## 10 Jan 2018

### Priest (8.2) Introduction to Non-Classical Logic, ‘The Semantics of FDE [First Degree Entailment]’, summary

[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 I

Prop0sitional Logic

8

First Degree Entailment

8.2

The Semantics of FDE

Brief summary:

In our semantics for First Degree Entailment (FDE), our only connectives are ∧, ∨ and ¬ (with A ⊃ B being defined as ¬A ∨ B.) FDE uses relations rather than functions to evaluate truth. So a truth-valuing interpretation in FDE is a relation ρ between propositional parameters and the values 1 and 0. We write 1 for p relates to 1, and 0 for p relates to 0. This allows a formula to have one of the following four value-assignment situations: just true (1, e.g.: 1), just false (0, e.g.: 0), both true and false (1 and 0, e.g.: 1, o), and neither true nor false (no such valuing formulations). In FDE, being false (that is, relating to 0) does not automatically mean being untrue (that is, not relating to 1), because it can still be related to 1 along with 0. For formulas built up with connectives, we use the same criteria as in classical logic to evaluate them, only here we can have formulas taking both values. In FDE, semantic consequence is defined as:

Σ ⊨ A iff for every interpretation, ρ, if 1 for all B ∈ Σ then 1

(144)

and logical truth or tautology as:

A iff φA, i.e., for all ρ, 1

(144)

Contents

8.2.1

[Connectives]

8.2.2

[Bivalence]

8.2.3

[Relations for Neither and for Both True and False]

8.2.4

[False is Not Untrue in FDE]

8.2.5

[The Truth-Evaluating ρ Relation]

8.2.6

[Evaluation Rules for Connectives]

8.2.7

[Example of Value Calculation]

8.2.8

[Semantic Consequence. Logical Truth or Tautology.]

Summary

8.2.1

[Connectives]

[Our only connectives in FDE are ∧, ∨ and ¬.

We define A ⊃ B as ¬A ∨ B.]

We are outlining the semantics for First Degree Entailment (FDE). The only connectives in the language of FDE are conjunction, disjunction, and negation. If we want the conditional, we instead use negation and disjunction.

The language of FDE contains just the connectives ∧, ∨ and ¬. A ⊃ B is defined, as usual, as ¬A ∨ B.

(142)

[contents]

8.2.2

[Bivalence]

[In classical semantics, an interpretation is a function that assigns 1 or 0 to a propositional parameter; not neither, and not both.]

[Recall first the notion of interpretation from section 1.3.1:

An interpretation of the language is a function, v, which assigns to each propositional parameter either 1 (true), or 0 (false). Thus, we write things such as v(p) = 1 and v(q) = 0.

(5)

We see from the wording that the definition seems to imply that we never assign to a propositional parameter no value at all or both values.]

In the classical propositional calculus, an interpretation is a function from formulas to the truth values 0 and 1, written thus: v(A) = 1 (or 0). Packed into this formalism is the assumption (usually made without comment in elementary logic-texts) that every formula is either true or false; never neither, and never both.

(142)

[contents]

8.2.3

[Relations for Neither and for Both True and False]

[We can use a relation that relates sentences to 1 or 0, and thus to neither or both, to get these additional two value situations.]

As we saw in chapter 7 on multi-valued logic, we have reason to think that these two options, 1 and 0, are inadequate by themselves. [There we had a third value, i, which was interpreted either to mean both true and false or neither true nor false. And so the truth valuation function v could assign, 1, 0, or i. The idea now is that to have the other two values (neither and both), it is natural to use a relation that relates formulas to 1 or 0. Such a relation then could hold between a formula and: 1, o, to 1 in one case and to o in another, or there may be no relation between the formula and the values (or rather, maybe we say that it relates the formula to an empty set. See for example Nolt’s Logics section 16.3.15.) I am not sure why this is more natural than simply adding a fourth value in a multivalued logic with an interpretation function. (This possibility is discussed in section 8.4). Perhaps it has certain advantages for calculating the non-classical value when connectives are involved. I am not sure yet how.  Another thing might be how sometimes a sentence that has both values is dealt with just insofar as it is true, or just insofar as it is false (putting aside its other value), and for that purpose it might be more convenient to use a relation where both values are explicitly given individually. Also, maybe it has something to do with the fact that functions cannot have more than one output, so if we wanted both true and false given individually, perhaps using relations is better. I am guessing.]

