they do not contain any free variables, and x and only x occurs free in ‘P’ and ‘Q’” (267). [Perhaps the idea here is that x is free in P but not free in (∀x)P.] Agler then says that we will determine the truth values of quantified wffs on the basis of non-quantified ones. [The basic idea of that procedure seems similar to what we did before. So here we would substitute names in for the variables, and that would give us formulations like we dealt with before. Then we compare the objects designated by these names with the objects in the extension for the predicate. But we do this comparison in a particular way for quantifiers. For universal quantification, we want to know if all the possible substitutions of names for the variables give us objects found in the predicate’s interpretation, and for existential quantification, we would just need to find at least one. Agler will say that we need to be careful with this procedure however.]
In what follows, we will define truth values of quantified formulas by relying on the truth values of simpler nonquantified formula. This method requires a little care for, at least initially, we might say that ‘(∃x) Px’ is true if and only if ‘Px’ is true, given some replacement of x with an object or a name (object constant) is true. Likewise, a wff like ‘(∀x)Px’ is true if and only if ‘Px’ | is true, given that every replacement of x with an object or name yields a true proposition.
(Agler 267-268)
[Agler then explains why this can be problematic. I do not follow it well, but I will try. Suppose we have Sx, from our model above. We want to know if (∀x)Sx is true or false. We have the following so far:
D = {Alfred, Bill, Corinne}
I (a) = {Alfred}
I (b) = {Bill}
I (c) = {Corinne}
I (Sx) = {<Alfred>, <Bill>}
Now, previously we were only truth-evaluating predicates taking constants. The predicates taking variables we evaluated by listing its tuples. But now we are dealing with formulations like (∀x)Sx. To evaluate this, we need think about all the possible constants that can be substituted for x. So for this we might do it in either of two ways. The first way would be to simply put the objects in the formula, so:
v(S(Alfred)) = T
v(S(Bill)) = T
v(S(Corinne)) = F
Since it is false for Corinne, then
v(∀x)Sx = F
But the problem with this, Agler says, is that structurally speaking, we cannot substitute “Alfred” for “x”, because one is an object and one is a variable. I am not exactly sure why that is problematic, but we can see that they are very different things. This leaves us with another option, which would be to substitute the names assigned to the objects into the variables. So:
v(Sa) = T
v(Sb) = T
v(Sc) = F
v(∀x)Sx = F
Agler says this is also problematic, because it assumes we have name assignments for all objects. In our model, we in fact do have name assignments for all objects. But it would seem that this is actually not something to be expected in all models. So suppose this is our model and evaluation:
D = {Alfred, Bill, Corinne}
I (a) = {Alfred}
I (b) = {Bill}
I (Sx) = {<Alfred>, <Bill>}
v(Sa) = T
v(Sb) = T
v(∀x)Sx = T
The problem here is that we have an object, Corinne, but we were unable to evaluate it using our method on account of there being no name assigned for it, and thus we came to the incorrect conclusion that v(∀x)Sx is true. One solution would seem to be to take the time to assign all the objects their own names. Agler says however this is not the solution. I am not sure why, but perhaps it is impractical in many cases. Yet the proper solution is a very hard for me to conceptually grasp, even if the procedure is easy to follow. Let me first give my overall impression of what is going on. It seems that instead of assigning a different name for every object, we think of every object being potentially assignable to one name. So really in the end we will need to compare the objects in the domain with those in the predicate’s extension, using the sort of procedure we noted before, namely, seeing for universal quantification if all the objects in the domain are in the predicate’s extension, and for existential quantification seeing if at least one is. With that in mind, we need to see the technical procedure for making every object assignable to one name. The way we do this is by conceiving of variant interpretation functions for that same letter. So consider our model we just used above, which did not have all the names assigned:
D = {Alfred, Bill, Corinne}
I (a) = {Alfred}
I (b) = {Bill}
I (Sx) = {<Alfred>, <Bill>}
v(∀x)Sx = ?
Instead of assigning ‘c’ to Corinne, to fill out the name assignments, we instead will make variant interpretations of ‘a’:
I (a) = Alfred
I2 (a) = Bill
I3 (a) = Corinne
Each one of these is called an “a-variant interpretation”. So to evaluate v(∀x)Sx, we ask, does every  substitution of ‘a’ or of every a-variant interpretation make the predication true? If so, then v(∀x)Sx=T. But since in this example it does not, that would make it false. Now, there is another complication. Here is the rule we just used:
v(∀x)P = T iff for every name ‘a’ not in ‘P’ and every a-variant interpretation ‘P(a/x) = T.’
v(∀x)P = F iff for at least one ‘a’ not in ‘P’ and every a-variant interpretation ‘P(a/x) =  F.’
What is a little confusing is this idea of a name ‘a’ not being in ‘P’. One way to read this is like in the other formulation where we spoke of an interpretation of ‘a’ being in R, which means in the interpretation for Rx. But that does not seem to work here. So what does it mean? In one of Agler’s class handouts, he gives a little more information about this. He writes:
The general idea is that ‘(∀x)P’ is true if and only if | ‘P(a/x)’ is true for every way of interpreting ‘a’ and ‘(∃x)P’ is true if and only if ‘P(a/x)’ is true for every way of interpreting ‘a’.
