Priest (1.2) “Multiple Denotation, Ambiguity, and the Strange Case of the Missing Amoeba”, ‘1.2. LP Semantics’, summary

 

by Corry Shores

 

[Search Blog Here. Index tabs are found at the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[Priest, “Multiple Denotation, Ambiguity, and the Strange Case of the Missing Amoeba”, entry directory]

 

[The following is meant to be summary, but is in large part my best guesswork of the text’s meaning, made on the basis of other texts by Priest and using examples from Agler. So please consult the original text rather than trust my notes below. Also, proofreading is incomplete, so you will find typos and other mistakes.]

 

 

 

 

Summary of

 

Graham Priest

 

“Multiple Denotation, Ambiguity, and the Strange Case of the Missing Amoeba”

 

1

Formal Considerations

 

1.2

LP Semantics

 

 

Brief summary:

We formulate a paraconsistent LP semantics, meaning that we can have interpretations where a formula is evaluated as both true and false, but not as neither. We do this using a truth-evaluating relation R rather than a truth-evaluating function v. We assign the base formulas true and/or false by writing R(α, 1) and/or R(α, 0). For formulas with universally quantified variables, they are related to 1 if all substitutions of those variables for members of the domain (or constants assigned to those members) yield the value of 1, and they are 0 otherwise. For existentially quantified variables, they are truth-value-related to 1 if there is at least one substitution that yields 1, and they are 0 otherwise. Formulas built up by means of connectives are evaluated according to classical rules, but given that the base formulas can have relations to both values, the more complex formulas can yield both values too. Validity “is defined in terms of truth preservation in all interpretations and evaluations of the variables.” An interpretation of a predicate is classical if the intersection of its extension and anti-extension is an empty set. LP is a proper sub-logic of classical logic, and all the logical truths of LP are those of classical logic. Identity in this LP semantics is understood as the set of couples (from the domain) whose two terms are [taken to be] the same. We cannot however have any non-identical couplings be excluded from the identity predicate/relation’s anti-extension [because in our LP semantics there can be no items that are neither identical nor non-identical to themselves or to another item. However, (as far as I can tell, at least, and I still need to have this confirmed) we can have a pairing of something and itself be in both the extension and anti-extension of the identity predicate/relation, meaning that thing is both identical and not identical to itself.] Also, substitutivity of identicals holds in this semantics.

 

 

Contents

 

1.2

LP Semantics

 

1.2.1

[Truth Evaluation and Validity in an LP Propositional and First-Order Semantics with a Relation-Based Structure for Truth-Evaluations (Like First Degree Entailment, Only without the Option of No Valuation.)]

 

1.2.2

[Familiar Formulations for Evaluating Complex Formulas. LP. Classical Predicates. LP as a Proper Sub-Logic of Classical.]

 

1.2.3

[Identity and Substitutivity in LP. ]

 

 

 

 

Summary

 

 

1.2

LP Semantics

 

 

1.2.1

[Truth Evaluation and Validity in an LP Propositional and First-Order Semantics with a Relation-Based Structure for Truth-Evaluations (Like First Degree Entailment, Only without the Option of No Valuation.)]

 

[An LP semantics can be formulated by evaluating with relations instead of functions. This evaluating relation R allows for a formula to have a truth-relation to 1, to have a truth-relation to 0, or to have both such relations. But it is important that we stipulate that every formula be assigned to at least one value (for otherwise it would not be an LP sort of system but one more like First Degree Entailment.) For propositions built up using connectives, we relate them to particular truth-values using a set of rules which will allow us to handle instances where there is more than one truth-value for a formula. The criteria for the evaluations are like what we would expect for a classical sort of evaluation, but in some cases they may yield two truth relations, depending on what the calculations produce. In a first-order language, we have function d, which assigns to each constant a member of the domain and to each predicate its extension and anti-extension. We then say that a formula is 1 if its interpretation is in the extension and 0 if its interpretation is in the anti-extension. For formulas with universally quantified variables, they are related to 1 if all substitutions of those variables for members of the domain (or constants assigned to those members) yield the value of 1, and 0 otherwise. For existentially quantified variables, they are truth-value-related to 1 if there is at least one substitution that yields 1, and they are 0 otherwise. Validity “is defined in terms of truth preservation in all interpretations and evaluations of the variables.”]

 

We now will consider the semantics of LP (see section 7.4 of Priest’s Introduction to Non-Classical Logic). According to a certain view, paradoxical sentences are thought to have more than one truth value. In order to accommodate these cases, we can keep our classical semantics, except now instead of evaluating truth values using a function, we do so instead with a relation. [See Introduction to Non-Classical Logic section 8.2 on the semantics of First Degree Entailment, which seems similar to what we do in the following, except here we cannot have the situation where a formula lacks a relation to 1 or 0.] So for propositional logic, we first define the relation of each atomic proposition to at least one truth value [thereby allowing for both truth values, but not allowing for neither truth values.] Then for formulations built up using connectives, we combine “the truth values of its components in all possible ways” (362). [For how this might work, let us draw from section 8.2.7 of Priest’s Introduction to Non-Classical Logic. I will modify the notation to be more like that of the current text. Let us begin with the simplest case to illustrate the notion of “combining the truth values of its components in all possible ways”. In this first example, there will only be one component and thus not yet any combinations to make, but still there will be the potential for multiple assignments. So suppose we only have one proposition, p. And suppose its values is defined as both 1 and 0. That means we will have:

R(p, 1)

and

R(p, 0)

And suppose we want to determine the value of ¬p. What are the possible ways to make that evaluation? First let us look at our rule for negation, given in the simplified form in section 1.2.2 below:

R_s(¬α, 1) iff R_s(α, 0)

R_s(¬α, 0) iff R_s(α, 1)

So, according to the evaluation of p that is R(p, 1), ¬p is 0:

R(¬p, 0)

But that is only one way to fulfill the criteria. For p, we also have R(p, 0), so in accordance with that evaluation, ¬p is 1.

R(¬p, 1)

So ¬p is both true and false:

R(¬p, 0)

and

R(¬p, 1)

Now let us look at an example from Priest’s Introduction to Non-Classical Logic section 8.2.7 where we need to deal with “combining the truth values of its components in all possible ways”, in other words, where we have more than one component and where we have more than one valuation for a component. Here first is a quotation:

As an example of how these conditions work, consider the formula ¬p ∧ (q ∨ r). Suppose that pρ1, pρ0, qρ1 and rρ0, and that ρ relates no parameter to anything else. Since p is true, ¬p is false; and since p is false, ¬p is true. Thus ¬p is both true and false. Since q is true, q ∨ r is true; and since q is not false, q ∨ r is not false. Thus, q ∨ r is simply true. But then, ¬p ∧ (q ∨ r) is true, since both conjuncts are true; and false, since the first conjunct is false. That is, ¬p ∧ (q ∨ r)ρ1 and ¬p ∧ (q ∨ r)ρ0.

(Priest’s, Introduction to Non-Classical Logic, p.143)

Here we have the following relations, which I will reformulate for our current conventions:

R(p, 1)

R(p, 0)

R(q, 1)

R(r, 0)

And we want to evaluate:

R(¬p ∧ (q ∨ r), ???)

Since p has two values, and the others have one, there are two combinations of values for the whole formula. Put simplistically (with the values standing-in for the value assignments of the base formulas), they are:

Combination 1:

¬(1) ∧ (1 ∨ 0)

0 ∧ 1

0

 

Combination 2:

¬(0) ∧ (1 ∨ 0)

1 ∧ 1

1

So

 R(¬p ∧ (q ∨ r), 0)

and

R(¬p ∧ (q ∨ r), 1)

But the evaluations as we will see will be made using a more formal procedure.] Priest then defines the way truth-values are evaluated in a more precise way:

R(¬α, z) iff ∃x(R(a, x)& z = ⊖x)

R(α ∧ β, z) iff ∃x∃y(R(a, x)& R(β, y)& z= x⊗y)

R(α ∨ β, z) iff ∃x∃y(R(a, x)& R(β, y)& z= x⊕y)

(362)

[I may not understand these well, so let us work through them.