As we saw in the last chapter, there are reasons to doubt this assumption. If one does, it is natural to formulate an interpretation, not as a function, but as a relation between formulas and truth values. Thus, a formula may relate to 1; it may relate to 0; it may relate to both; or it may relate to neither. This is the main idea behind the following semantics for FDE.

(142)

[contents]

8.2.4

[False is Not Untrue in FDE]

[In FDE, being false (relating to 0) does not automatically mean being untrue, because it can still be related to 1 along with 0.]

[In classical logic, which is bivalent, not being true automatically makes something false, and vice versa. But now, suppose under our new assumptions the formula is related to 0. Does that mean that it is untrue, which would be so if it did not relate to 1? Not necessarily. Some sentences can relate to both. So being 0 does not mean not relating to 1. In other words, in FDE, true does not mean unfalse, and false does not mean untrue.]

Note that it is now very important to distinguish between being false in an interpretation and not being true in it. (There is, of course, no difference in the classical case.) The fact that a formula is false (relates to 0) does not mean that it is untrue (it may also relate to 1). And the fact that it is untrue (does not relate to 1) does not mean that it is false (it may not relate to 0 either).

(143)

[contents]

8.2.5

[The Truth-Evaluating ρ Relation]

[In FDE, a (truth-value) interpretation is a relation ρ between propositional parameters and the values 1 and 0. We write 1 for p relates to 1, and 0 for p relates to 0.]

We call our truth-evaluating relation ρ. It is a relation holding “between propositional parameters and the values 1 and 0” (143). Priest then writes:

In mathematical notation, ρP ×{1, 0}, where P is the set of propositional parameters.

(143)

[At this point, let us bring some more clarity on the topic of relations. We will start with David Agler’s Symbolic Logic section 6.2. I here quote from my “Brief summary” of that section:

The purpose of the language of predicate logic is to express logical relations holding within propositions. There is the simple relation of predication to a subject, which would be a one-place predicate. There are also the relations of items within a predicate, as in “... is taller than ...”, which in this case is a two-place predicate, and so on. To say John is tall we might write Tj, and to write John is taller than Frank we could write Tjf. The number individuals that some predicate requires to make a proposition is called its adicity. And  when all the names have been removed from a predicational sentence, what remains is called an unsaturated predicate or a rheme. We might also formulate those above propositions using variables rather than constants, as in Tx and Txy. When dealing with variables, the domain of discourse D is the set of items that can be substituted for the variables in question, and this possible substitutions are called substitution instances for variables or just substitution instances. The domain is restricted if it contains only certain things and it is unrestricted if it includes all things. We may either explicitly stipulate what the domain is, which is common in formal logic, or the context of a discussion might implicitly determine the domain, and this domain can fluidly change as the discussion progresses.

(from the “Brief summary” at “Agler (6.2) Symbolic Logic: Syntax, Semantics, and Proof, "The Language of RL", summary”)

Let us next turn to Suppes’ Introduction to Logic. To get to the notion of relation, we first need the concept of ordered couples. This is from the “Brief summary” to section 10.1:

An ordered couple is two objects given in a fixed order. We list the items in a series, separated by commas and placed between angle brackets, for example: ⟨x, y⟩. [...] We can have ordered triples, quadruples, and so on. Generally speaking, we can have any-numbered ordered n-tuples. We define them all on the basis of ordered couples. An ordered triple, for example, would be:

x, y, z⟩ = ⟨⟨x, y⟩, z

and an ordered n-tuple:

x1, x2, ..., xn⟩ = ⟨⟨x1, x2, ..., xn-1⟩, xn

(from the “Brief summary” at “Suppes (10.1) Introduction to Logic, “Ordered Couples”, summary”)

Now here is the brief summary for section 10.2 on the definition of relations:

The order of the terms in a relation is often important. So we cannot just think of a predicate taking more than one term as relating members of some set. For, that set needs to have a fixed order. For example, if the relation is love, then it matters who loves whom, as the feeling is not always mutual. We designate such ordered n-tuples with angle brackets: ⟨x, y⟩. The set that can be substituted for the first term is the domain, for the second term, the counterdomain, and the union of both the domain and the counterdomain is the field. (The following quotes Suppes:)

(I) A binary relation is a set of ordered couples.

According to this definition the relation of loving is the set of ordered couples ⟨x, y⟩ such that x loves y. The relation of being less than is the set of all ordered couples ⟨x, y⟩ of numbers such that, for some positive number z,

x + z = y.

