**Philosophy of **

**Logics**

### Week 3: Paracomplete and

### Paraconsistent Logics

### Introduction

In this week, we explore two variations on the classical logic of our basic connectives. They are non-classical rivals of the

classical logic; they agree with the classical logic on what the basic connectives are, but it disagrees on how they work.

We will begin the discussion with some philosophical

motivation (unsettledness) for the first system, and then turn to an informal characterization of it. After doing so, we will repeat the same procedure for the second system of logic.

### Paracomplete Logic: Unsettledness

Why is it that legal courts stipulate a legal meaning for ‘child’, ‘adult’, and so on? Why not simply use the ordinary meanings of such terms? The answer is not that such terms are

meaningless; rather, such terms are not sufficiently settled over all cases. Because matters of law require---or, at least, strive towards---precision, courts simply stipulate a new word (say, ‘ child*’), one with precise meaning to take the place of our otherwise vague words.

Of course, wise courts try to preserve the original, ordinary meanings as far the meanings go (e.g. , they don’t declare

that a 91 year old man is a legal child); however, they stipulate a new, precise (or more precise) term for legal matters.

### Paracomplete Logic: Unsettledness

### (continued)

The relevant sense of ‘unsettledness’, whatever exactly it

might be, is not an epistemic sense. We aren’t ignorant of the precise meaning of ‘ child’, but rather the ordinary meaning is not fully precise and fully exhaustive. The ordinary meaning of ‘child’ doesn’t exhaustively cover all objects; it fails to

exhaustively divide objects into those of which ‘child’ is true and those of which ‘child’ is false. Instead, the ordinary

meaning seems to leave gaps:

there seem to be some objects such that the predicate ‘child’ neither definitely applies nor definitely fails to apply.

### Sorites Paradox

The ancient sorites puzzle (pronounced ‘so- right-tees’) is

often invoked to highlight the apparent unsettledness of (much of) our language. Consider the following argument.

P1. 1 grain of sand is not a heap (of sand)

P2. If 1 grain of sand is not a heap, 2 grains aren’t a heap. P3. If 2 grains of sand aren’t a heap, 3 grains aren’t, either. ...

c. Therefore, a billion zillion grains of sand is not a heap. The ordinary meaning of ‘heap’ is such that a small difference makes no difference in its application. Adding only one grain of sand to something that is definitely not a heap does not

### How to Solve the Puzzle?

One response to the sorites is that the classical logic gets things wrong. This is not surprising. The classical theory is motivated by the idea of a precise language. If, as the

foregoing suggests, our language has pockets of imprecision ---that is, meaningful but nonetheless unsettled sentences--- then the classical theory is too narrow for an account of

### Incomplete Assignments

What the foregoing considerations motivate is not a rejection of classical assignments; rather, such considerations motivate a broadening of our account of cases. In addition to complete and consistent assignments, considerations of ‘unsettledness’ motivate incomplete assignments, assignments according to which, for some A, neither A nor ~A is true.

We will use the term ‘paracomplete’ for any system of logic that recognizes incomplete assignments---assignments

according to which, for some statement A, neither A nor ~A is true.

### Paracomplete Assignments

*(Paracomplete Assignments) An assignment a is *

*paracomplete if and only if a is complete and consistent *
(i.e., classical), or incomplete and consistent.

Since classical assignments are complete and consistent, any classical assignment counts as a paracomplete assign- ment. On the other hand, some paracomplete assignments

---namely, the incomplete ones---are not classical.

Q1. According to Assignment 1, (p) “Tom is bald” is true and ~p is not true. Is this paracomplete? Incomplete? Classical?

Q2. According to Assignment 2, neither p nor ~p is true. Is this assignment paracomplete? Incomplete? Classical?

Q3. How about Assignment 3, according to which both p and ~p are true? Is it paracomplete? Incomplete? Is it classical?

