The Possibility of Alternative Logic

Kinds of Alternative Logics

Deviant Logics versus extensions. Enhancements to Classical logic opposed to rivals.

What is a Deviant or Extension? Let L_A be an alternative logic to L_B. 

Extensions

If L_A has at least the same theorems as L_B then it is an extension. Predicate Calc extends Propositional. Modal extends non-modal. 

Rivals

If L_B has different theorems from L_A then it is a rival. S5 is a rival to K, Free logic is a rival to non-free, Paraconsistent is rival to classical.

(This isn't exactly right because there are logics which are plainly rivals but are not deviant in that sense. e.g supervaluation on formulas whose parts have unassigned t.v. will yield all the same truths as classical logic. Never mind.)

The Possibility of Alternatives

Don't 'rival' logics just indicate that the meanings of the logical words have changed? So they aren't really rivals at all?

Inevitability of Logical Translation

Consider the anthropologist's collection of statements with logical terms. Quine says that we can never be justified in translating these as anything other than classical logical terms. Any logical deviancy must be considered a bad translation. 

Alterations will be restricted to the translations of the non-logical word, because we are more certain about logic than about any of the evidence that could be used to challenge logic. (C.f. Moore's argument against skepticism: observing a hand is more certain than the rational steps to skepticism about what observing the hand tells us.)

This argument only tells us that we will translate into our own logic. If we are using classical we'll translate that way; if not, not.

Theory Dependence of Meaning

Are logical terms defined in their meaning by the theorems they support and truth valuations they take? If so then talking about deviant logics is not talking about the same thing as classical logics. 

Compare to various ideas about planets. Do we and Greeks talk about the same thing when we talk about planets? They meant lights on crystal spheres centred on Earth. We don't. (Feyerabend and Kuhn are famous proponents of the idea that because theory determines meaning rival scientific theories are 'incommensurable.') 

1. Tonk

It may simply not be the case that meanings can be determined by theory. Consider Prior's 'tonk' connective defined as 

A |- A tonk B 

A tonk B |- B

What does tonk 'mean' here?

But there are other formal reasons to think that tonk is badly defined.

2. Core Meaning

There is a core of meaning that allows common 'meaning' (whatever 'meaning' is.)

Our core of meaning for science terms is observational. Is there a logical analogue to that? If not then maybe logic is completely theory-dependent in a way science isn't.

Putnam: there is a core of meaning - Law of Excluded Middle is non-core. 

the logical words 'or' and 'not' have a certain core meaning which is ... independent of the principle of excluded middle. Thus in a certain sense the meaning does not change if we go over to a three-valued logic or to Intuitionist logic. Of course, if by saying that a change in the accepted logical principles is tantamount to a change in the meaning of the logical connectives, what one has in mind is the fact that changing the accepted logical principles will affect the global use of the logical connectives, then the thesis is tautological and hardly arguable. But if the claim is that a change in the accepted logical principles would amount merely to redefining the logical connectives, then in the case of Intuitionist logic, this is demonstrably false.

Is it possible to modify meanings of logical terms? How would you argue that it isn't?

Anyway, even if these rivals involve meaning changes, there may be good reasons for it.

Justifying Alternatives in General

Against Alternatives

Kant thought logic was completed science. He thought the same of Geometry and Physics. It might be that those are bad analogies and that logic is a different kind of thing from those scientific theories. K says logic is the laws of thought. What does he mean?

1.   Positive? (Descriptive) 

      In that case how is error possible? He says only through corruption of the mind by senses. 

      Really? How do the senses cause formal fallacies?

2.   Normative? (Prescriptive)

      What is normativity? How do we identify them? Presumably through study of the way things work.

      In that case our discovery of 'normative' laws is the same process as the discovery of positive laws. Could be wrong.

Frege thought logic was self-evident. He thought his own logical principles were self-evident and they turned out to be contradictory (and explosive, too - so badly contradictory.)