The obvious extension of (I) is that a relation which holds among three things is a set of ordered triples, and a relation which holds among n things is a set of ordered n-tuples.

A relation is called ‘n-ary’ if its members are n-tuples. For the special cases n = 2 and n = 3 we use special names, speaking of ‘binary’ and ‘ternary’ relations.

Since a relation is a set of ordered n-tuples, we can also use the “∈” notation to indicate that certain things stand in a given relation. Thus we can write:

⟨John, Mary⟩ ∈ L, instead of:

John L Mary

to indicate that John loves Mary. Similarly we can write:

⟨George, Mary, Elizabeth⟩ ∈ P,

P(George, Mary, Elizabeth)

to indicate, let us say, that George and Mary are the parents of Elizabeth.

It is necessary to remember that an ordered couple is not a relation, but the set consisting of the ordered couple is. For instance,

⟨Thomas Aquinas, 4⟩ is not a relation;

{⟨Thomas Aquinas, 4⟩} is a relation;

{{⟨Thomas Aquinas, 4⟩}} is not a relation.

The last example of the three is not a relation because the only member of the set is itself a set, which is not an ordered couple.

(quoting Suppes 211)

If R is a binary relation, then the domain of R – in symbols: D(R) – is the set of all things x such that, for some y, ⟨x, y⟩ ∈ R. Thus if M is the | relation which consists of all couples ⟨x, y⟩ such that x is the mother of y, then the domain of M is the set of all women who are not childless. If

R1 = {⟨Λ, Plato⟩, ⟨Jane Austen, 101⟩, ⟨the youngest bride in Tibet, Richelieu⟩},

then

D(R1) = {Λ, Jane Austen, the youngest bride in Tibet}.

The counterdomain (or converse domain) of a binary relation R (in symbols : C(R)) is the set of all things y such that, for some x, ⟨x, y⟩ ∈ R. The counterdomain of the relation M considered just above is the set of all people – since everyone has a mother. If B is the relation which consists of all couples ⟨x, y⟩ such that x is the brother of y, then the domain of B is the set of all men who have at least one brother or sister, and the counterdomain is the set of all people who have at least one brother. We have for the relation R1 defined above:

C(R1) = {Plato, 101, Richelieu}.

The field of a binary relation R (in symbols: F(R)) is the union of its domain and its counterdomain. Thus z belongs to the field of a binary relation R if and only if either ⟨x, z⟩ ∈ R for some x or ⟨z, y⟩ ∈ R for some y. The field of the relation B considered just above is the set of all people who belong to families containing at least two children, at least one of which is male. As another example,

F(R1) = {Λ, Jane Austen, the youngest bride in Tibet, Plato, 101, Richelieu}.

(quoting Suppes 212, and also “Brief summary” at “Suppes (10.1) Introduction to Logic, “Ordered Couples”, summary”)

And finally, recall the notion of Cartesian product, first from Suppes Introduction to Logic section 10.1 (quoting):

It is often useful to consider the set of all ordered couples which can be formed from two sets in a fixed order. The Cartesian (or cross) product of | two sets A and B (in symbols: A × B) is the set of all ordered couples ⟨x, y⟩ such that xA and y B. For example, if

A = {1, 2}

B = {Gandhi, Nehru}

then

A × B = {⟨1, Gandhi⟩, ⟨1, Nehru⟩, ⟨2, Gandhi⟩, ⟨2, Nehru⟩}.

(Suppes, Introduction to Logic, 209-210)

And this is from Priest’s Introduction to Non-Classical Logic (our current text) section 0.1.10.

Given n sets X1, . . . , Xn, their cartesian product, X1×· · ·×Xn, is the set of all n-tuples, the first member of which is in X1, the second of which is in X2, etc. Thus, ⟨x1, . . . , xn⟩ ∈ X1×· · ·×Xn if and only if x1 X1 and . . . and xn Xn. A relation, R, between X1×· · ·×Xn is any subset of X1×· · ·×Xn. | ⟨x1, . . . , xn⟩ ∈ R is usually written as Rx1 . . . xn.