### Cautions!

*In discussing classical logic, we said that (i) A is false *

*according to a IFF ~A is true according to a, and that (ii) ~A is *
*true according to a IFF A is not true according to a. *

We retain the first classical idea, (i). The trouble, however, is
that untruth and falsity come apart in the paracomplete logical
theory. Our paracomplete theory acknowledges assignments
*that are entirely incomplete with respect to A, that is , some *
*assignment a such that neither A is true nor ~A is true, *

*respectively, according to a. The upshot is that we can no *

longer understand false-according-to-an-assignment simply as not-true-according-to-an-assignment. In other words, we give up (ii), at least for the paracomplete logic.

### Paracomplete Constraint for Atomic

### Statements

To make things clearer, let us adopt a shorter notation to

accommodate the broader, paracomplete theory. In particular,

*we use ‘a⊨*_{1}*A’ to abbreviate ‘A is *

*true-according-to-a' and a⊨*_{0}*A’ to abbreviate ‘A is false-according-to-a'. *

While we still won’t give explicit truth (or falsity!) conditions for atomic statements, we will give the following constraint on

them: (Paracomplete Constraint for Atomic Statements) For
*any atomic statement A and any assignment a, exactly one of *
the following obtains:

*a⊨*_{1}*A and a⊭*_{0}*A.*
*a⊭*_{1}*A and a⊨*_{0}*A.*
*a⊭*_{1}*A and a⊭*_{0}*A.*

### Paracomplete Truth & Falsity Cond. for

### Molecular Sentences

*(Basic Paracomplete Truth and Falsity Conditions) Where a is *
*any assignment, and A and B any statements, the *

paracomplete conditions for the basic connectives are as follows:

*Conjunction: a⊨*_{1}*A&B IFF a⊨*_{1}*A and a⊨*_{1}*B.*

*Conjunction: a⊨*_{0}*A&B IFF a⊨*_{0}*A or a⊨*_{0}*B.*

*Disjunction: a⊨*_{1}*AvB IFF a⊨*_{1}*A or a⊨*_{1}*B.*

*Disjunction: a⊨*_{0}*AvB IFF a⊨*_{0}*A and a⊨*_{0}*B.*

*Negation: a⊨*_{1}*~A IFF a⊨*_{0}*A.*

### Paracomplete Consequence; its

### Relation to Classical Consequence

*A paracompletely entails B IFF there’s no paracomplete *

*assignment according to which A is true but B not true.*

Let us use ‘⊢_{bc}’ for basic classical entailment and ‘⊢_{K3}’ for our
given basic paracomplete entailment relation.* One notable
fact about the relation between the two entailment relations,
namely, the basic classical and basic paracomplete relations
is the following:

*(CP) For any sentences A*_{1}*...A _{n} and B, if A*

_{1}

*...A*⊢

_{n}_{K3}

*B then*

*A*

_{1}

*...A*⊢

_{n}_{bc}

*B. (Why?)*

*The name ‘K3’ is standard for our basic paracomplete consequence relation. The K3 relation (or, generally, K3

logical theory) is so called for the logician Stephen Kleene’s most famous 3-valued logic.

### Paracomplete Consequence; its

### Relation to Classic (continued)

Philosophically, CP suggests that, while our language may have an ‘unsettled’ or ‘gappy’ fragment, it may also enjoy an entirely precise, classical fragment. For example, it might be that the mathematical or scientific fragment of our language is precise and, in effect, even though our broad language---

perhaps due to a bit of vague language or the like--- is ‘unsettled’ or ‘gappy.’

### No Tautology in K3

Another notable feature of the basic paracomplete theory is
that, contrary to the classical theory, there are no tautologies!
*Recall that, in general, sentence A is a tautology IFF A is true *
according to all assignments. On the paracomplete theory, our
assignments are either classical or incomplete but consistent.
As above, the paracomplete theory has a broader range of

assignments than the classical theory. The result of such

### No Tautology in K3 (continued)

Of course, given the philosophical motivation of the

paracomplete theory---for example, unsettledness or gaps--- one would expect that Excluded Middle would fail, and it does. To see why, remember that

*a⊨*_{1}*A and a⊭*_{0}*A;*
*a⊭*_{1}*A and a⊨*_{0}*A; or*
*a⊭*_{1}*A and a⊭*_{0}*A,*

*and check out whether Av~A is true or not in each of the *
above cases.