R(¬α, z) iff ∃x(R(a, x)& z = ⊖x)

Let us begin with:

R(¬α, z)

We are giving the evaluation for the negation of a proposition. This means that z is a truth value, but it is not determined yet which one (the formula will work for both.) Now this part:

∃x(R(a, x)& z = ⊖x)

I want to first note that in order for me to get this work, I need to see it not that “R(a, x)& z” equals “⊖x” but rather that it is a conjunction of “R(a, x)” and “z = ⊖x”. Probably that is evident by the syntax, but I am unaware of those conventions. So we are assuming that the variables are truth values, so the x of ∃x I think would be saying that ‘there is a truth value ...’ (possibly a different one, because it has a different letter symbol). And with regard to this other truth value, it is related to the unnegated form of the proposition; and also, the first value we mentioned (the one assigned to the negated form of the proposition), would be derived if we applied the negation function to the second truth value we mentioned (the one assigned to the unnegated form of the proposition). So suppose our z here is understood to be 0, and our x is understood to be 1. That means a formula with negation, like ¬α, relates to 1 only if there is another value, 0, which relates to the unnegated form α and which is obtained when the negation function operates on the 1 value (see section 1.1.1 for how these connective functions work). So far, we have only defined what makes a negated form false. But the rule for what makes it true I think is built in, because the z value can be thought of as 1, and the x as 0. So it gives the rule for both cases. The next one was for conjunction:

R(α ∧ β, z) iff ∃x∃y(R(a, x)& R(β, y)& z= x⊗y)

So the conjunction gets a certain value only under the following conditions. Here we will relate a certain truth-value to the whole conjunction. We will do so on the following basis. We suppose that the first conjunct is related to a value and the second one is also related to a value (without specifying whether those values are the same or not), and we also suppose in conjunction with this that the conjunction operation on the two assigned values will yield the value that we assign to the whole conjunction. So suppose our z value is 1, and thus we are giving the rule for the circumstances under which the conjunction would be related to 1. But let us now jump to z= x⊗y. If z is 1, then x⊗y needs to equal 1, and the only way that can happen is if both x and y are related to 1. (So even though we have three variables, in this case they will need to take the same value in the end). And the formula says before that, in this case, the first conjunct is related to the first truth-value, which is here 1, and the second to the second value, which is also 1. Now for the criteria for the conjunction to relate to 0. In that case, x⊗y would need to yield 0. This can happen if either term is 0 or if both are. And again, one of those values relates to the first conjunct and the other to the second. So this leaves the following possibilities, that z is o, x is 1, and y is 0, that z is o, x is 0, and y is 1, and that z is o, x is 0, and y is 0. What does not work under the assumption would be that z is 0, x is 1 and y is 1, because with that being the case, the conjunction function would not yield a value that equals z, which is a stipulation here. And finally we have:

R(α ∨ β, z) iff ∃x∃y(R(a, x)& R(β, y)& z= x⊕y)

Here if z is 1, then either the first disjunct is 1, or the second is, or both are. And z is 0 if both disjuncts are 0, to give a shorthand of the above sort of explanation.] “Validity is defined in terms of truth preservation:”

Σ ⊨ α iff for all v, iff R(β, 1) for all β ∈ Σ then R(α, 1)

[Let us look at how it is defined for First Degree Entailment in his Introduction to Non-Classical Logic section 8.2.8:

Semantic consequence is defined, in the usual way, in terms of truth preservation, thus:

Σ ⊨ A iff for every interpretation, ρ, if Bρ1 for all B ∈ Σ then Aρ1

and:

⊨ A iff φ ⊨ A, i.e., for all ρ, Aρ1

(Priest, Introduction to Non-Classical Logic, 144)

Now, were we to make these parallel, I would have expected:

Σ ⊨ α iff for all R, iff R(β, 1) for all β ∈ Σ then R(α, 1)

(with boldface for emphasis)

One possibility is that they are equivalent, in other words, that since the R serves for truth valuation, we might as well say v, which practically does the exact same thing and is what we are more used to anyway. The other possibility is that v means a variable in formula a. So perhaps

Σ ⊨ α iff for all v, iff R(β, 1) for all β ∈ Σ then R(α, 1)

means something like, ‘for every substitution of the variables in all of the formulas ...’. If we stick strictly with the definition of v as a valuating function given previously (rather than distinguishing a function v when syntactically it works as a function and a v variable when syntactically it works as a variable), then I suppose we have something like the first option I considered here where it is understood as equivalent to R of this text or to ρ in the other text. I would also note something. In the version found on Priest’s website, there is a hand marking it seems through the v.

But the version at JSTOR (and presumably in the print version itself, but I have not checked) there is no line there:

I assume this is a stray marking, but it came to my attention while working on that part. I also note a discrepancy between this formulation and the one in Introduction to Non-Classical Logic. Here there are two biconditionals “iff”, but in the other text there is just one.

Σ ⊨ α iff for all v, iff R(β, 1) for all β ∈ Σ then R(α, 1)

(362)

Σ ⊨ A iff for every interpretation, ρ, if Bρ1 for all B ∈ Σ then Aρ1

(Priest, Introduction to Non-Classical Logic, 144)

Regarding the formulation with a biconditional then a conditional, it reminds me a little of what Frederic Schuller says in this course video around 28 minutes or so. So suppose for some reason there are no relation-evaluation configurations at all that make all the premises true in a classical situation, like with the formula

p ∧  ¬p ⊨ q

then in the formula

Σ ⊨ A iff for every interpretation, ρ, if Bρ1 for all B ∈ Σ then Aρ1

we assign the part reading

for every interpretation, ρ, if Bρ1 for all B ∈ Σ

as 0. For, Bρ0 for all B ∈ Σ. When the antecedent of a conditional is 0, then the whole conditional would be 1. So the part reading:

for every interpretation, ρ, if Bρ1 for all B ∈ Σ then Aρ1

is 1. That side is biconditionally conjoined to

Σ ⊨ A

Now, since one side of the biconditional is 1, that makes the other side 1, and so the inference would be valid. (Yet in fact I am not at all sure how this works.) But in our current text, as we noted, we are using the biconditional “iff” twice.

Σ ⊨ α iff for all v, iff R(β, 1) for all β ∈ Σ then R(α, 1)

I do not know how to translate that symbolically, because I do not how the structure “if and only if Q, then P” works. Is it “P if and only if Q”? Until I learn the proper understanding of that structure, I will not know really how to grasp the way it works.] Let us return now to the text. We will now formulate truth evaluation for our first-order language:

For the first-order case, we now take an interpretation to be a pair, ⟨D, d⟩, where D is a non-empty domain of objects, d assigns each constant a member of the domain, and each n-place predicate, P, a pair ⟨E_p, A_p⟩, (the ex- | tension and anti-extension of P, respectively), such that E_p ∪ A_p = Dⁿ. The relation between formulas and truth values is now relative to an evaluation of the free variables, s. In the atomic case, this is defined by the clauses:

R_s(Pt₁ ... t_n, 1) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ E_p

R_s(Pt₁ ... t_n, o) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ A_p

(362-363)

[So like in section 1.1.2, an interpretation involves a domain of objects D and a function d that assigns each constant to a member in the domain. But unlike before, d works a bit differently for predicates. We first should note the idea of a predicate’s extension and its anti-extension. The following comes from Priest’s One section P.5:

In first-order logic, a predicate has an extension—the set of things of which it is true—and an anti-extension—the set of things of which it is false. In classical logic, the anti-extension need not be mentioned explicitly, since it is simply the complement of the extension. However, in a paraconsistent logic, the extension of a predicate does not determine the anti-extension. It therefore needs a separate specification.

(Priest, One, p.xx, boldface mine)

We will also need the notion of the union of sets, but let us mention the intersection, because we need that later. This comes from the “Brief summary” of the post on Suppes’ Introduction to Logic section 9.5.

Certain operations can be performed on sets. If we find all the members shared in common between two sets, we are finding their intersection (∩):

(x)(x ∈ A ∩ B ↔ x ∈ A & x ∈ B)

When two intersecting sets share no members in common, that is, when they are mutually exclusive sets, their intersection is the empty set. The set containing all the members in total from two sets is their union (∪):

(x)(x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B)

(Suppes, Introduction to Logic, pp.184-185)

So we are going to take apart the line:

d assigns [...] each n-place predicate, P, a pair ⟨E_p, A_p⟩, (the ex- | tension and anti-extension of P, respectively), such that E_p ∪ A_p = Dⁿ.

(362-363)

Let us return to our example from David Agler’s Symbolic Logic: Syntax, Semantics, and Proof section 6.4.2, as modified for section 1.1 of our current text, but now I will further modify it for section 1.2 (our current section):

D = {Alfred, Bill, Corinne}

C = {a, b, c}

d(a) = Alfred

d(b) = Bill

d(c) = Corinne

D¹ = {⟨Alfred⟩, ⟨Bill⟩, ⟨Corinne⟩}

D² = {⟨Alfred, Alfred⟩, ⟨Alfred, Bill⟩, ⟨Alfred, Corinne⟩, ⟨Bill, Alfred⟩, ⟨Bill, Bill⟩, ⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩, ⟨Corinne, Bill⟩, ⟨Corinne, Corinne⟩}

[Sx: x is short]

d(Sx) = ⟨ { ⟨Alfred⟩, ⟨Bill⟩}, {⟨Corinne⟩} ⟩

[in other words, the extension of S is Alfred and Bill, and the anti-extension is Corinne]

[Lxy: x loves y]

d(Lxy) = ⟨ { ⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩}, { ⟨Alfred, Alfred⟩, ⟨Alfred, Bill⟩, ⟨Alfred, Corinne⟩, ⟨Bill, Alfred⟩, ⟨Bill, Bill⟩, ⟨Corinne, Bill⟩, ⟨Corinne, Corinne⟩ } ⟩

[In other words, the extension of L is {⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩} and the anti-extension is all the rest.]

