Philosophy 101 (philpapers-induced) #12: Logic: classical or non-classical?

Philosophy 101 (philpapers-induced) #12: Logic: classical or non-classical? September 4, 2020

Having posted the Philpapers survey results, the biggest ever survey of philosophers conducted in 2009, several readers were not aware of it (the reason for re-communicating it) and were unsure as to what some of the questions meant. I offered to do a series on them, so here it is – Philosophy 101 (Philpapers induced). I will go down the questions in order. I will explain the terms and the question, whilst also giving some context within the discipline of Philosophy of Religion.

This is the twelfth post, after…

#1 – a priori

#2 – Abstract objects – Platonism or nominalism?

#3 – Aesthetic value: objective or subjective

#4 – Analytic-Synthetic Distinction

#5 – Epistemic justification: internalism or externalism?

#6  – External world: idealism, skepticism, or non-skeptical realism?

#7 – Free will: compatibilism, libertarianism, or no free will?

#8 – Belief in God: theism or atheism?

#9 – Knowledge claims: contextualism, relativism, or invariantism?

#10: Knowledge: Empiricism or Rationalism?

#11 – Laws of nature: Humean or non-Humean?

The question for this post is: logic: classical or non-classical? Here are the results:

Logic: classical or non-classical?

Accept or lean toward: classical 480 / 931 (51.6%)
Other 308 / 931 (33.1%)
Accept or lean toward: non-classical 143 / 931 (15.4%)

I won’t be explaining all the different subsets of logic here as that would entail multiple book-length volumes. We should start this piece by defining what logic is. Logic is a branch of both mathematics and philosophy and is broadly the appraisal of arguments in the context of formal or informal language taken together with a deductive system and/or model-theoretic semantics. There is no settlement to the definition of logic, hence this piece, right. Wikipedia sums it up in simple terms:

Logic (from Greekλογικήlogikḗ, ‘possessed of reasonintellectualdialecticalargumentative‘)[1][2][i] is the systematic study of valid rules of inference, i.e. the relations that lead to the acceptance of one proposition (the conclusion) on the basis of a set of other propositions (premises). More broadly, logic is the analysis and appraisal of arguments.[3]

There is no universal agreement as to the exact definition and boundaries of logic, hence the issue still remains one of the main subjects of research and debates in the field of philosophy of logic (see § Rival conceptions).[4][5][6] However, it has traditionally included the classification of arguments; the systematic exposition of the logical forms; the validity and soundness of deductive reasoning; the strength of inductive reasoning; the study of formal proofs and inference (including paradoxes and fallacies); and the study of syntax and semantics.

A good argument not only possesses validity and soundness (or strength, in induction), but it also avoids circular dependencies, is clearly stated, relevant, and consistent; otherwise it is useless for reasoning and persuasion, and is classified as a fallacy.[7]

There are a wide range of logics – a plurality of them. In philosophy, language and semantics are very closely related to logic and classical logic looks to codify this relationship such that it is truth-functional. It’s all about deductive validity.

Classic Logic

Classical Logic (or Standard Logic) is not actually derived from the classical period but came out of the 19th and 20th centuries (starting with Frege, but including an awful lot of other logicians).

Essentially, inferences have premises and conclusions, but language is a bit irregular, so we can get into a spot of bother.

There are different views on what language, natural language (as in, what people speak, as opposed to formal languages that are designed for a functional purpose), is and how it is underwritten. Natural languages are very irregular. The question here is about the relationship between natural and formal languages (such as propositional calculus, that seeks to set out propositions and conclusions in an almost mathematical manner using connectives). Some might say that natural languages have underlying logical forms, that can be displayed in formal languages. One might say that a good declarative sentence in natural language may contain propositions that can again be reflected in formal language. Others maintain that natural language’s vagueness means that it should be replaced with formal languages in a very regimented way. In this way, there can be no doubt as to the truth or certain propositions in a context of bivalence (things are either true or false).

This is where such logic shares much in common with mathematics.

Formal languages look to evade ambiguities (amphibolies), such as:

John is married, and Mary is single, or Joe is crazy.

This could be formalised as (A&B)∨C) or (A&(BC), for example.

Modal logic is a form of classic logic where modal denotes possibility or necessity, and formal symbols are often utilised to give it structure. However, there are some who maintain that modal logical stretches into non-classical territory, or at least extends classical theories.