### No Tautology in K3 (continued)

That there are no logical truths, on the paracomplete theory,

is slightly more diffìcult to see. The basic idea, however, is that
there is a paracomplete assignment according to which every
*atomic statement A is neither true nor false. With suitable *

attention to the truth and falsity conditions for molecular

sentences, one can see that such an assignement will be one according to which no statement---atomic or molecular---is

true or false. Hence, no statement is true according to every assignment and, thus, there are no logical truths.

### Semantic Values

There are various ways of modeling our given (paracomplete) assignments. We will stick to an approach that is similar to the approach taken towards modeling classical assignments. In particular, we will take our assignments to be (modeled by) certain functions ---ones obeying various constraints--- from

our set *S of sentences into our set V of ‘semantic values'. *

What’s different, now , is that our set of ‘semantic values' to be {1, n, 0} (where ‘1’ may be thought to mark the semantic status of being true; ‘0’ to mark the semantic status of being false; ‘n’ to mark the semantic status of being neither true nor false).

### Assignments are Functions!

To be clearer, our semantics begins with a set of semantic

values , namely, *V={1, n, 0}. In turn, we let our assignments be *

*functions v from S into V , so that we have v(A) = 1 or v( A) = n *

or v(A) = 0 for every sentence A and any such assignment v.
*As with the classicallanguage, let us defìne truth as follows: A *
**is true-according-to v if and only “ v(A)=1. Similarly, A is **

### Paracomplete Assignment

*We say that a function v, from _{S into V, is a paracomplete }*

assignment (in the narrow sense) if and only if it ‘obeys’ the following truth conditions:*

* Such an assignment is paracomplete in the narrow sense of
*“conforming to basic paracomplete logic.” However, note that *
there are other forms of paracomplete logic.

~ 0 1 n n 1 0 & 1 n 0 1 1 n 0 n n n 0 0 0 0 0 v 1 n 0 1 1 1 1 n 1 n n 0 1 n 0

### Paracomplete Consequence

*(Basic Paracomplete Consequence) B is a consequence of A *
in our basic paracomplete logic (namely, K3) IFF there is no
*paracomplete assignment v such that v(A)=1 but v(B)≠1.*

*(General Basic Paracomplete Consequence) B is a *

consequence of A_{1}...A_{n} in our basic paracomplete logic (K3)

*IFF there is no paracomplete assignment v such that *

### Defined Connectives

We will use the same defined connectives here, letting ‘A->B’ be shorthand for ~AvB , and similarly letting ‘A<->B’ be

shorthand for (A->B)&(B->A). Their truth-conditions are defined as follows: -> 1 n 0 1 1 n 0 n 1 n n 0 1 1 1 <-> 1 n 0 1 1 n 0 n n n n 0 0 n 1

### Valid Argument Forms in K3

• Modus Ponens: A->B,A⊢_{K3}B

• Modus Tollens: A->B,~B⊢_{K3}~A

• Disjunctive Syllogism: A v B,~A⊢_{K3}B

• Contraposition: A->B⊢_{K3}~B->~A

• Explosion: A,~A⊢_{K3}B

• v-Introduction: A⊢_{K3}A v B
• &-Introduction: A, B⊢_{K3}A&B
• &-Elimination: A & B⊢_{K3}A

• De Morgan: ~(A v B)⊣⊢_{K3}~A&~B

• De Morgan: ~(A & B)⊣⊢_{K3}~A v ~B

• Double Negation Elimination (DNE): ~~A⊢_{K3}A.

### No Tautologies in K3!

• Excluded Middle: ⊬_{K3} A v~A

• Non-Contradiction: ⊬_{K3}~(A&~A)*

* This is an unsatisfactory feature of K3. After all, we wanted to reject (the Law of) Excluded Middle, but not (the Law of) Non-Contradiction.

Note: It is easy to see that LEM and LNC are equivalent under the DN and De Morgan rules. Hence, a logical system which rejects LEM but endorses LNC will have to reject DN or De Morgan rules.