Now what about the part reading

E_p ∪ A_p = Dⁿ

? For predicate S, it is D¹ and for predicate L, it is D². We said that the union of sets is set containing all the members in total from two sets. So let us fill out the values:

E_S ∪ A_S = D¹

{⟨Alfred⟩, ⟨Bill⟩} ∪ {⟨Corinne⟩} = {⟨Alfred⟩, ⟨Bill⟩, ⟨Corinne⟩}

The next part:

The relation between formulas and truth values is now relative to an evaluation of the free variables, s. In the atomic case, this is defined by the clauses:

R_s(Pt₁ ... t_n, 1) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ E_p

R_s(Pt₁ ... t_n, o) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ A_p

(363)

Let us consider a sort of example based somewhat on the ideas in section 11.2 of Priest’s In Contradiction.

As I write, my pen is touching the paper. As I come to the end of a word I lift it off. At one time it is on; at another it is off (that is, not on). Since the motion is continuous, there must be an instant at which the pen leaves the paper. At that instant, is it on the paper or off?

(Priest, In Contradiction, p.160)

So suppose our domain has one object, the pen, or “e”. And suppose we have one predicate, O, for “is on the paper” or the like. Let us set this up to fill out the formulations.

D = {pen}

d(e) = {pen}

[O: “is on the paper”]

d(O) = {⟨pen⟩}

(I am not sure here if “at the instant of change” belongs to the pen, and thus something like:

D = {pen at t₁}

or if it belongs to the predicate and thus something like:

[O: “is on the paper at t₁”]

or if it is somehow exterior to these two terms and added by other means in other formulations. I will put that matter aside for the time being.) First take a classical situation where we think that at the instant of change, it is still on the paper. Then O’s extension would be {e} and its anti-extension would I think be an empty set, because there are no objects in the domain that do not hold for the predicate. So let us evaluate the truth of the formula saying that the pen is on the paper.

D = {pen}

d(e) = {pen}

[O: “is on the paper”]

d(O) = {⟨pen⟩}

 

E_O = {⟨pen⟩}

A_O = {⟨∅⟩}

 

R_s(Pt₁ ... t_n, 1) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ E_p

R_s(Pt₁ ... t_n, o) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ A_p

 

R_s(Oe, 1) iff ⟨d(e)⟩ ∈ E_O

R_s(Oe, 0) iff ⟨d(e)⟩ ∈ E_A

 

R_s(Oe, 1) iff ⟨pen⟩ ∈ {⟨pen⟩}

R_s(Oe, 0) iff ⟨pen⟩ ∈ {⟨∅⟩}

 

R_s(Oe, 1)

But now suppose instead we want to take the paraconsistent view that it is both on and off the paper at the instant of change. The extension of the predicate would still include the pen, because “is on the paper” still holds for it. But I would think that the anti-extension would include the pen too. For, since it is also off the paper, in light of that fact, we would also need to say that the predicate as well does not hold for the pen. And so the pen would be in both the extension and anti-extension of predicate O, even though in a classical system, this is not permissible. I would think then the evaluation would go as follows:

D = {pen}

d(e) = {pen}

[O: “is on the paper”]

d(O) = {⟨pen⟩}

 

E_O = {⟨pen⟩}

A_O = {⟨pen⟩}

 

R_s(Pt₁ ... t_n, 1) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ E_p

R_s(Pt₁ ... t_n, o) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ A_p

 

R_s(Oe, 1) iff ⟨d(e)⟩ ∈ E_O

R_s(Oe, 0) iff ⟨d(e)⟩ ∈ E_A

 

R_s(Oe, 1) iff ⟨pen⟩ ∈ {⟨pen⟩}

R_s(Oe, 0) iff ⟨pen⟩ ∈ {⟨pen⟩}

 

R_s(Oe, 1)

R_s(Oe, 0)

Let us continue with the current Priest text:

Truth values (relative to s) are then extended to all sentences. The recursive clauses for the connectives are the same as in the propositional case (relativised to s).

(363)

So, I would assume that, for our above example:

R_s(Oe, 1)

 

R_s(¬Oe, 0)

The next part on quantifiers is complicated:

For the quantifiers, if F is a map from D into {0,1} such that R_s(v/a)(α, F(a)) for all a ∈ D, let us say that F tracks α. Then:

R_s(∀vα, z) iff ∃F(F tracks α & z=⊗{F(a); a ∈ D})

R_s(∃vα, z) iff ∃F(F tracks α & z=⊕{F(a); a ∈ D})

(363)

So let us go part-by-part.

F is a map from D into {0,1}

By being a “map,” I am assuming that F is a function or at least that it operates like one. So it will assign the members of D either a 1 or a 0. For the part reading:

R_s(v/a)(α, F(a))

let us first just suppose for the moment that we have:

R(α, F(a))

Here we have formula alpha (α) and constant/domain-item a. F(a) is a 1 or a 0. So the R relation in R(α, F(a)) assigns a 1 or a 0 to alpha, by means of function F, which assigns a 1 or a 0 to a. Let us look at the actual formulation now.

F is a map from D into {0,1} such that R_s(v/a)(α, F(a)) for all a ∈ D

This I think is saying that we are making the relational truth assignment relativized to s (that is, given the particular variable substitution that s happens to make in some certain case), where function s substitutes each member of the domain for the variable, and the value for each substitution is determined by how F assigns a 1 or a 0 for that given substitution. (This all seems to imply that the F function is defined such that it assigns a 1 when the formula’s predicate should be holding for that particular item and a 0 when the predicate should not be holding.) Now, insofar as F makes such assignments (that are in accordance with the intended interpretations of the predicates), then we say that F tracks alpha. Now with that terminology being defined, we then evaluate formulas with universal and existential quantification in the following way:

R_s(∀vα, z) iff ∃F(F tracks α & z=⊗{F(a); a ∈ D})

R_s(∃vα, z) iff ∃F(F tracks α & z=⊕{F(a); a ∈ D})

So again the formulations seem to be made general enough that they function to correctly assign 1 or 0, even though there is no mention of those particular values. This will be easier to unpack if we arbitrarily assign to z either 1 or 0. So let’s say:

R_s(∀vα, 1) iff ∃F(F tracks α & 1=⊗{F(a); a ∈ D})

R_s(∀vα, 0) iff ∃F(F tracks α & 0=⊗{F(a); a ∈ D})

So, supposing that the v here means a variable in formula alpha, these seem to be saying the following. We relate a formula with a universally quantified variable to 1 only if there is a function that assigns to each substitution for the variable either a 1 or 0 (in such a way that the truth-value assigned to the particular substitution corresponds to the truth or falsity of the formula after the substitution is made) and also if the generalized conjunction function yields a 1 when performed on all those truth-value assignments for each substitution. So let us take an Agler-like example that we set up in section 1.1.1 (and note for extension and anti-extension I am using domain items, but perhaps I should be using constant letters standing for those items):

D = {Alfred, Bill, Corinne}

C = {a, b, c}

d(a) = Alfred

d(b) = Bill

d(c) = Corinne

 

[Sx: x is short]

d(Sx) = {⟨Alfred⟩, ⟨Bill⟩}

 

E_S = {⟨Alfred⟩, ⟨Bill⟩}

A_S = {⟨Corinne⟩}

 

R_s(∀vS, ???)

 

R_s(∀vα, 1) iff ∃F(F tracks α & 1=⊗{F(a); a ∈ D})

R_s(∀vα, 0) iff ∃F(F tracks α & 0=⊗{F(a); a ∈ D})

 

R_s(Pt₁ ... t_n, 1) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ E_p

R_s(Pt₁ ... t_n, o) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ A_p

 

F-tracking instance 1:

 

R_s1(Sa, 1) iff ⟨d(a)⟩ ∈ E_S

R_s1(Sa, o) iff ⟨d(a)⟩ ∈ A_S

 

R_s1(Sa, 1) iff ⟨Alfred⟩ ∈ {⟨Alfred⟩, ⟨Bill⟩}

R_s1(Sa, o) iff ⟨Alfred⟩ ∈ {⟨Corinne⟩}

 

R_s1(Sa, 1)

 

F-tracking instance 2:

 

R_s2(Sb, 1) iff ⟨d(b)⟩ ∈ E_S

R_s2(Sb, o) iff ⟨d(b)⟩ ∈ A_S

 

R_s2(Sb, 1) iff ⟨Bill⟩ ∈ {⟨Alfred⟩, ⟨Bill⟩}

R_s2(Sb, o) iff ⟨Bill⟩ ∈ {⟨Corinne⟩}

 

R_s2(Sb, 1)

 

F-tracking instance 3:

 

R_s3(Sc, 1) iff ⟨d(c)⟩ ∈ E_S

R_s3(Sc, o) iff ⟨d(c)⟩ ∈ A_S

 

R_s3(Sc, 1) iff ⟨Corinne⟩ ∈ {⟨Alfred⟩, ⟨Bill⟩}

R_s3(Sc, o) iff ⟨Corinne⟩ ∈ {⟨Corinne⟩}

 

R_s3(Sc, 0)

 

R_s(∀vS, 1) iff ∃F(F tracks α & 1=⊗{F(a); a ∈ D})

R_s((∀vS, 0) iff ∃F(F tracks α & 0=⊗{F(a); a ∈ D})

 

{F(a); a ∈ D}

 

{1, 1, 0}

 

⊗X=0 iff 0 ∈ X

[and 1 otherwise]

 

⊗{1, 1, 0} = 0

 

R_s(∀vS, 1) iff ∃F(F tracks α & 1=0)

R_s((∀vS, 0) iff ∃F(F tracks α & 0=0)

 

R_s(∀vS, 0)

And similarly for the existential quantifier. Validity is defined in the following way:

Validity is defined in terms of truth preservation in all interpretations and evaluations of the variables, as usual.