The end result of lots of discussion that could be had over language, syntax and semantics is that:

We now introduce a deductive system, D, for our languages. As above, we define an argument to be a non-empty collection of sentences in the formal language, one of which is designated to be the conclusion. If there are any other sentences in the argument, they are its premises.[1]

Classical logic is bound by certain rules:

  1. Law of excluded middle and double negation elimination

  2. Law of noncontradiction, and the principle of explosion

  3. Monotonicity of entailment and idempotency of entailment

  4. Commutativity of conjunction

  5. De Morgan duality: every logical operator is dual to another

While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics.[4][5] In other words, the overwhelming majority of time spent studying classical logic has been spent studying specifically propositional and first-order logic, as opposed to the other forms of classical logic.

As the SEP concludes:

Logic and reasoning go hand in hand. We say that someone has reasoned poorly about something if they have not reasoned logically, or that an argument is bad because it is not logically valid. To date, research has been devoted to exactly just what types of logical systems are appropriate for guiding our reasoning. Traditionally, classical logic has been the logic suggested as the ideal for guiding reasoning (for example, see Quine [1986], Resnik [1996] or Rumfitt [2015]). For this reason, classical logic has often been called “the one right logic”. See Priest [2006a] for a description of how being the best reasoning-guiding logic could make a logic the one right logic.

That classical logic has been given as the answer to which logic ought to guide reasoning is not unexpected. It has rules which are more or less intuitive, and is surprisingly simple for how strong it is. Plus, it is both sound and complete, which is an added bonus. There are some issues, though. As indicated in Section 5, there are certain expressive limitations to classical logic. Thus, much literature has been written challenging this status quo. This literature in general stems from three positions.

Which neatly gets us onto non-classical logic(s).

Non-classical logic

Non-classical logics (or alternative logics) differ from propositional or predicate logic in several ways, and often because of extensions, deviations, and variations, which then has an impact of how we interpret logical truth and consequence.

If there are more than two truth values (the bivalence of True and False), such as in Ł3 logic, where you can have T, F and #, where # is an indeterminate proposition of either: both true and false or neither true nor false. This affects how truth tables might look.

“I am bald” can be indeterminate for someone who is perhaps not completely bald, i.e. has a few hairs here and there. We could assign a truth value of # to this. Here, we might return to the vagueness or ambiguity of natural languages.

Here is a list of different non-classical logics (from da Costa, Newton (1994), “Schrödinger logics”, Studia Logica53 (4): 533):

It depends how you define things and so some people include extensions of classical logic (such as modal logic) in with properly “deviant” logics.

For these logics, some of the aforementioned constraints do not hold, such as the law of the excluded middle for intuitionistic logic.

Discussion

When I first saw this question, I thought it was a false dichotomy, that you could use different logics for different ends and that one does not ultimately have to be true to the exclusion of the other. Indeed, in looking at the Philpapers survey discussion, the creators had this same problem in mind:

Various respondents wondered how to interpret this question, given that it’s not obvious that there has to be a fact of the matter about whether classical or nonclassical logic is correct. Still, plenty of people think there is a fact of the matter, and those that don’t (including ourselves) seem to have by and large chosen an “other” option.

As the SEP concluded in its article on classical logic:

Thus, much literature has been written challenging this status quo. This literature in general stems from three positions. The first is that classical logic is not reason-guiding because some other single logic is. Examples of this type of argument can be found in Brouwer [1949], Heyting [1956] and Dummett [2000] who argue that intuitionistic logic is correct, and Anderson and Belnap [1975], who argue relevance logic is correct, among many others. Further, some people propose that an extension of classical logic which can express the notion of “denumerably infinite” (see Shapiro [1991]). The second objection to the claim that classical logic is the one right logic comes from a different perspective: logical pluralists claim that classical logic is not the (single) one right logic, because more than one logic is right. See Beall and Restall [2006] and Shapiro [2014] for examples of this type of view (see also the entry on logical pluralism). Finally, the last objection to the claim that classical logic is the one right logic is that logic(s) is not reasoning-guiding, and so there is no one right logic.

Suffice it to say that, though classical logic has traditionally been thought of as “the one right logic”, this is not accepted by everyone. An interesting feature of these debates, though, is that they demonstrate clearly the strengths and weaknesses of various logics (including classical logic) when it comes to capturing reasoning.

Io think the way this question was answered probably reflects what most philosophers are comfortable with (i.e., what they know) and the fact that there isn’t necessarily a bivalence in the question itself!

 


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