### How about the Sorites Paradox?

Consider the Sorities Paradox again:

P1. 1 grain of sand is not a heap (of sand)

P2. If 1 grain of sand is not a heap, 2 grains aren’t a heap. P3. If 2 grains of sand aren’t a heap, 3 grains aren’t, either. ...

c. Therefore, a billion zillion grains of sand is not a heap.
This argument is valid in K3, but perhaps we just do not want
*to say that each P(k>1) is clearly true. Perhaps, for some k, *
the antecedent manages to be true but the consequent fails to
be so. In that case, the truth-table says that the whole

conditional is neither true nor false. Consequently, the above argument turns out to be valid but not factually correct.

### Paraconsistent Logic:Overdeterminacy

In addition to indeterminacy, some philosophers have thought that our language also exhibits overdeterminacy. A sentence is said to be overdetermined if it is both true and false. Falsity is truth of negation: a sentence A is false IFF its negation ~A is true. So, to say that a sentence is overdetermined is to say that it is true and that its negation is true. Here’s an example:

(L) L is false.

As you may remember, L appears to be both true and false. Why? Either L is true or it is false. If L is true, then L is false and so it is true and false at the same time. If L is false, then L is true and so it is true and false at the same time. In either

### Paraconsistent Logic:Overdeterminacy

### (continued)

Why can’t we simply say that L is another case of

‘indeterminacy’, a ‘gappy’ sentence that is neither true nor false? If we do, then the argument that it is both true

and false breaks down. (Why?) Such a thought is correct, but the problem lingers. Consider, in particular, this sentence:

(D) D is either false or gappy.

Either D is true, false, or gappy. Case 1: D is true. Thus, D is false-or-gappy. (Why?) Case 2: D is false. Then it is false or gappy. Hence, D is true. (Why?) Case 3: D is gappy. Thus, D is false or gappy. (Why?) Hence, D is true. In any case, D is true but false-or-gappy at the same time.

### Paraconsistency

What is worth noting is that in both the classical and paracomplete theories, Explosion holds: A, ~A⊢B.

According to those theories, then, if there’s some assignment according to which both A and ~A are true, then that

assignment is the trivial one. (Why?)

We will use the term ‘paraconsistent’ for any system of

logic that recognizes inconsistent but non-trivial assignments
*---assignments according to which, for some A, both A and ~A *
*are true, but not all A are true.*

Definition. (Paraconsistent) A system of logic is para-

*consistent IFF it recognizes some assignment a such that *

### Paraconsistent Assignment

Of course, a paraconsistent (system of) logic might recognize only inconsistent assignments (at least some of which are

non-trivial), but the motivation for such a logic is not obvious. Similarly, a paraconsistent logic might reject incomplete

assignments, and only recognize classical assignments or complete but inconsistent assignments.

For our purposes, we will take neither of those routes.

Instead, we will discuss a paraconsistent system of logic that retains our previous assignments---both classical and

incomplete but consistent assignments---but further expands the range by acknowledging inconsistent assignments. In this way, our paraconsistent theory is another broadening---from complete and consistent, to incomplete and consistent, to inconsistent.

### Falsity and Untruth

Since we’re retaining incomplete cases, we thereby retain the same ‘breakdown’ between falsity and untruth. As such, we

will employ the same notation, using ‘a⊨_{1}A’ to abbreviate

‘A is true-according-to-a', and ‘a⊨_{0}A’ for ‘A is

### Paraconsistent Constraint for Atomic

### Sentences

(Paraconsistent Constraint for Atomic Sentences) For any
*atomic A and any assignment a, exactly one of the following *

obtains: *a⊨*_{1}*A and a⊭*_{0}*A.*

*a⊭*_{1}*A and a⊨*_{0}*A.*
*a⊭*_{1}*A and a⊭*_{0}*A.*
*a⊨*_{1}*A and a⊨*_{0}*A.*

*For present purposes, we don’t need to know how atomic *

*sentences get to be true-according-to a or false-according-to *

*a; we just need to know that, for any assignment a, every *

*atomic is either true-according-to a, false-according-to a, *
*neither true-according-to a nor false-according-to a, or both *
*true-according-to a and false-according-to a.*