(363)

(See section 1.1.2 for a discussion of how validity would work, but here only with the relations structures.)  Here is the quotation in full]:

Over recent years, some logicians have suggested that certain sentences, notably, paradoxical ones, may have more than one truth value. Classical semantics are easily modified to accommodate this possibility. Instead of taking an evaluation to be a function, we take it to be a relation, R, between formulas and truth values, such that each formula relates to at least one truth value. In the propositional case, the relation may be taken as defined on atomic formulas in the first instance. It can then be extended to all formulas according to the following simple idea: the truth values of a compound are exactly those that can be obtained by combining the truth values of its components in all possible ways. More precisely:

R(¬α, z) iff ∃x(R(a, x)& z = ⊖x)

R(α ∧ β, z) iff ∃x∃y(R(a, x)& R(β, y)& z= x⊗y)

R(α ∨ β, z) iff ∃x∃y(R(a, x)& R(β, y)& z= x⊕y)

Validity is defined in terms of truth preservation:

Σ ⊨ α iff for all v, iff R(β, 1) for all β ∈ Σ then R(α, 1)

For the first-order case, we now take an interpretation to be a pair, ⟨D, d⟩, where D is a non-empty domain of objects, d assigns each constant a member of the domain, and each n-place predicate, P, a pair ⟨E_p, A_p⟩, (the ex- | tension and anti-extension of P, respectively), such that E_p ∪ A_p = Dⁿ. The relation between formulas and truth values is now relative to an evaluation of the free variables, s. In the atomic case, this is defined by the clauses:

R_s(Pt₁ ... t_n, 1) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ E_p

R_s(Pt₁ ... t_n, o) iff ⟨f(t₁), ..., f(t_n)⟩ ∈ A_p

Truth values (relative to s) are then extended to all sentences. The recursive clauses for the connectives are the same as in the propositional case (relativised to s). For the quantifiers, if F is a map from D into {0,1} such that R_s(v/a)(α, F(a)) for all a ∈ D, let us say that F tracks α. Then:

R_s(∀vα, z) iff ∃F(F tracks α & z=⊗{F(a); a ∈ D})

R_s(∃vα, z) iff ∃F(F tracks α & z=⊕{F(a); a ∈ D})

Validity is defined in terms of truth preservation in all interpretations and evaluations of the variables, as usual.

(362-363)

[contents]

 

 

1.2.2

[Familiar Formulations for Evaluating Complex Formulas. LP. Classical Predicates. LP as a Proper Sub-Logic of Classical.]

 

[The rules for evaluating formulas that are made more complex by connectives or quantifiers are as you would expect for a classical situation, only now using a truth-evaluating relation. These semantics are a version of the paraconsistent logic LP. An interpretation of a predicate is classical if the intersection of its extension and anti-extension is an empty set. LP is a proper sub-logic of classical logic, and all the logical truths of LP are those of classical logic.]

 

Priest then provides a more familiar set of formulations for the truth conditions for the connectives and quantifiers, which can be directly derived from the above more general formulations.

R_s(¬α, 1) iff R_s(α, 0)

R_s(¬α, 0) iff R_s(α, 1)

 

R_s(α ∧ β, 1) iff R_s(α, 1) and R_s(β, 1)

R_s(α ∧ β, 0) iff R_s(α, 0) or R_s(β, 0)

 

R_s(α ∨ β, 1) iff R_s(α, 1) or R_s(β, 1)

R_s(α ∨ β, 0) iff R_s(α, 1) and R_s(β, 1)

 

R_s(∀vα, 1) iff for every a ∈ D, R_s(v/a)(α, 1)

R_s(∀vα, 0) iff for some a ∈ D, R_s(v/a)(α, 0)

 

R_s(∃vα, 1) iff for some a ∈ D, R_s(v/a)(α, 1)

R_s(∃vα, o) iff for all a ∈ D, R_s(v/a)(α, 0)

(363)

Priest notes that what we have just outlined above provides one version to the semantics of LP, which is a paraconsistent logic (see section 7.3 of Introduction to Non-Classical Logic.) Priest then writes:

If the interpretation of a predicate, P, is such that E_P ∩ A_P = ∅, I will call it classical.

(364)

[First recall what we noted above about the intersection, from the brief summary to the post on section 9.5 of Suppes’ Introduction to Logic:

If we find all the members shared in common between two sets, we are finding their intersection (∩):

(x)(x ∈ A ∩ B ↔ x ∈ A & x ∈ B)

(based on Suppes, Introduction to Logic, p.184)

Suppes has us consider the following two sets: A the set of all men and B the set of all animals that weigh more than ten tons (184). He writes,

In this case we notice that A ∩ B is the empty set (despite the fact that A ≠ Λ, and B ≠ Λ, since some whales weigh more than ten tons). When A ∩ B = Λ, we say that A and B are mutually exclusive.
(Suppes, Introduction to Logic, 184)

No also recall from section 1.2.2 above the example of the instant of change, where we first considered a classical view that would say the pen is either on the paper or off the paper, but never both or neither. For the notion that the pen is on the paper at the moment of change, we assigned the extension and anti-extension as:

E_O = {⟨pen⟩}

A_O = {⟨∅⟩}

I would think that the intersection of these sets is the empty set, because they share no members in common, and thus these sets are mutually exclusive.

E_O = {⟨pen⟩}

A_O = {⟨∅⟩}

 

If the interpretation of a predicate, P, is such that E_P ∩ A_P = ∅, I will call it classical.

 

E_O ∩ A_O = ∅

 

[Classical]

The intersection of the two sets in the paraconsistent interpretation (the pen is both on and not on the paper) would not be an empty set, I think, but rather contain one member in both, namely, the pen. Next he writes:

As is easy to see, if all the predicates of an interpretation are classical, then the interpretation is essentially a functional interpretation of classical logic. Hence, LP is a sub-logic of classical logic. It is, however, a proper sub-logic, since it is paraconsistent.

(364)

Unfortunately I do not grasp this part. I do not know what it means to be a sub-logic. Does LP prove fewer formulas (have fewer tautologies)? If so, is that what makes it a sub-logic? If not, what makes it a sub-logic? His final point is:

I note, without proof, that the logical truths of LP are exactly those of classical logic.

(364)

He cites chapter 5 of In Contradiction. I have not read it yet. But it seems that possibly a good place to look for such a proof might be Section 5.5. Now finally here is a quotation of the whole paragraph:]

It is easy to check that the recursive truth conditions may be put in an equivalent, but slightly more familiar, form as follows:

R_s(¬α, 1) iff R_s(α, 0)

R_s(¬α, 0) iff R_s(α, 1)

 

R_s(α ∧ β, 1) iff R_s(α, 1) and R_s(β, 1)

R_s(α ∧ β, 0) iff R_s(α, 0) or R_s(β, 0)

 

R_s(α ∨ β, 1) iff R_s(α, 1) or R_s(β, 1)

R_s(α ∨ β, 0) iff R_s(α, 1) and R_s(β, 1)

 

R_s(∀vα, 1) iff for every a ∈ D, R_s(v/a)(α, 1)

R_s(∀vα, 0) iff for some a ∈ D, R_s(v/a)(α, 0)

 

R_s(∃vα, 1) iff for some a ∈ D, R_s(v/a)(α, 1)

R_s(∃vα, o) iff for all a ∈ D, R_s(v/a)(α, 0)

|

The above semantics are one version of the semantics for the paraconsistent logic LP.¹ If the interpretation of a predicate, P, is such that E_P ∩ A_P = ∅, I will call it classical. As is easy to see, if all the predicates of an interpretation are classical, then the interpretation is essentially a functional interpretation of classical logic. Hence, LP is a sub-logic of classical logic. It is, however, a proper sub-logic, since it is paraconsistent. I note, without proof, that the logical truths of LP are exactly those of classical logic.²

(363-364)

1. They are formulated this way in Priest (1984). Note that if we allow the possibility that sentences may relate to no truth value as well, but leave everything else the same, we get a logic with truth value gaps satisfying the Fregean principle: gap-in/gap-out. In particular, then, the alternative truth conditions just given are no longer equivalent. If we preserve these truth conditions instead, we obtain First Degree Entailment.

2. See Priest (1987), ch. 5.

(364)

[contents]

 

 

1.2.3

[Identity and Substitutivity in LP. ]

 

[Identity in this relation-based truth evaluatory LP semantics is understood as the set of couples (from the domain) whose first and second term are (taken to be) the same. We cannot however have any non-identical couplings be excluded from the identity predicate/relation’s anti-extension, because in our LP semantics there can be no items that are neither identical nor non-identical to another item. (However, we can have pairings of visually identical terms that are in both the extension and anti-extension of the identity predicate/relation, meaning that it is both identical and non-identical with itself.) Substitutivity of identicals holds in this semantics.]

