§1. Introduction
On the informal fallacy theoretic view of fallacies a fallacy is any argument that appears valid notwithstanding invalidity, or any argument liable to be taken-true notwithstanding falsity. However, the informal fallacy theoretic approach to fallacy detection is wrongheaded and self-defeating. Taxonomies of informal fallacies overwhelmingly either misidentify true, valid, and sound arguments as false, invalid, and unsound, or fail to detect genuine falsities, invalidities, and unsoundness.
In this post we make a case for formal fallacy theory contra informal fallacy theory; and, using the No True Scotsman Fallacy as a case study, show that informal fallacy theoretic analysis does much worse at fallacy detection than formal fallacy theoretic analysis. In particular, we show that the informal fallacy theoretic label No True Scotsman Fallacy misidentifies valid arguments as invalid.
§2. Case Study: No True Scotsman Fallacy
On the informal fallacy theoretic analysis of the No True Scotsman Fallacy <NTSF> no argument with the form of argument 1.1. is valid. Alternatively, arguments with the form of argument 1.1 are always invalid, and, so, argument 1.2 is always valid.
ARGUMENT 1.1.: ∀S: ((S) ↔ ((α(S)) & (β(S)))). <Never valid on NTSF>.
ARGUMENT 1.2.: ∀S=:⊥ ⇒ ((S) ↔ ((α(S)) & (β(S)))). <Always valid on NTSF>.
However, on a formal analysis with standard first order predicate logic several arguments with the form of argument 1.1 are valid, and, so, argument 1.2 is invalid. A great many legitimate definitions and conditional assertions with universal generalizations occurring in premise position have the form of argument 1.1., and constitute countermodels for argument 1.2.
Consider the definition of a whole tone scale, given below <See Definition 1.3>. It states that any given scale, W, is a whole tone scale, (W), if and only if it is a sextuple of notes from the octave, (Oγ(W)), and each of the notes in the sextuple is two semitones apart from the other (Oδ(W)).
DEFINITION 1.3.: ∀W: ((W) ↔ ((Oγ(W)) & (Oδ(W)))).
Now, we prove a theorem showing that argument 1.2. is false. The upshot of this theorem [See Argument 1.4] is that the No True Scotsman Fallacy isn’t always invalid, and so is not a fallacy.
ARGUMENT 1.4: ∀S: ((S) ↔ ((α(S)) & (β(S)))).
PROOF: By argument 1.2. ∀S =:⊥ ⇒ ((S) ↔ ((α(S)) & (β(S)))), or ¬((S) ↔ ((α(S)) & (β(S)))). This implies ¬(((S) → ((α(S)) & (β(S)))) & (((α(S)) & (β(S))) → (S))). However, by definition 1.3., ∀W: ((W) ↔ ((Oγ(W)) & (Oδ(W)))). Substituting all W and W with S and S, and of Oγ and Oδ with α and β respectively, we get: ∀S: ((S) ↔ ((α(S)) & (β(S)))). This implies (((S) → ((α(S)) & (β(S)))) & (((α(S)) & (β(S))) → (S))). So, contra argument 1.2., we have proved ∀S: ((S) ↔ ((α(S)) & (β(S)))).■
§3. Discussion
Argument 1.4 establishes beyond all doubt that arguments identifiable as No True Scotsman Fallacies aren’t always fallacious; in other words, the No True Scotsman Fallacy isn’t a fallacy. Definition 1.3 provides one of many valid argument forms whose instances are invariably misidentified as fallacious on the informal fallacy theoretic approach to fallacies.
It may be objected by defenders of informal fallacy theory that even though not all arguments with the form of argument 1.1 are invalid some certainly are. We are inclined to agree. But, if there are variously valid and invalid instances of arguments with the form of argument 1.1., then the informal fallacy theoretic label No True Scotsman Fallacy does nothing that helps tell apart valid from invalid instances of such arguments.
By contrast, the formal analysis offered here as an alternative tells us:
i.> All argument instances with the form of argument 1.1. are biconditionals with a universal generalisation occurring in premise position.
ii.> Argument instances with the form of argument 1.1. can only fail to obtain when
a.> there exists no individual possessing all properties/predicates associated with the individual occurring in a premise that is a universal generalisation,
or
b.> there exists at least one individual lacking some or all of the the properties predicated of the individual occurring in premises that are universal generalisations.
§4. Concluding Remarks
If some conditional arguments with universal generalisations occurring in premise position, or in the conclusion, fail then they fail not because all such arguments are instances of the No True Scotsman Fallacy. Instead, the failure of such arguments is due to the fact that any one or both of conditions ii. a. and b. supplied in the previous paragraph obtain.
This post showed that the informal fallacy theoretic label ‘No True Scotsman Fallacy’ is at best pleonastic, and at worst false. It is pleonastic whenever it is blindly applied to invalid arguments premised on [an] invalid universal generalisation[s]; and, it is false whenever it is slapped onto valid arguments which happen to be premised on valid universal generalisations.
In future posts we’ll use other informal ‘fallacies’ as case studies and demonstrate the weaknesses and failings of the informal fallacy theoretic approach to fallacy detection vis-à-vis formal fallacy theoretic approaches.
Absolute Generality 