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Is there a way to prove there's only one viable definition of the connectives, which is their truth functional definitions?

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    If it's a different definition, it's a different connective. Commented Jun 16 at 17:56
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    Can you explain what you mean by "viable"? There are lots of definitions of the connectives out there, and they all seem "viable" to me. Commented Jun 17 at 6:34
  • No. For a given logic, there may be many distinct but logically equivalent ways to define its connectives. But there are many logics, and these have different connectives with different definitions. In some cases it is possible to translate connectives from non-classical logics into their classical counterparts, but not in all cases.
    – Bumble
    Commented Jun 17 at 11:47

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I would say that the plethora of notions of disjunction make the answer to this a resounding no. You can find an overview of them on the Stanford Encyclopedia of Philosophy, but let me give a sketch of the difference between the classical and intuitionistic notions.

In classical logic, the disjunction φ ∨ ψ holds when φ holds or ψ holds, which I assume is the "truth functional definition" you are talking about. Under this understanding, ¬(¬φ ∧ ¬ψ) is equivalent to φ ∨ ψ: if it isn't the case that neither holds, then at least one holds, and the disjunction holds.

In intuitionistic logic, disjunction has a stronger meaning: φ ∨ ψ holds when φ holds or when ψ holds and you know which is the case. In this setting, ¬(¬φ ∧ ¬ψ) is weaker than φ ∨ ψ, since in the former case you can't tell which of the two holds.

This can of course be stated much more formally, for example that in an intuitionistic derivation system, ⊢ φ ∨ ψ implies ⊢ φ or ⊢ ψ, while in a classical one no such result holds.

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  • +1 in general, but the description of intuitionistic disjunction is a little misleading. Some intuitionistic treatments of disjunction entail “you know which is the case” in some sense, but others certainly don’t — for instance, most intuitionistic foundations don’t let you generally define a function f : { xX | φ₀(x) ∨ φ₁(x) } → {0,1} such that if f(x) = i then φᵢ(x) holds. The general idea is that a proof of disjunction should provide a way to determine which disjunct holds — but the disjunction simply holding doesn’t necessarily tell us. Commented Jun 18 at 11:07
  • @PeterLeFanuLumsdaine I have to admit, I'm mostly used to type-theoretic foundations where something like this does hold. How would you describe what disjunction means in a more general intuitionistic context?
    – Komi Golov
    Commented Jun 18 at 12:37
  • Even in type-theoretic foundations, it can go both ways — if you interpret disjunction as disjoint sum (as in classic BHK/propositions-as-types) then as you say you can extract a witness, but if you interpret disjunction as truncated sum (like in HoTT with propositions-as-hprops), then like in IFOL-based foundations, that information isn’t retained. Re “what it means”, my first instinct (being used to intuitionistic thinking) is to answer that intuitionistic φ ∨ ψ means exactly that either φ holds or ψ holds, while classical disjunction just means the disjuncts don’t both fail. (cont’d) Commented Jun 18 at 15:39
  • (cont’d) The less facetious and more serious answer is that I feel explanations like “in system X, disjunction means Y” are almost always misleading. What’s usually behind them is that system X has some intended model M, and in that model, disjunction is interpreted as Y. But unless X is designed exactly to capture one specific interpretation, disjunction in X won’t have a single specific meaning — X specifies our methods of proof for disjunction, and its models show what meanings are compatible with that. Commented Jun 18 at 15:40
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Outside of its utility within the language it is expressed, there is not an absolute method of endorsement. That is because logical operators are constituents of logical systems. In fact, we can even have two valid or operators in a single system; classical logic has the inclusive and the exclusive disjunction. The reason this situation is tenable is that they are variants on each other (disjunction) but each useful in its own context. If classical logic is a formal semantics abstracted from natural language, then how the operators are grounded spring from their utility in natural language semantics itself.

What this position implicitly advocates is a pluralist's notion of logic. See Logical pluralism. There have been philosophical cases made by persons such as Brouwer and Carnap that there is no "one-true logic". Those of us who are philosophical psychologists reject arguments put forth by thinkers such as Frege which are anti-pscyhologist in nature. There is often a need expressed tacitly by thinkers that there exists some "true" logic, and the rest of logics are "false". But the opposing view is that logics are not true or false at all, but rather are models and tools for thinking. In this conception, all models are wrong, but some models are useful.

So, in formal systems, what counts is not the endorsement of political entities as to whether or not this system is correct or incorrect insomuch as the recognition that formal logics are artificial languages endorsed by convention. This is distasteful to those of Platonic bent, who believe they are discovering an a real, mind-independent abstraction rather than using their mind for constructing one in the vein of a Sprachspiel. But on this view, logical operators derive their viability from within the logic system in which they operate, and one's logic is merely a question of preference and utility.

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Sounds like a question category theory could answer. I’ll ask some people I know.

I think you’re asking about a more essential “motivating criterion” from which the truth functions would follow.

One of the simplest is the mere fact that we take truth as bivalent, and we study binary or “dyadic” functions (functions with two arguments).

From there, it’s just plain combinatorics. The number of ways the set {0, 1} can be mapped into {0, 1} is 2^2. The number of ways that itself can be mapped into the set {0,1} again is 2^2^2. From thence do we get the standard result that there are actually 16 possible binary operators for logic, Boolean operators.

The next thing that should be explored is the relationships between each operator. This could shed light on why it appears to be so common to use ‘and’, ‘or’, ‘if-then’, etc.

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    What motivates the second mapping, which takes us from 4 to 16?
    – Tom
    Commented Jun 16 at 18:23
  • Because Boolean functions are binary functions - they have two arguments. Denoting the set of truth-values {0,1} as 2, every logical connective is a function of the form ∘: 2 × 2 → 2. By currying, this function is equivalent to ∘: 2 → 2 → 2. The number of ways a set A can be mapped to a set B is |B| to the power of |A|. Commented Jun 17 at 2:51
  • Every element in A represents a “choice” of an element in B. Element a in A can be mapped to any element b in B. That means that there are |B| choices for a single element of A. When we have to make multiple choices, we can multiply the number of options for each choice, to get the total number of choices. If we have 2 options for our first choice, and 3 options for our second choice, then there are 2 x 3 = 6 options for choosing from the first choice, then the second. So, we have |B| options, and we are going to “make a choice” |A| times - once for each element of A. So it’s B x B … (A times) Commented Jun 17 at 3:03

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