### Paraconsistent Truth & Falsity Cond.

### for Molecular Sentences

*(Basic Paraconsistent Truth and Falsity Conditions) Where a *
*is any assignment, and A and B any statements, the para- *

consistent conditions for the basic connectives are as follows:
*Conjunction: a⊨*_{1}*A&B IFF a⊨*_{1}*A and a⊨*_{1}*B.*

*Conjunction: a⊨*_{0}*A&B IFF a⊨*_{0}*A or a⊨*_{0}*B.*

*Disjunction: a⊨*_{1}*AvB IFF a⊨*_{1}*A or a⊨*_{1}*B.*

*Disjunction: a⊨*_{0}*AvB IFF a⊨*_{0}*A and a⊨*_{0}*B.*

*Negation: a⊨*_{1}*~A IFF a⊨*_{0}*A.*

*Negation: a⊨*_{0}*~A IFF a⊨*_{1}*A.*

Note: These conditions are exactly our basic paracomplete conditions. The difference, of course, is that now some of our assignments are inconsistent. This fact shouldn’t be

*surprising, since, as above, our paraconsistent logic expands *
the range of assignments beyond our previous ones.

### Paraconsistent Consequence

*As you know, B is a consequence of A IFF there’s no *

*assignment according to which A is true but B not true. In the *
present approach, assignments are paraconsistent ones,

which are either classical assignments, or incomplete

### Paraconsistent & Classical

### Consequence

For purposes of comparison, let us use ‘⊢_{bc}’, as before, for

basic classical consequence, and use ‘⊢_{FDE}’ for our given basic

paraconsistent consequence relation.* One notable fact about the relation between the two consequence relations ---the

basic classical and basic paraconsistent relations--- is the result that we previously had:

*CP2. For any sentences A*_{1}*...A _{n} and B, if A*

_{1}

*...A*⊢

_{n}_{FDE}

*B then*

*A*

_{1}

*...A*⊢

_{n}_{bc}

*B.*

Compare this with CP. Why is CP2 true?

*‘FDE’ is the now-fairly-standard name of this particular paraconsistent logic; the name is for what Anderson

and Belnap called a logic of first degree entailment (FDE), or the ‘logic of tautological entailments.’

### Paraconsistent & Classical

### Consequence (continued)

CP2 tells us something important about the relation between our basic classical consequence relation and our broader

paraconsistent consequence relation: namely, that the latter is
*a proper part of the former. In other words, the basic classical *
consequence relation is ‘stronger’ than our basic

paracomplete one: the former has the latter as a proper part. CP2 tells us that any arguments that are valid according to the paraconsistent logic are valid according to the classical logic. As it turns out, the converse doesn’t hold; there are

some arguments that are valid according to the classical logic but not the paracomplete logic. (Can you think of one?) The reason is that the latter theory recognizes more assignments, and hence recognizes more ‘potential counterexamples’.

### Semantics

As before, the key ingredients of logical consequence are

*assignments and truth-according-to-an-assignment (and also *
*falsity-according-to-an-assignment) conditions. Our*

concern is with the basic paraconsistent theory. The question is: how shall we model our paraconsistent assignments?

### Assignments as Functions (again)

We will take our assignments to be (modeled by) certain

functions---ones obeying various constraints---from our set *S *

of sentences into our set *V of ‘semanticvalues’. One *

difference, of course, is that we will expand our set of ‘semantic values' from {1, n, 0} to

{1, b, n, 0},

where b, now, may be thought to mark

the semantic status of being both true and false.

The idea, intuitively, is that the paraconsistent theory retains the previous semantic values; however, it also recognizes more options---namely, ‘overdeterminacy’ or ‘gluts’.

### Designated Values

Our semantics begins with a set of semantic values, namely,
V={1, b, n, 0}. The difference from the basic paracomplete
logic is in the fact that we now need to be explicit about our
set *D of so-called designated values. Intuitively, the *

designated values can be thought of as different ‘ways of being true'. In our previous logical systems of logic, no

sentence could be both true and false; they were only true, if true at all.