(Priest xxviii-xxix)

(And recall also the explanation in 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)

There is something I would like to understand about this, so let us assume we have two propositions in P, namely, p and q. So I would think then that:

P ×{1, 0}

p, 1⟩, ⟨p, 0⟩, ⟨q, 1⟩, ⟨q, 0⟩

So in other words, we would not have something like:

p, 1⟩, ⟨p, 0⟩, ⟨p, {1, 0}⟩, ⟨p, ∅⟩,

q, 1⟩, ⟨q, 0⟩, ⟨q, {1, 0}⟩, ⟨q, ∅⟩

(I am saying this, because it would not seem to be a possible sort of outcome from the specified mathematical procedure. In other words, if something has two values, it is understood as involving two separate relational pairs, as in

p, 1⟩, ⟨p, 0⟩

and thus we would write:

1

0

and not something like

{1, 0}

{∅}

((But maybe something like {∅} is possible, because Suppes in his Introduction to Logic has in on of the examples we saw above:

R1 = {⟨Λ, Plato⟩, ⟨Jane Austen, 101⟩, ⟨the youngest bride in Tibet, Richelieu⟩},

then

D(R1) = {Λ, Jane Austen, the youngest bride in Tibet}.

(Suppes, Introduction to Logic, p.211. See the entry.)

where Λ is the empty set.)) I mention this, because later I will encounter an issue for evaluating sematic consequence. Also, I am not sure if for our 1 and 0 if we say there are two ρ relations, or if there is one, but two instantiations of it, or what.) Now I want to try to give a visual presentation to how the formulation works, using our above example. I would think that we would fill out the formula in the following way:

ρP ×{1, 0}

P ×{1, 0}

P = {p, q}

{p, q} ×{1, 0} = {⟨p, 1⟩, ⟨p, 0⟩, ⟨q, 1⟩, ⟨q, 0⟩}

ρ = p, 1⟩, ⟨p, 0⟩

{⟨p, 1⟩, ⟨p, 0⟩} ⊆ {⟨p, 1⟩, ⟨p, 0⟩, ⟨q, 1⟩, ⟨q, 0⟩}

Here is the quotation:]

An FDE interpretation is a relation, ρ1 between propositional parameters and the values 1 and 0. (In mathematical notation, ρP ×{1, 0}, where P is the set of propositional parameters.) Thus, 1 means that p relates to 1, and 0 means that p relates to 0.

(143. Note, the superscript of ρ1 is not a part of the symbol but is rather a footnote number.)

1. Not to be confused with the reflexive ρ of normal modal logics.

(143)

[contents]

8.2.6

[Evaluation Rules for Connectives]

[For formulas built up with connectives, we use the same criteria as in classical logic to evaluate them, only here we can have formulas related to both values.]

We now will define how we calculate the values for ρ interpretations of formulas built up with connectives. What is interesting to note is that they are the same rules as in classical logic. The only differences is that when a formula relates to 1, it might also in addition to that have a relation to 0.

Given an interpretation, ρ, this is extended to a relation between all formulas and truth values by the recursive clauses:

A1 iff 1 and 1

A0 iff 0 or 0

A1 iff 1 or 1

A0 iff 0 and 0

¬1 iff 0

¬0 iff 1

Note that these are exactly the same as the classical truth conditions, stripped of the assumption that truth and falsity are exclusive and exhaustive. Thus, a conjunction is true (under an interpretation) if both conjuncts are true (under that interpretation); it is false if at least one conjunct is false, etc.

(143)

[contents]

8.2.7

[Example of Value Calculation]

[We calculate the values of complex formulas (built up with connectives) recursively starting from the  base formula value assignments.]

[Priest will show how these evaluations work. Suppose we have just the following relations:

1

0

1

0

And we have the formula

¬p ∧ (q r)

We want to know what is its truth value? That is to say, to what value(s) does ρ relate it to?

¬p ∧ (q r)ρ???

Let us begin with (q r). q is simply true, and r is simply false. That means the whole disjunction is simply true.

A1 iff 1 or 1

A0 iff 0 and 0

(q r)

1

0

(q r)ρ1

Next we have ¬p. p relates to both true and false. So insofar as p relates to true, ¬p is false. And insofar as p relates to false, ¬p is true. Thus ¬p relates both to true and false.

¬1 iff 0

¬0 iff 1

1

0

¬1

¬0

Evaluating the last part is tricky. Perhaps the best way to articulate this is to say: Insofar as ¬p is related to 1, ¬p ∧ (q r) is related to 1. And insofar as ¬p is related to 0, ¬p ∧ (q r) is related to 0.

A1 iff 1 and 1

A0 iff 0 or 0

¬p ∧ (q r)

¬1

¬0

(q r)ρ1

¬p ∧ (q r)ρ1

¬p ∧ (q r)ρ0

Here is the whole quotation:

As an example of how these conditions work, consider the formula ¬p ∧ (q r). Suppose that 1, 0, 1 and 0, 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, qr 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.