 

[This next part on identity I have trouble following, so I want to look first at similar passages in Priest’s Introduction to Non-Classical Logic section 26.2 (not yet summarized):

We now add identity to the language, starting with the relational semantics. In an interpretation, ⟨D, v⟩, v^ε (=) = {⟨d, d⟩: ∈ D}. The anti-extension of = can be any subset of D².

(Priest’s Introduction to Non-Classical Logic, p.486)

So we have our D. Let us return to our Aglerian example:

D = {Alfred, Bill, Corinne}

C = {a, b, c}

d(a) = Alfred

d(b) = Bill

d(c) = Corinne

D¹ = {⟨Alfred⟩, ⟨Bill⟩, ⟨Corinne⟩}

D² = {⟨Alfred, Alfred⟩, ⟨Alfred, Bill⟩, ⟨Alfred, Corinne⟩, ⟨Bill, Alfred⟩, ⟨Bill, Bill⟩, ⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩, ⟨Corinne, Bill⟩, ⟨Corinne, Corinne⟩}

I am not exactly sure how to write the equality component here, but let us think of the extension for the equality relation/predicate (or whatever it is) as:

E_vε(=) = {⟨Alfred, Alfred⟩, ⟨Bill, Bill⟩, ⟨Corinne, Corinne⟩}

What is the anti-extension? Under a classical mode of thinking, it would seem to be all those cases where the first term is not the same as the second. So:

A_vε(=) = {⟨Alfred, Bill⟩, ⟨Alfred, Corinne⟩, ⟨Bill, Alfred⟩, ⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩, ⟨Corinne, Bill⟩}

So here, the anti-extension would be a subset of D².  Now, in the text in Introduction to Non-Classical Logic, it reads: “The anti-extension of = can be any subset of D².” The fact that it can be “any” subset suggests that in First Degree Entailment, we can have an interpretation where one or more of the items that are in the classical anti-extension are instead missing from it. So suppose the above set is lacking ⟨Alfred, Bill⟩. That would seem to suggest that Alfred and Bill are neither identical (as they are not in the extension of the identity predicate/relation) nor non-identical (as they are not in the anti-extension). With that in mind, let us return to our current text, but first recall the notion of the union of sets from Suppes’ Introduction to Logic section 9.5. (the following comes from the “Brief summary” of that post):

The set containing all the members in total from two sets is their union (∪):

(x)(x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B)

(Suppes, Introduction to Logic, pp.184-185)

Now back to our current Priest text:

To extend the machinery to identity is not difficult. We simply require identity to be a logical (semi-)constant: E₌ is always {⟨x, x⟩; x ∈ D}. (A₌ can be anything, except, of course, that E₌ ∪ A₌ = D².)

This is very similar to the First Degree Entailment passages, but I found this part to be bit tricky:

A₌ can be anything, except, of course, that E₌ ∪ A₌ = D².

I am guessing that this is the stipulation that prevents there from being any couples of visibly non-identical terms that are in neither the extension nor the anti-extension. In other words, I am guessing that this stipulates that the union of the extension and the anti-extension be the same as the set of all couples that can be made from the domain. This would seem to exclude the possibility that there are any pairings that are neither identical nor are non-identical. But what about the possibility that for example ⟨Alfred, Alfred⟩ is in both the extension and anti-extension of the identity relation/predicate? Suppose that is so, and take our definition of set union again:

The set containing all the members in total from two sets is their union (∪):

(x)(x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B)

(based on but not quoting Suppes, Introduction to Logic, pp.184-185)

So as far as I can tell, ⟨Alfred, Alfred⟩ for example can be in both the extension and anti-extension of the identity relation/predicate. That would seem to suggest that Alfred is both identical to himself and not identical to himself. Priest then writes:

The logical truths in the language with identity are, again, exactly the same as those of classical logic. I note also that the semantics verify the standard substitutivity principle for identity: a = b, α(a) ⊨ α(b).

(364)

As I did not understand how to compare systems in terms of their logical truths, I will leave this part also to be filled in later after I learn it. (He again cites In Contradiction Chapter 5, so I would think that it is related to the above mention.) The last point is that these LP semantics allow for the  substitutivity of items that have been designated as equal/identical. But as for making sense of this part’s more technical features, I am not sure. What would need to take place I think is that for all evaluations that make a=b and α(a) true (and maybe specifically make them ‘at least true’ but I am not sure), then α(b) will be true too. I do not know what α(a) and α(b) mean exactly, but I would think that they mean a formula where a certain variable is substituted by either a or b. So if constant a is designated as being identical to b, then all valuations making that equality true along with making the substitution of a for some variable in some particular formula true will also make that formula true were it substituted by b.]

To extend the machinery to identity is not difficult. We simply require identity to be a logical (semi-)constant: E₌ is always {⟨x, x⟩; x ∈ D}. (A₌ can be anything, except, of course, that E₌ ∪ A₌ = D².) The logical truths in the language with identity are, again, exactly the same as those of classical logic. I note also that the semantics verify the standard substitutivity principle for identity: a = b, α(a) ⊨ α(b).

(364)

[contents]

 

 

 

From:

 

Priest, Graham. 1995. “Multiple Denotation, Ambiguity, and the Strange Case of the Missing Amoeba.” Logique et Analyse 38: 361-373.

Available at his website:

http://grahampriest.net/publications/papers/#1995

And at JSTOR:

http://www.jstor.org/stable/44084549

 

 

 

Also cited:

 

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

 

Grandy, Richard. 1979 [first published 1977]. Advanced Logic for Applications. Dordrecht: Reidel.

 

Kaynak, Baran. 2011. “Classical Relations and Fuzzy Relations.” Slide presentation. Available at:

https://www.slideshare.net/barankaynak/classical-relations-and-fuzzy-relations

Slide 8:

https://www.slideshare.net/barankaynak/classical-relations-and-fuzzy-relations/8

 

 

Priest, Graham. 2006 [first edition published 1987]. In Contradiction: A Study of the Transconsistent, Expanded Edition. Oxford/New York: Oxford University.

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

Priest, Graham. 2014. One: Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness. Oxford: Oxford University.

 

Schuller, Frederic P. Class Video Lecture 1 . “Logic of Propositions and Predicates.” From his course, “Geometric Anatomy of Theoretical Physics / Geometrische Anatomie der Theoretischen Physik” at the Institute for Quantum Gravity of the University of Erlangen-Nürnberg. Available at youtube at:

Class 1: “Logic of Propositions and Predicates” :
https://youtu.be/aEJpQ3d1ltA

28 minute mark for the conditional’s evaluation scheme and its relation to ex falso quodlibet:

https://youtu.be/V49i_LM8B0E?t=28m

Discussed on this blog here:

http://piratesandrevolutionaries.blogspot.com/2016/03/the-axioms-of-axiomatic-set-theory-from.html

 

Suppes, Patrick. 1957. Introduction to Logic. New York: Van Nostrand Reinhold / Litton Educational.

 

 

.

Priest (12.3) Introduction to Non-Classical Logic, ‘Semantics [of first classical first-order logic]’, summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[Priest, Introduction to Non-Classical Logic, entry directory]

 

[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other distracting mistakes, because I have not finished proofreading.]

 

 

Summary of

 

Graham Priest

 

An Introduction to Non-Classical Logic: From If to Is

 

Part II

Quantification and Identity

 

12

Classical First-order Logic

 

12.3

Semantics

 

 

 

Brief summary [much of the following is quotation]:

Regarding the semantics of classical first-order logic, we say that:

An interpretation of the language is a pair, ℑ = ⟨D, v⟩. D is a non-empty set (the domain of quantification); v is a function such that:

• if c is a constant, v(c) is a member of D

• if P is an n-place predicate, v(P) is a subset of Dⁿ

(Dⁿ is the set of all n-tuples of members of D, {⟨d₁, ..., d_n⟩: d₁, ..., d_n ∈ D}. By convention, ⟨d⟩ is just d, and so D¹ is D.)

(Priest 264)

All formulas have a truth value. To evaluate them, since they use variables which are substitutable by constants,

we extend the language to ensure that every member of the domain has a name. For all d ∈ D, we add a constant to the language, k_d, such that v(k_d) = d. The extended language is the language of ℑ, and written L(ℑ). The truth conditions for (closed) atomic sentences are:

v(Pa₁ ... a_n) = 1 iff  ⟨v(a₁),  ..., v(a_n)⟩ ∈ v(P) (otherwise it is 0)

The truth conditions for the connectives are as in the propositional case (1.3.2).

(Priest 265)

v(¬A) = 1 if v(A) = 0, and 0 otherwise.
v(A ∧ B) = 1 if v(A) = v(B) = 1, and 0 otherwise.
v(A ∨ B) = 1 if v(A) = 1 or v(B) = 1, and 0 otherwise.
v(A ⊃ B) = 1 if v(A) = 0 or v(B) = 1, and 0 otherwise.
v(A ≡ B) = 1 if v(A) = v(B), and 0 otherwise.