(Agler, Handout 6, p.12-13. Then, after giving the variant interpretation material, he continues:)
We might then replace our intuitive definition of when ‘(∀x)P’ is true from
‘(∀x)P’ is true if and only if ‘P(a/x)’ is true for every way of interpreting ‘a
to
‘(∀x)P’ is true iff for every a-variant interpretation, it is the case that the ‘P(a/x)’ is true.
However, we need to make one further caveat. We said that for any name ‘a’, an interpretation Ia is ‘a-variant’, ‘a-varies’, or is an ‘a-variant interpretation’ if and only if Ia interprets ‘a’ (i.e. it assigns it an object in D) and it either does not differ from I or it differs only in the interpretation it assigns to ‘a’. Part of the idea here is that we want to hold constant our interpretation of all other formula (e.g. n-place predicate terms or other names) and consider the various ways of interpreting ‘a’ relative to the domain. For this to occur, the substituted name ‘a’ should not already occur in P since this name presumably already has an interpretation. In other words, determining the truth value of ‘(∀x)Pxa’ requires us to hold our interpretation of ‘P’ and ‘a’ fixed, and to cash out the truth value of quantified part of the formula by considering the various ways (the variant interpretations) a substituted name (other than ‘a’) could be interpreted in the domain.
(Agler Handout 6, p.13)
So that clarifies what is meant by the name ‘a’ not being in P. It does not mean: “the name ‘a’ not being in the interpretation of P” but means literally being in the formula, like Pxa. If I follow, the idea seems to be that for Pxa, we are not evaluating for the constant ‘a’ but rather we are concerned with what happens for all the variants on ‘a’ substituted in for the x. I will now quote this section.]
However, this will not work without some further elaboration. On the one hand, we cannot replace variables with objects from the domain since variables are linguistic items, and replacing a variable with an object won’t yield a wff, or even a proposition. On the other hand, we cannot replace variables with names from our logical vocabulary because this falsely assumes that we have a name for every object in the domain. It might be the case that some objects in the domain are unnamed.
The solution to this problem is not simply to expand our logical vocabulary so that there is a name for every object but to consider the multitude of different ways in which a single name can be interpreted relative to the domain, that is, to consider the many different ways that an object in the domain can be assigned to a name. To see this more clearly, consider the following domain:
D: {John, Vic, Liz}
Now let’s consider the following interpretation I of ‘a’ relative to D:
I (a): John
This is a perfectly legitimate interpretation, but we might think of a variant interpretation of I, such as
I1(a): Vic
Further, we might even think of another variant interpretation of I, such as
I2(a): Liz
Let’s say that for any name ‘a,’ an interpretation ‘Ia’ is a-variant or a-varies if and only if ‘Ia’ interprets ‘a’ (i.e., it assigns it an object in D) and either does not differ from I or differs only in the interpretation it assigns to ‘a’ (i.e., it doesn’t differ on the interpretation of any other feature of RL). Thus, I, I1, and I2 are all a-variant interpretations of I since they all assign ‘a’ to an object in the domain and either do not differ from I or differ only in the interpretation they assign to ‘a.’
Using the notion of a variant interpretation, we can define what it means for a quantified formula to be true or false.
7    v(∀x)P = T iff for every name ‘a’ not in ‘P’ and every a-variant interpretation ‘P(a/x) = T.’
       v(∀x)P = F iff for at least one ‘a’ not in ‘P’ and at least one a-variant interpretation ‘P(a/x) = F.’
8     v(∃x)P = T iff for at least one name ‘a’ not in ‘P’ and at least one a-variant interpretation ‘P(a/x) = T’.
      v(∃x)P = F iff for every name ‘a’ not in ‘P’ and every a-variant interpretation ‘P(a/x) = F. |
Using the notion of a variable interpretation, we have a solution to the problem of defining quantified formulas by replacing variables with names from our logical vocabulary since our solution does not falsely assume that we have a name for every object in the domain. Instead, it assumes that there are many ways in which a name can be interpreted, and so while there may not always be a name for every object in the domain, there is always at least one variant interpretation that assigns a name to the previously unnamed object. Using this notion, a universally quantified proposition ‘(∀x)P’ is true if and only if for every name ‘a’ not in ‘P’ and every a-variant interpretation, it is the case that ‘P(a/x)’ is true. In other words, it is true for every formula that is the result of replacing x with ‘a’; for instance,‘(∀x)Px’ is true if and only if ‘Pa,’ ‘Pb,’ ‘Pc,’ and so on are true. Likewise, an existentially quantified proposition ‘(∃x)P’ is true if and only if for at least one name ‘a’ not in ‘P’ and at least one avariant interpretation, it is the case that ‘P(a/x)’is true (the formula that is the result of replacing x with ‘a’).
(Agler 268-269)


Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.

Also quoted from:
Agler, David. “Handout #6 – Predicate Logic – Symbols, Syntax, Semantics, Translation.”
Some changes to the book quotations may have been made, as based on:
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