(143)

[contents]

8.2.8

[Semantic Consequence. Logical Truth or Tautology.]

[In FDE, semantic consequence is defined as: “Σ ⊨ A iff for every interpretation, ρ, if 1 for all B ∈ Σ then 1”; and logical truth or tautology is defined as “⊨ A iff φA, i.e., for all ρ, 1”]

[Recall from section 1.1.6 the notion of semantic validity:

It is also standard to define two notions of validity. The first is semantic. A valid inference is one that preserves truth, in a certain sense. Specifically, every interpretation (that is, crudely, a way of assigning truth values) that makes all the premises true makes the conclusion true. We use the metalinguistic symbol ‘⊨’ for this. What distinguishes different logics is the different notions of interpretation they employ.

(3)

And from section 1.3.3 the notion of semantic consequence:

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.

(5)

And the notion of logical truth or tautology from section 1.3.4:

A is a logical truth (tautology) (⊨ A) iff it is a semantic consequence of the empty set of premises (φA), that is, every interpretation makes A true.

(5)

In FDE, semantic consequence is a matter of truth preservation. [Of course there is added complexity, and I may not understand it well. It is

Σ ⊨ A iff for every interpretation, ρ, if 1 for all B ∈ Σ then 1

What I am not sure about is if a conclusion is a semantic consequence of certain premises so long as every premise is at least true (and thus may also be false in addition). And a formula is a logical truth or tautology if all interpretations make it at least true. I am not sure how this works. Suppose we have:

p ¬p

In classical logic we have two possible interpretations, that p is 1 or 0. If p is 1, then the disjunction is 1, because then at least one conjunct is 1. If p is 0, then ¬p  is 1, and thus the disjunction is 1, for the same reason. Maybe the idea for FDE is the following, but this is a guess. Suppose we want to know if p ¬p  is a tautology. Maybe next we need to evaluate it for all possible ρ relation configurations.

A1 iff 1 or 1

A0 iff 0 and 0

¬1 iff 0

¬0 iff 1

Configuration 1:

1

(p ¬p)ρ1

Configuration 2:

pρ0

(p ¬p)ρ1

Configuration 3:

1

pρ0

(p ¬p)ρ1

For the neither value, I do not know what to do with the connectives. So suppose there is no relation of p to a truth value, or if the relation relates it to an empty set.

Configuration 4:

[No listed formula]

or

Then how do we evaluate the negation?

¬pρ???

The first rule of negation says:

¬1 iff 0

But p is not related to 0, so ¬p is not related to 1. The second rule

¬0 iff 1

But p is not related to 1, so ¬p is not related to o. Thus

[No listed formula]

or

¬

Now for the disjunction:

(p ¬p )ρ???

The first rule is:

A1 iff 1 or 1

But neither p nor ¬p is related to 1, so (p ¬p ) is not related to 1. And

A0 iff 0 and 0

But neither (and thus not both) p nor ¬p is related to 0, so (p ¬p ) is not related to 0. So as far as I can tell,

Configuration 4:

[No listed formulas]

or

(p ¬p)ρ

This is the rule for tautology:

A iff φA, i.e., for all ρ, 1

So as far as I can tell, (p ¬p) in FDE is not a tautology. But I do not understand these evaluations yet. And I would think that under the reasoning I just gave, nothing could be a tautology, because any formula could possibly be given a neither value. Thus in my interpretation of “for all ρ”, we do not regard what I was writing as ρ∅ as counting for this criteria, that is to say, we should not count the neither value as constituting an instance of ρ. Thus maybe:

A iff φA, i.e., for all ρ, 1

(p ¬p)

Configuration 1:

1

(p ¬p )ρ1

Configuration 2:

pρ0

(p ¬p )ρ1

Configuration 3:

1

pρ0

(p ¬p )ρ1

[No configuration 4, because:]

[no ρ relation for p, thus]

[no ρ relation for (p ¬p )]

FDE (p ¬p) ???

But this probably does not work either, because in FDE, would not p ¬p be invalid? I will need to think more about tautologies in a system where you can have neither value. Here is the quotation:]

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

Σ ⊨ A iff for every interpretation, ρ, if 1 for all B ∈ Σ then 1

and:

A iff φA, i.e., for all ρ, 1

(144)

[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

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

.