(Priest 5)

For the quantifiers:

v(∀xA) = 1 iff for all d ∈ D, v(A_x(k_d)) = 1 (otherwise it is 0)

v(∃xA) = 1 iff for some d ∈ D, v(A_x(k_d)) = 1 (otherwise it is 0)

(Priest 265)

We define validity in the following way:

Validity is a relationship between premises and conclusions that are closed formulas, and is defined in terms of the preservation of truth in all interpretations, thus: Σ ⊨ A iff every interpretation that makes all the members of Σ true makes A true.

(Priest 265)

We find the following equivalences in classical first-order logic:

v(¬∃xA) = v(∀x¬A)

v(¬∀xA) = v(∃x¬A)

v(¬∃x(Px ∧ A)) = v(∀x(Px ⊃ ¬A)

v(¬∀x(Px ⊃ A) = v(∃x(Px∧¬A))

(Priest 265)

And lastly we note [something regarding denotation, namely] that

If C is some set of constants such that every object in the domain has a name in C, then:

v(∀xA) = 1 iff for all c ∈ C, v(A_x(c)) = 1 (otherwise it is 0)

v(∃xA) = 1 iff for some c ∈ C, v(A_x(c)) = 1 (otherwise it is 0)

(Priest 265)

 

 

 

Contents

 

12.3.1

[Interpretations of Constants and Predicates]

 

12.3.2

[Truth Evaluation for Classical First-Order Logic]

 

12.3.3

[Validity in Classical First-Order Logic]

 

12.3.4

[Some Equivalences]

 

12.3.5

[Constants, Denotation, and Quantification]

 

 

 

 

Summary

 

12.3.1

[Interpretations of Constants and Predicates]

 

[In the following we will look at interpretations for classical first-order logic. To give illustration to some of these concepts, let me first take an example from David Agler’s Symbolic Logic: Syntax, Semantics, and Proof section 6.4.2. We have the following domain:

D = {Alfred, Bill, Corinne}

(Agler 265)

We have an interpretation function I. It assigns objects in the domain to names (considered constants also, I think).

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)

Then we have n-place predicates:

Sx: x is short

(Agler 265)

Lxy: x loves y

(Agler 265)

To these n-place predicates, the interpretation function I assigns sets of n-tuples:

I(Sx): {⟨Alfred⟩, ⟨Bill⟩}

I(Lxy): {⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩}

(Agler 265, angle-brackets for the singletons added by me for my own convenience, but perhaps erroneously. See the text of this section explaining their omission.)

Agler then describes how we make the truth-value assignments with the v valuation function:

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,’ ... ‘a_n’ be a finite set of names in RL, and let ‘a_i’ be a randomly selected name.

(1) v(Ra_i) = T if and only if the interpretation of ‘a_i’ is in ‘R.’

This says that we can assign a value of true to ‘Ra_i’ if and only if our interpretation of ‘a_i’ 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)

I am a little confused about the use of Ra_i, because this seems to be a 1-place predicate only. I wonder if we can modify this for larger valences. So maybe:

v(Ra₁, ... a_n) = T if and only if the n-tupled interpretations of a₁, ... a_n are in ‘R.’

Suppose we have

v(Lac)

The interpretation of a and c would be Alfred and Corinnne. But recall the interpretation for L

I(Lxy): {⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩}

Here we have Alfred and Corinne. But they are not in the same order as Lac. I am not sure how to formally work with this. I think back when I made the summary, I had:

I(Lac) = {⟨Alfred, Corinne⟩}

This is not written in the Agler text, but it is based on the above formulation for I(Lxy) and this line:

I (Sx): objects in D that are short (i.e.,{Alfred, Bill})

(Agler 266)

But maybe for Priest’s formulations we are not allowed to use the v function for a specific formula with constants, and rather the formulations need to refer to the interpretations of the constants taken by themselves but not to formulas with constants in them. This is why I wrote for the Agler reformulation:

the n-tupled interpretations of a₁, ... a_n

meaning that the interpretations made in that order and then combined into an n-tuple. I am not sure if that works, but note that in the next section, 12.3.2, the evaluation is written as:

v(Pa₁ ... a_n) = 1 iff  ⟨v(a₁),  ..., v(a_n)⟩ ∈ v(P) (otherwise it is 0)

where it says

⟨v(a₁),  ..., v(a_n)⟩

is what I mean I suppose by the n-tupled interpretations of a₁, ... a_n. Here Priest is not saying

v(Pa₁ ... a_n) = 1 iff  v(Pa₁ ... a_n) ∈ v(P) (otherwise it is 0)

So probably a formulation like

I(Lac) = {⟨Alfred, Corinne⟩}

would not work here. (Also, it brings out my confusion between the uses of the v function.) At any rate, supposing that we have determined the n-tuples corresponding to the predicate formulations, we would see that in our example

{⟨Alfred, Corinne⟩}

is not a member of

I(Lxy): {⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩}

Thus

v(Lac) = F

Now let us turn to Priest’s mode of formulation. He writes:

An interpretation of the language is a pair, ℑ = ⟨D, v⟩. D is a non-empty set (the domain of quantification); v is a function such that:

• if c is a constant, v(c) is a member of D

• if P is an n-place predicate, v(P) is a subset of Dⁿ

(Dⁿ is the set of all n-tuples of members of D, {⟨d₁, ..., d_n⟩: d₁, ..., d_n ∈ D}. By convention, ⟨d⟩ is just d, and so D¹ is D.)

(Priest 264. Note, the first formulation should look like:

)

I am not certain, but what Priest is calling an “interpretation” seems to be what Agler was calling a “model”:

a model consists of a domain (D) and an interpretation function (I).

(Agler 264)

So returning to Priest, the domain D is the “domain of quantification”. I am assuming this is the domain of objects to which the quantifiable variables may be assigned, but I am not sure. Let us suppose that it is, as I am not sure what else it would be. (I am not sure if the constants are in this domain. From what is said later, it seems not, but I really do not know.) The v function seems to be like Agler’s I function. Again from Priest:

 v is a function such that:

• if c is a constant, v(c) is a member of D

• if P is an n-place predicate, v(P) is a subset of Dⁿ

In our Agler example,

I (c) = Corinne

So I would think that here with the Priest notation we would write:

v(c) = Corinne

or maybe

v(c) = {Corinne}

But I am not sure. The next idea we should cover is the idea behind Dⁿ. Priest says:

Dⁿ is the set of all n-tuples of members of D, {⟨d₁, ..., d_n⟩: d₁, ..., d_n ∈ D}. By convention, ⟨d⟩ is just d, and so D¹ is D.

So let us look at our Agler example:

D = {Alfred, Bill, Corinne}

D¹ = {⟨Alfred⟩, ⟨Bill⟩, ⟨Corinne⟩}

D² = {⟨Alfred, Alfred⟩, ⟨Alfred, Bill⟩, ⟨Alfred, Corinne⟩, ⟨Bill, Alfred⟩, ⟨Bill, Bill⟩, ⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩, ⟨Corinne, Bill⟩, ⟨Corinne, Corinne⟩}

As he says, “By convention, ⟨d⟩ is just d, and so D¹ is D,” so:

D¹ = {⟨Alfred⟩, ⟨Bill⟩, ⟨Corinne⟩} =

D = {Alfred, Bill, Corinne}

(I wonder if one way to understand these formulations are as Cartesian products of sets made upon iterations of themselves. Consider slide 8 of Baran Kaynak’s presentation “Classical Relations and Fuzzy Relations”

The elements in two sets A and B are given as A ={0, 1} and B ={a,b, c}.

Various Cartesian products of these two sets can
be written as shown:

A × B ={(0,a),(0,b),(0,c),(1,a),(1,b),(1,c)}

B × A ={(a, 0), (a, 1), (b, 0), (b, 1), (c, 0), (c, 1)}

A × A = A²={(0, 0), (0, 1), (1, 0), (1, 1)}

B × B = B²={(a, a), (a, b), (a, c), (b, a), (b, b), (b,
c), (c, a), (c, b), (c, c)}

(Kaynak, slide 8)

At any rate,) consider again:

D² = {⟨Alfred, Alfred⟩, ⟨Alfred, Bill⟩, ⟨Alfred, Corinne⟩, ⟨Bill, Alfred⟩, ⟨Bill, Bill⟩, ⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩, ⟨Corinne, Bill⟩, ⟨Corinne, Corinne⟩}

And recall our two-place predicate from the Agler example:

I(Lxy): {⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩}

Is

{⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩}

a subset of

{⟨Alfred, Alfred⟩, ⟨Alfred, Bill⟩, ⟨Alfred, Corinne⟩, ⟨Bill, Alfred⟩, ⟨Bill, Bill⟩, ⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩, ⟨Corinne, Bill⟩, ⟨Corinne, Corinne⟩}

? Yes, it is. So recall this part of Priest’s formulation:

if P is an n-place predicate, v(P) is a subset of Dⁿ

So with the Agler example, we might write:

if L is an 2-place predicate, v(L) is a subset of D²

(Or maybe we should write:

if Lxy is an 2-place predicate, v(Lxy) is a subset of D²

) And in fact it is, as we saw. Let me now repeat the whole quotation for a clean presentation of it:]

An interpretation of the language is a pair, ℑ = ⟨D, v⟩. D is a non-empty set (the domain of quantification); v is a function such that:

• if c is a constant, v(c) is a member of D

• if P is an n-place predicate, v(P) is a subset of Dⁿ

(Dⁿ is the set of all n-tuples of members of D, {⟨d₁, ..., d_n⟩: d₁, ..., d_n ∈ D}. By convention, ⟨d⟩ is just d, and so D¹ is D.)