In our current, paraconsistent logic, some sentences can be only true, as before, but they can also be true and false. As such, we have expanded our semantic values, and we now

designate both 1 and b. It is the designated values in terms of

which we define truth-according-to-an assignment. Our set *D *

### Truth-according-to an Assignment

With _{V so given, and designated values so specified, we let}

*our assignments be functions v from S into V, so that we have *

*v(A) = 1 or v(A) = b or v(A) = n or v(A) = 0, for every sentence *
*A and any assignment v. (Note that ‘v’ is just a functional *

notation for an assignment.)

*We define truth-according-to-an-assignment as follows: A *

*is true-according-to v IFF v(A)∈D, i.e., v(A) = 1 or v(A) = b.*

*What about falsity-according-to-an-assignment? Notice that *
*we are now also considering the idea that A may be true and *
false. Any sentence that is true and false is false. Our

definition of falsity according to an assignment, then, is as
*follows: A is false-according-to v IFF v(A) = 0 or v(A) = b.*

### Paraconsistent Assignment

*We say that a function v, from _{S into V, is a paraconsistent }*

assignment (in the narrow sense) IFF it ‘obeys’ the following truth conditions:*

* This is so only in the “narrow” sense, because there are other types of assignment discussed as a part of alternative systems of paraconsistent logic.

~ 0 1 b b n n 1 0 & 1 b n 0 1 1 b n 0 b b b 0 0 n n 0 n 0 0 0 0 0 0 v 1 b n 0 1 1 1 1 1 b 1 b 1 b n 1 1 n n 0 1 b n 0

### Paraconsistent Consequence

(Basic Paraconsistent Consequence) B is a logical

*consequence of A in our basic paraconsistent logic (i.e. FDE) *
*IFF the're is no paraconsistent assignment v such that v(A)*

*∈D but v(B)∉D. *

*(General Basic Paraconsistent Consequence) B is a *

consequence of A_{1}...A_{n} in our basic paraconsistent logic

*(namely, FDE) IFF there is no paraconsistent assignment v *
*such that v(A*_{1}*),...,v(A*_{n})∈_{D but v(B)∉D.}

### Defined Connectives

We will use the same defined connectives here, letting ‘A->B’ be shorthand for ~AvB , and similarly letting ‘A<->B’ be

shorthand for (A->B)&(B->A). However, their truth-conditions are defined in a suitably modified way:

-> 1 b n 0 1 1 b n 0 b 1 b 1 b n 1 1 n n 0 1 1 1 1 <-> 1 b n 0 1 1 b n 0 b b b 1 b n n 1 n n 0 0 b n 1

### Valid Argument Forms in FDE

• Modus Ponens: A->B,A⊬_{FDE}B *

• Modus Tollens: A->B,~B⊬_{FDE}~A

• Disjunctive Syllogism: A v B,~A⊬_{FDE}B **

• Contraposition: A->B⊣⊢_{FDE}~B->~A

• Explosion: A,~A⊬_{FDE}B

• v-Introduction: A⊢_{FDE}A v B
• &-Introduction: A, B⊢_{FDE}A&B
• &-Elimination: A & B⊢_{FDE}A

• De Morgan: ~(A v B)⊣⊢_{FDE}~A&~B

• De Morgan: ~(A & B)⊣⊢_{FDE}~A v ~B

• Double Negation Elimination (DNE): ~~A⊢_{FDE}A.

• Double Negation Introduction (DNI): A⊢_{FDE}~~A

*This is worrisome, becaue MP seems to be the most basic pattern of human reasoning. ** The failure of DS is important in explaining why Explosion does not hold.

### No Tautologies in FDE!

• Excluded Middle: ⊬_{FDE} A v~A

### Summary

● We studied two non-standard systems of logic, K3 (a.k.a. Strong Kleene) and FDE (a.k.a. First Degree Entailment). ● K3 recognizes paracomplete (incomplete and consistent)

assignments as well as classical (complete and consistent) assignments.

● As a result, there are no tautologies in K3, but most of valid argument in classical logic are also valid in K3.

● FDE recongnizes inconsistent assignments as well as paracomplete and classical assignments.

● Consequently, some of the most familiar arguments forms are not valid any longer in FDE.

● Note that K3 is not the only form of paracomplete logic; similarly, FDE is not the only form of paraconsistent logic.