People disagree over which are self-evident. Certainty is thus subjective, and cannot be a useful criterion for distinguishing the good logic from all others.

For Alternatives

Logics are for understanding good arguments. 

Justifying extensions is the easiest.

Modal logics useful for identifying the formal patterns in good modal arguments.

Note that there are assumptions made about the metaphysical nature of necessity. If logic is 'topic-neutral' then this might be a problem. Alternative modal logics correspond to the alternative intuitions about modal nature, or what arguments hold.

Can be used for shorthand for other notions - temporal, epistemic, etc.

Rivals are harder. 

Quine admits the possibility of rivals - we have a web of beliefs, perceptions at the edge, connections to other concepts of increasing generality and abstraction, and logic at centre. Every element of the web is modifiable - but the closer to the centre the greater the ramifications of a change, and the higher the evidence required to justify it. Belief that I see a snake is easier to modify than belief that there are snakes. C.f. also Aristotle -> Newton -> Einstein. 

Practically, he thinks, we will never be justified in modifying classical logic.

Classical logic derived from arguments in macro-world in which A v ~A holds. But doesn't hold in micro-world. So Quantum logic removes it as a theorem - i.e. it isn't true in all cases. Intuitionistic logic.

What are criteria for deciding to modify logic? Same as criteria for modifying any other theory. Simplicity, economy (minimise belief change,) generality (inc. explanatory power.)

Note that no theory can be either verified or falsified absolutely. Consider latter. Suppose you have an inconsistent theory. How do you fix it? There are still those who think Earth is flat: ships disappear over horizon because of light bending, light bends (down-up) because of ethereal nature, etc... Compare to Continental Drift. Galileo's observations through a telescope (Feyerabend, Against Method.)

Why prefer simplicity, economy, and generality? (Aesthetic? Pragmatic?)

How are we to judge simplicity, economy, and generality? (As above.)

Note. This argument assumes that inconsistencies are bad, and need to be fixed. So it appeals to a logic that tells us that. Can an argument justify a logic that doesn't have all the necessary parts for the argument to go through? Don't we need the logic to tell us what theoretical parts need to be altered.

Yes. We have to rebuild the boat while we are sailing it. (Neurath)

Justifying Alternatives in Particular

Many-valued and the Liar or sorites

Paraconsistency and the Liar.

Can there be justifications for more than one deviant logic at the same time? 

Which shall we choose? Is there no such thing as a single true logic? C.f. Quine's indeterminacy of meaning argument.

Or do we just have to accept no one true universal logic, but there are true logics in specific fields of discourse.

The latter is ok for extensions - talk about time in a temporal logic, talk about quantified staetments using predicate calc. Rival logics suggest a universal replacement (3-valued, paraconsistent, ...,) but some rivals, like quantum logic can be restricted to a particular topic. (Can they? Schrodinger's Cat is pretty large.)

All this talk of logics of restricted range is opposed to the Rylean idea of logic as topic neutral; all form and no content.

Are these alternatives then really to be thought of as logics or, rather, as calculi? 

Applications: they have applications in specific areas but are not committed to them only. C.f. Quantum, Fuzzy, 3-value, temporal, ...

Derivations: they may be intended to deal with argument forms in particular areas - but is that really a handicap if there are non ad hoc ways of integrating that topic area with the rest of the universe?

Further Reading

Beall, J. C. & G. Restall 'Logical Pluralism' Australasian Journal of Philosophy, 78 (2000) 475–493. (pdf file, 19 pages.)

Haack, S. Deviant Logic (Cambridge: CUP, 1974)

Putnam, H. 'Is Logic Empirical?' Boston Studies in the Philosophy of Science, vol. 5, (eds. R. S. Cohen & M. W. Wartofsky) (Dordrecht: D. Reidel, 1968), pp. 216-241.

Quine, W. V. O. 'Two Dogmas of Empiricism', Philosophical Review 60 (1951): 20-43