(Priest 264. Note, the first formulation should look like:

)

[contents]

 

 

12.3.2

[Truth Evaluation for Classical First-Order Logic]

 

[Recall from section 12.2.4 that a formula is “bound if it occurs in a context of the form ∃x ... x ... or ∀x ... x”, and it is free otherwise. And a “formula with no free variables is said to be closed”. Priest now writes:

Given an interpretation, truth values are assigned to all closed formulas.

(264)

Now, suppose we consider our Agler example

v(Lac) = F

(see Agler p.267)

Here do not have any variables, but we have a truth-value. Perhaps since there are no variables, that means there are no free variables, and thus that it would be by definition closed and hence eligible for a truth-value assignment by means of v. But I would like to mention a confusion I have regarding v. According to 12.3.1 above, v assigns either singular members of D (to constants) or n-tuple subsets of D (to n-place predicates). But here in this section, it will assign truth-values. I do not understand that yet. Either it is a different v function that assigns to formulas values from another domain (like the domain of values V in the structure ⟨V, D, {f_c; c ∈ C}⟩ from section 7.2.2.) Or perhaps “1” and “0” are members of D, and the n-tuples of Dⁿ with 1 and 0 as members are normally not extensions of predicates. I do not know how this works yet. At any rate, we suppose that by means of v we can assign 1 or o to a formula. He next writes:

To state the truth conditions, we extend the language to ensure that every member of the domain has a name.

(264)

I am not exactly sure why this is necessary. I would guess that it has something to do with evaluating quantifiers. For example, in order to evaluate all substitutions of a formula’s variable, we might need to make sure that the formula holds for all members of the domain. And maybe we also need to note that the domain members themselves cannot be substituted for the variables (we cannot say, LAlbert,Corinne for example) and instead only names or constants can (in other words, we can only say Lac in this case). Thus we need our language to include a set of constants that will fully exhaust the set of objects in the domain. This might be what is meant then in what follows here:

For all d ∈ D, we add a | constant to the language, k_d, such that v(k_d) = d. The extended language is the language of ℑ, and written L(ℑ).

(364-365)

(Note, some of these issues are discussed more directly in section 12.3.5 below.) Let us now look at how we evaluate closed atomic sentences:

v(Pa₁ ... a_n) = 1 iff  ⟨v(a₁),  ..., v(a_n)⟩ ∈ v(P) (otherwise it is 0)

And let us consider our Agler example:

v(Lxy) = 1 iff  ⟨v(x), v(y)⟩ ∈ v(L) (otherwise it is 0)

or maybe:

v(Lxy) = 1 iff  ⟨v(x), v(y)⟩ ∈ v(Lxy) (otherwise it is 0)

(Here I am supposing that the first v assigns 1 or 0 to formulas, and the second v assigns sets of n-tuples to n-place predicates. I mention this to compare with the Agler text, where I assigns to objects (or n-tuples of objects) in the domain and v assigns to truth values.). Let us now make the formula under truth evaluation specific:

v(Lac) = 1 iff  ⟨v(a), v(c)⟩ ∈ v(Lxy) (otherwise it is 0)

expanded further

v(Lac) = 1 iff  ⟨v(a), v(c)⟩ ∈ {⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩} (otherwise it is 0)

and expended yet further

v(Lac) = 1 iff  ⟨Alfred, Corrine⟩ ∈ {⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩} (otherwise it is 0)

Now, since

⟨Alfred, Corrine⟩ ∉ {⟨Bill, Corinne⟩, ⟨Corinne, Alfred⟩}

we would have:

v(Lac) = 0

We now will examine how to evaluate formulas made more complex by means of the connectives. He writes:

The truth conditions for the connectives are as in the propositional case (1.3.2).

(365)

Recall from section 1.3.2 the following evaluation rules:

Given an interpretation of the language, v, this is extended to a function that assigns every formula a truth value, by the following recursive clauses, which mirror the syntactic recursive clauses:
v(¬A) = 1 if v(A) = 0, and 0 otherwise.
v(A ∧ B) = 1 if v(A) = v(B) = 1, and 0 otherwise.
v(A ∨ B) = 1 if v(A) = 1 or v(B) = 1, and 0 otherwise.
v(A ⊃ B) = 1 if v(A) = 0 or v(B) = 1, and 0 otherwise.
v(A ≡ B) = 1 if v(A) = v(B), and 0 otherwise.

(5)

So suppose we take:

v(¬Lac)

Since

v(Lac) = 0

That means

v(¬Lac) = 1

And so on for the rest. Next we learn how to evaluate quantified formulas. He writes:

For the quantifiers:

v(∀xA) = 1 iff for all d ∈ D, v(A_x(k_d)) = 1 (otherwise it is 0)

v(∃xA) = 1 iff for some d ∈ D, v(A_x(k_d)) = 1 (otherwise it is 0)

(265)

Let us go through it part by part to be sure we know their senses.

v(∀xA)

Recall from section 12.2.3:

If A is any formula, and x is any variable, then ∀xA, ∃xA are formulas. I will omit outermost brackets in formulas.

(264)

So

∀xA

means the formula A, which has variable x, and we are universally quantifying over x. I am not certain, but suppose we take the Agler example from above

Lxy

Let us then write:

∀x∀yLxy

I am not sure if we can do that, but I am trying to understand what ∀xA means. So perhaps ∀x∀yLxy would mean something like, everyone loves everyone (including themselves). (Or “for all x and for all y, x loves y.”) So, ∀xA might mean, for all x, A holds. Or something like that. Then

v(∀xA)

means the truth-value that this statement would be assigned. Similarly for v(∃xA). Recall again our formulations:

v(∀xA) = 1 iff for all d ∈ D, v(A_x(k_d)) = 1 (otherwise it is 0)

v(∃xA) = 1 iff for some d ∈ D, v(A_x(k_d)) = 1 (otherwise it is 0)

The d ∈ D we already said was all the objects d in the domain D. And recall from above that k_d is a constant, which I think is like a name for the object, like “a” for “Alfred”. Now recall from section 12.2.4 that:

A_x(c) is the formula obtained by substituting c for each free occurrence of x in A.

(264)

In our current passage, we have something similar:

A_x(k_d)

So this would seem to mean that we have substituted k_d for all free occurrences of  x in A. Let us think of our Agler examples, using a simpler one:

I (Sx) = {⟨Alfred⟩, ⟨Bill⟩}

So we might write:

S_x(a)

This would be Sa, or Alfred is short (I guess technically it means, a is a member of the interpretation of S, and we can secondarily note that a’s interpretation is Alfred. Thus Alfred by extension is short.). And so on for b and c. But Priest’s formulation is:

for all d ∈ D, v(A_x(k_d))

In our Agler example, we have three d’s: Alfred, Bill, and Corinne. And so we have three k_d’s: a, b, and c. We need then to do the truth-valuation for each of them:

v(A_x(k_d))

So:

v(S_x(a)) = 1

v(S_x(b)) = 1

v(S_x(c)) = 0

Now, again we have:

v(∀xA) = 1 iff for all d ∈ D, v(A_x(k_d)) = 1 (otherwise it is 0)

v(∃xA) = 1 iff for some d ∈ D, v(A_x(k_d)) = 1 (otherwise it is 0)

Let us formulate it for our example:

v(∀xSx) = 1 iff for all d ∈ D, v(S_x(k_d)) = 1 (otherwise it is 0)

v(∃xSx) = 1 iff for some d ∈ D, v(S_x(k_d)) = 1 (otherwise it is 0)

Since not all our v(S_x(k_d)) equal 1, that means:

v(∀xSx) = 0

and since at least one v(S_x(k_d)) equals 1, then

v(∃xSx) = 1

. Let me quote it all now in full.]

Given an interpretation, truth values are assigned to all closed formulas. To state the truth conditions, we extend the language to ensure that every member of the domain has a name. For all d ∈ D, we add a | constant to the language, k_d, such that v(k_d) = d. The extended language is the language of ℑ, and written L(ℑ). The truth conditions for (closed) atomic sentences are:

v(Pa₁ ... a_n) = 1 iff  ⟨v(a₁),  ..., v(a_n)⟩ ∈ v(P) (otherwise it is 0)

The truth conditions for the connectives are as in the propositional case (1.3.2). For the quantifiers:

v(∀xA) = 1 iff for all d ∈ D, v(A_x(k_d)) = 1 (otherwise it is 0)

v(∃xA) = 1 iff for some d ∈ D, v(A_x(k_d)) = 1 (otherwise it is 0)

(264-265)

[contents]

 

 

12.3.3

[Validity in Classical First-Order Logic]

 

[Next we define validity. Recall from section 1.3.3:

Let Σ be any set of formulas (the premises); then A (the conclusion) is a semantic consequence of Σ (Σ ⊨ A) iff there is no interpretation that makes all the members of Σ true and A false, that is, every interpretation that makes all the members of Σ true makes A true. ‘Σ ⊭ A’ means that it is not the case that Σ ⊨ A.

