Philosophy of
Logics
Week 13: Intuitionist &
Relevant Logics
Liar Paradox Again
Remind yourself of this sentence: (c) c is false.
Claim: If c or not-c, then c and not-c.
Proof: Suppose that c or not-c. If c, then c is false, and so c and not-c. If not-c, then c is not false, and so c and not-c. Therefore, in any case, c and not-c.
LEM or LNC?
Corollary: One of LEM and LNC should be abandoned.
LEM (Law of Excluded Middle): For any proposition p, p or not-p.
LNC (Law of Non-Contradiction): For any proposition p, it is not the case that p and not-p.
What is Intuitionist Logic?
Intuitionistic logic is not just a formal theory.
Historically, it has been deeply motivated by the idealist tradition of the western philosophy
(Kant, Hegel, Fichte, Bradley, etc.). This
tradition has manifested itself in the form of
anti-realism in metaphysics, and intuitionism in phil. of math.
Metaphysics: Realism vs. Anti-Realism Phi. of Math: Platonism vs. Intuitionism
Anti-Realism: Definition
Do you remember this definition?
(Anti-Realism) Everything that determines the truth is somehow dependent on mind or lang.
as opposed to:
(Realism) There exists reality independent from mind and language, and this reality determines the truth, but in general, not vice-versa.
Anti-Realism: Truth Theories
There exists a connection between the theories of truth and realism vs. anti-realism debate:
● Realists often accept the correspondence
theory, according to which truth is determined by the match between a prop. and a fact, not by any epistemological capability.
● Anti-realists usually adopt a criterial theory of truth (coherence, pragmatic, or redundancy th.), according to which p is true (false) only if it is verified (refuted) by a criterion of truth.
Anti-Realism: Motivation
Why has anti-realism become popular in the first place? Descartes's skeptical scenarios suggest that, if there is a gap between mind and reality, then it is perhaps impossible to know anything.
By contrast, if the reality is nothing but, in some sense, the creation of a human mind, and so
there is no gap between mind and reality, then perhaps we can know something. So
Anti-Realism: VP => not-LEM
Many antirealists are committed to this principle: (Verification Principle) (a) For any prop. p, p is true only if we can verify p's truth. (b) For any prop. p, p is false only if we can refute p. VP will entail the denial of LEM if some p is
neither verifiable nor refutable. Is there such p? What are the means of verification (refutation)?
p is a posteriori => Experience. p is a priori => Proof.
Intuitionist Logic
So, when applied to logic, VP takes this form: (a) p is true only if p is provable.
(b) p is false only if p is refutable (by a proof).
So if p is neither provable nor refutable, then p is neither true nor false.
Intuitionistic logic, which rejects LEM, was inspired by VP (<=the antirealist tradition of Kant, Hegel, etc. <= Descartes skeptical
scenarios.) Originally, it was suggested and developed by Heyting in the early 20th C.
Intuitionistic Logic (continued)
I cannot discuss all the details of IL here.
However, the crucial idea is that either (i) we analyze truth and falsity as follows:
1. p is true IFF p is provable, and 2. p is false IFF p is refutable.
or (ii) simply replace the notions of truth and falsity with those of provability and refutability. For our purpose, (i) and (ii) are not so different. We take approach (ii).
Bochvar's 3-value logic.
In Bochvar's 3-value logic, proposition p has one of three truth-values: true, false, neither. Each value can be expressed in that logic:
p is true => p. p is false => ~p.
p is neither true nor false => ~(Tp v T~p).
% To know what "T" means, see p. 207 of the book (English edition).
Intuitionistic logic.
In contrast, there is no means in IL to say that p is neither provable nor refutable:
p is provable=> p. p is refutable => ~p.
p is neither provable nor refutable => ?
Can we write it as "~(p v ~p)"? No. In IL, to say that (i) ~(pv~p) is not to say that (ii) p is neither provable nor refutable, but to say that (iii) it is refutable that p is either provable or refutable. (iii) is stronger than (ii)!
How IL avoids the Revenge Problem
So IL has, in some sense, a restricted power of expression. We cannot express the proposition that p is neither provable nor refutable
(here-after: neither). But actually, this can be a merit: (d) d is refutable-or-neither.
d will be provable or refutable-or-neither. If d is
provable, d is provable and refutable-or-neither. If d s refutable-or-neither, d is provable and
refutable-or-neither. Paradox! "Fortunately," d is not expressible in IL.
Axioms of IL
The following wffs are Heyting's axioms for prop. IL: ● A → (B → A). ● (A → B) → ((A → (B → C)) → (A → C)). ● A → (B → A & B). ● A & B → A. ● A & B → B. ● A → A ∨ B. ● B → A ∨ B. ● (A → C) → ((B → C) → (A ∨ B → C)). ● (A → B) → ((A → ¬B) → ¬A). ● ¬A → (A → B).
How did I learn to love a
contradiction :-)
Alternatively, we may just accept the fact that c and not-c. Here is a defense of this choice:
"For any sentence s, we usually think that c cannot be true and false. That's why we feel
like a solution is necessary for the paradox. But perhaps this is a small, unexpected exception to LNC. As we saw, c can't be only true, c can't be only false. If so, is it neither? That sounds as much paradoxical as saying that it is both. Then why do we not learn to accept that there is an exception to LNC?"