(Introduction to Non-Classical Logic, p.5)

Here we say something similar, except we stipulate that this holds for closed formulas. I am not sure why that it is, but probably it can be shown why it is problematic for the validity to have a non-closed formula involved. I just myself do not know the reason. Let me quote:]

Validity is a relationship between premises and conclusions that are closed formulas, and is defined in terms of the preservation of truth in all interpretations, thus: Σ ⊨ A iff every interpretation that makes all the members of Σ true makes A true.

(265)

[contents]

 

 

12.3.4

[Some Equivalences]

 

Priest will note the following equivalences:

v(¬∃xA) = v(∀x¬A)

v(¬∀xA) = v(∃x¬A)

v(¬∃x(Px ∧ A)) = v(∀x(Px ⊃ ¬A)

v(¬∀x(Px ⊃ A) = v(∃x(Px∧¬A))

(Priest 265)

[Priest then gives the reasoning for the first two. The main ideas seem to be the following. Suppose that it is not the case that there is at least one thing for which the predicate holds. In our Agler example, suppose that there are no people who are short. That means that it is false that there is someone who is short. That furthermore means that it is false that everyone is short (this part of the reasoning I may not be following well. Given my ignorance, I have a confusion that I will mention now but correct later after I come to a proper understanding. I am speaking about the part reading “for all d in the domain of the interpretation, v(A_x(k_d)) = 0, i.e., v(¬A_x(k_d)) = 1. So, v(∀x¬A) = 1”. As far as I can tell, the first two formulations mean that the predicate does not hold for all the objects in the domain, but I would think that there could be two very different sorts of situations where that would be so. One is where it holds for none, as in our situation, and the other is where it holds for some, maybe even most. So I am not knowledgeable enough yet to interpret this. My best guess is that “for all d in the domain of the interpretation, v(A_x(k_d)) = 0” means not that it is false that the predicate holds for all members together ((such that it may still hold for at least one)) but rather that it means, given any substitution, the predicate will not hold for any of them taken individually. So in other words, “for all d in the domain of the interpretation” does not work the same way as a universal quantification sort of structure where you would say “for all x”. Maybe d is not a variable in that sense, and we are speaking of something more like a procedure where we make each d substitution and find that for every substitution, the predicate does not hold. But let me explicate my misunderstanding a little more so that I can better fix it. I do realize from Agler’s Introduction section 7.1 that ¬(∃x)P is equivalent to (∀x)¬P. And intuitively that makes sense. If there is not at least one object for which the predicate holds, then for all objects the negation of the predicate holds (If not one of the people are short, then all of them are not short). My confusion rather comes from trying to understand the senses of the formulations in the proof and then grasping how those senses relate logically. In Agler’s Introduction section 6.5, he lists

¬(∀x)Mx

as meaning “Not everything is moveable”, which from its sense I would think can mean at least one thing is not moveable. Another sense altogether would be that “nothing at all is moveable” (not even one), which Agler says is symbolized as:

(∀x)¬Mx

Also, Priest writes “v(A_x(k_d)) = 0”. In Agler’s Introduction section 6.4, he writes:

v(∀x)P = F iff for at least one ‘a’ not in ‘P’ and at least one a-variant interpretation ‘P(a/x) = F.’

(Agler 268)

So what I am not sure about is if we are to understand Priest’s “for all d in the domain of the interpretation, v(A_x(k_d)) = 0” to mean that the predicate does not hold for at least one object. Under that sense, it is possible that there is just one that it does not hold for, but it does hold for all the others, and thus it cannot be said that it holds for none. So to recap: in Priest’s explanation, where I get lost in following the senses is going from

for all d in the domain of the interpretation, v(A_x(k_d)) = 0, i.e., v(¬A_x(k_d)) = 1

(does it mean not all are short or none are short?)

to

v(∀x¬A) = 1

(No one is short)

It seems most likely that “for all d in the domain of the interpretation, v(A_x(k_d)) = 0, i.e., v(¬A_x(k_d)) = 1” means that given any item in the domain, the predicate will not hold for it (or perhaps, given any name for an item in the domain, it is not found in the interpretation of the predicate). And thus this has the sense of “the predicate holds for none (not even one).” The other three equivalences Priest leaves for exercises; but since I did not grasp the steps in the reasoning, I will try them later after I form a proper understanding]

Note that in any interpretation, we have the following:

v(¬∃xA) = v(∀x¬A)

v(¬∀xA) = v(∃x¬A)

v(¬∃x(Px ∧ A)) = v(∀x(Px ⊃ ¬A)

v(¬∀x(Px ⊃ A) = v(∃x(Px∧¬A))

For the first of these, suppose that v(¬∃xA) = 1. Then v(∃xA) = 0. So for all d in the domain of the interpretation, v(A_x(k_d)) = 0, i.e., v(¬A_x(k_d)) = 1. So, v(∀x¬A) = 1. Conversely, suppose that v(∀x¬A) = 1. Then for all d in the domain of the interpretation v(¬A_x(k_d)) = 1, that is, v(A_x(k_d)) = 0. Hence, v(∃xA) = 0, and v(¬∃xA) = 1. The other three cases are left as exercises.

(265)

[contents]

 

 

12.3.5

[Constants, Denotation, and Quantification]

 

[In this part, some of the ideas will be developed later in the text, so I will not try to completely explicate it. I will instead note some notions that I think will be very useful for understanding Priest’s text, “Multiple Denotation, Ambiguity, and the Strange Case of the Missing Amoeba.” We are to think that every object in the domain is assigned a “name” (using Agler’s term) in another set of constants, C. So using Agler’s example:

D = {Alfred, Bill, Corinne}

C = {a, b, c}

v(a) = Alfred

v(b) = Bill

v(c) = Corinne

What I note is that this seems to be a matter of denotation. In other words, denotation is the extensional reference of “names” in a set of constants to the objects in the domain. At any rate, we suppose that we have a set C of such constants, and for every object in the domain there is a constant in C  assigned to it. Another important thing to note in all of this is that, as we also mentioned above, the universal and existential quantifiers do not operate on members of the domain but rather on the constants/names that are assigned to the members of the domain. This notion seems to be reinforced in the footnote. If I gather properly (but probably I am misunderstanding something), it seems to be saying that we can formulate our set C of constants such that there are some objects in the domain for which there is no constant/name for them. As such, we could have one thing in the domain for which a predicate P should hold, but since there is no name for that thing, ∃xPx is not true. So suppose instead of the above model that we subtract the c constant to get:

D = {Alfred, Bill, Corinne}

C = {a, b}

v(a) = Alfred

v(b) = Bill

And suppose further that Corinne is the only female and that the predicate for being female is F. So

v(F) = {Corinne}

But suppose we also write:

∃xFx

Our rule in this section reads:

v(∃xA) = 1 iff for some c ∈ C, v(A_x(c)) = 1 (otherwise it is 0)

Now, our only constants are a and b.

v(F_x(a)) = 0

and

v(F_x(b)) = 0

That is all our constants. So

v(∃xFx) = 0

even though the predicate F holds for at least one member of our domain, Corinne. The problem it seems is that there is no way for the existentially quantified predicate formulation to express that state of affiars, because apparently the quantifications range over variables but not domain objects (I am not sure if they range over constants. It seems not, but I do not know). The last thing I would note is something I might be misunderstanding. My sense is that the constants/names k_d must completely name the objects in the domain, but the constants in C can be stipulated either to completely make those designations (as in the text body here) or not to (as in the footnote). That is my guess at the moment.]

Note also the following. If C is some set of constants such that every object in the domain has a name in C, then:

v(∀xA) = 1 iff for all c ∈ C, v(A_x(c)) = 1 (otherwise it is 0)

v(∃xA) = 1 iff for some c ∈ C, v(A_x(c)) = 1 (otherwise it is 0)

The proof is a simple corollary of the Denotation Lemma, and is given in 12.8.4. As we will see, counter-models read off from the open branch of a tableau are of this kind. Since appropriate versions of the Denotation Lemma can be proved for all the logics we will be concerned with in this | part of the book, the same is true for all of them. I will not keep mentioning the fact.²

(265-266)

2 If C is the set of the constants in the original language (that is, unaugmented by the special constants, k_d, of 12.3.2) and the truth conditions of quantifiers are given as in 12.3.5, we obtain a notion of quantification different from the more standard one employed here, and usually called substitutional quantification. It is a feature of substitutional quantification, that something in the domain can be in v(P), and yet ∃xPx is not true, just because the object in question has no name in the interpretation. In principle, all the logics we will meet in this book could be formulated with substitution quantification but we will not pursue this.

(266)

[contents]

 

 

 

 

 

From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

Also cited:

 

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

 

Kaynak, Baran. 2011. “Classical Relations and Fuzzy Relations.” Slide presentation. Available at:

https://www.slideshare.net/barankaynak/classical-relations-and-fuzzy-relations

Slide 8:

https://www.slideshare.net/barankaynak/classical-relations-and-fuzzy-relations/8

 

 

 

 

.