An Obstacle :-(
The problem is that anything follows from a contradiction: 1 (1) P&~P A 1 (2) P 1 &E 1 (3) PvQ 2 vI 1 (4) ~P 1 &E 1 (5) Q 3,4 DS
Thus, P&~P |- Q. (Medival logicians were aware of this, and called it "Ex Falso
An Obstacle (continued)
It is easy to see the similarity between the EFQ problem and the tonk problem:
1. If tonk introduction and tonk elimination are adopted, any Q can be proved from any P. 2. If a contradiction P&~P is logically true, then
any Q follows from it.
In both cases, the main problem is not just that your logic is inconsistent. Rather, it is that your logic becomes totally useless; for, every
Paraconsistent and Relevant Logics
Paraconsistent and relevant logics are
differently defined, and motivated by different problems.
1. PL: A logic which allows the possiblity of a true contradiction.
2. RL: A logic which disallows the derivation from A1,...,An to C whenever A1,...,An are
irrelevant to C.
How do we define irrelevance? EFQ offers a good exampl: There is NO common sentential letter shared by P&~P and Q.
PL and RL (continued)
So, while RL and PL are motivated by different considerations, RL can be, and often are, used to solve the EFQ problem of PL.
For this reason, while not all relevant logics are paraconsistent, the converse usually holds: many systems of paraconsistent logic adopt the rules and axioms of relevant logics to prevent everything from being derived by EFQ.
Due to limited time, we will focus on a ver. of relevant & paraconsistent logic, RPL. (Mostly due to Stephen Read's Relevant Logic.)
How is EFQ Blocked?
Think about the proof of EFQ again:
1 (1) P&~P A
1 (2) P 1 &E
1 (3) PvQ 2 vI
1 (4) ~P 1 &E
1 (5) Q 3,4 DS
Look at the derivation of 5 from 3 and 4. PvQ (3) means that one of P and Q is true. Since ~P (4), the reasoning goes, what is true cannot be
How is EFQ blocked? (continued)
Of course, it was an error to think "Since ~P
(4), ...what is true cannot be P, but must be Q." For, if P is both true and false, then PvQ will be true regardless of the truth-value of Q. Look at this:
P ~P Q PvQ
Both True False True
Hence, we cannot derive Q from ~P and PvQ; for there is a case in which the premises are
true but the conclusion is false. This means that Disjunctive Syllogism is invalid in RPL.
Obstacle again :-(
But this raises a different worry: Can we derive Q from P->Q and P? Think about this case:
P Q ~Q P&~Q P->Q (=~(P&~Q)) Both False True Both True
So you cannot derive Q from P and P->Q,
because there is a case in which both premises are true but the conclusion is not true.
We may give up Disjunctive Syllogism, but
Truth-functional Connectives
How do we solve these problem? First, note that, in classical logic, we have connectives "&," "v," and "->," as defined as follows:
● A&B=df. ~(A->~B) ● AvB=df (~A)->B
● (A->B) is true IFF A is false or B is true.
Note that these arguments are INVALID in RPL ● AvB,~A//B
● A->B,A//B ● ~A//~(A&B)
Intensional Connectives
In addition to those truth-functional connectives, we introduce these new connectives:
● A × B=df~(A⇒~B) ● A + B=df(~A)⇒B ● A ⇒ B=df?
The following arguments are VALID in RPL: ● A+B,~A//B
● A⇒B,A//B ● ~A//~(A×B)
Relevant Implication
But how do we understand "=>"? Its meaning can be PARTIALLY explicated in this manner:
(A1⇒(A2⇒...(An ⇒C)...)) is true IF (i) one can merge A1,A2,...,An into a consistent bunch of information B (I-bunch), and (ii) one can
derive C from B, using all parts of B
(derivational utility). (See Read's Relevant L) NOTE: This is not meant to be a full definition of the truth-condition of relevant conditionals. Not even close. (Can you tell me why?)
Relevant Implicaton (continued)
Example 1: In RPL, P⇒(~P⇒Q) is not
necessarily true, because you cannot merge P and ~P into a consistent bunch of information. Example 2: In RPL, P⇒(Q⇒P) is not
necessarily true, because, although P and Q can be merged into a consisent bunch of
information B, one does not derive P from B using all parts of B.
Since the antecedents are intuitively irrelevant to the consequent in both examples, this is a desirable result.
Axioms of R (standard)
1.A → A Identity
2.(A → B) → ((B → C) → (A → C)) Suffixing
3.A → ((A → B) → B) Assertion
4.(A → (A → B)) → (A → B) Contraction 5.(A & B) → A,(A & B) → B &E
6.A → (A∨B), B → (A∨B) vI
7.((A → B) & (A → C)) → (A → (B & C)) &I 8.((A∨B) → C)↔((A → C) & (B → C)) vE 9.(A & (B∨C)) → ((A & B)∨(A & C)) Dist.
10.(A → ~B) → (B → ~A) Contraposition
11.~~A → A Double Neg.
Summary
● The liar paradox indicates that we have to give up LEM or LNC.
● IL is a logic in which LEM is not a theorem. ● Historically, IL was inspired by antirealism,
which, in turn, was a response to skepticism. ● Differing from Bochvar's 3 valued logic, the
revenge problem does not occur in IL.
● PL is a logic in which LNC is not a theorem. ● The problem is, if there is but one true
contradiction, how to prevent it from "spreading."
Summary
● RL is often adopted as a solution to this problem.
● RPL has two types of connectives, one type truth-functional and the other intensional.
● The standard rules of inference do not work with the truth-functional connectives.
● Fortunately, most such rules do work with the intensional connectives.
● Relevant implication is defined in terms of "consistent merger" and "depending on all parts."
