No True Scotsman

"Imagine some Scottish chauvinist settled down one Sunday morning with this customary copy of The News of the World. He reads the story under the headline, 'Sidcup Sex Maniac Strikes Again'. Our reader is, as he confidently expected, agreeably shocked: 'No Scot would do such a thing!' Yet the very next Sunday he finds in that same favourite source a report of the even more scandalous on goings of Mr Angus MacSporran in Aberdeen. This clearly constitutes a counter example, which definitvely falsifies the universal proposition originally put forward. ('Falsifies' here is, of course, simply the opposite of 'verifies'; and it therefore means 'shows to be false'.) Allowing that this is indeed such a counter example, he ought to withdraw; retreating perhaps to a rather weaker claim about most or some. But even an imaginary Scot is, like the rest of us, human; and we none of us always do what we ought to do. So what in fact he says is: 'No true Scotsman would do such a thing!'." -Antony Flew in "Evasion and Falsification"

The philosopher Antony Flew presents what is to be the "No True Scotsman" fallacy.  If the reasoner re-characterizes the situation solely in order to escape refutation of the proposition, then this error is a type of ad hoc rescue of one's generalization.

All arguments or discussions are between at least two individuals or at least two positions. In all arguments, there are at least two positions being argued or discussed. Take an argument between an individual who believes that absolute physics of Newton is true & another individual who believes that relative physics of Einstein is true, or "there exists a law of nature" or "there doesn't exist a law of nature". The argument is based on two propositions, which are that "absolute physics of Newton is true" & "absolute physics of Newton isn't true". The argument is based on the foundation of P & ~P. But logic forbids that P & ~P are true, or forbids the truth of "absolute physics of Newton is true & absolute physics of Newton isn't true".

Structure of Discussion
(1) P; (2) ~P; (3) P&~P; (4) ~(P&~P); (5) Pv~P  [Standard Notation of Symbolic Logic]
(1) p; (2) Np; (3) KpNp; (4) NKpNp; (5) ApNp  [Polish Notation of Symbolic Logic]

Step One: An individual proposes that (1) is true. p.
Step Two: Another individual proposes that (2) is true. Np.
Step Three: Discussion between both individuals contain (3). KpNp.
Step Four: Discussions can't contain (3) because discussion follows (4). NKpNp.
Step Five: Discussions must contain either (2) or (3) because discussions follows (5). ApNp
Step Six: Examine if both (1) and (2) are Falsifiable.

The processes of discussion or arguments, so far, show that there is a problem to be solved by the individuals in the discussion. One of the propositions must be rejected & the argument doesn't show which proposition must be rejected. We may either reject "absolute physics of Newton is true" or we may reject "relative physics of Einstein is true"; We may either reject "there is a law of nature is true" or we may reject "there isn't a law of nature is true". But nothing shows which one, in the argument, is to be discarded as false. Is "relative physics of Einstein is true" false or is "there exists a law of nature is true" false?

No True Scotsman fallacy can't be committed when propositions are analytic and necessary. Analytic and necessary propositions are impossible to be false. No True Scotsman fallacy can be committed when propositions are synthetic and contingent, or so one would think.

There is a criterion of demarcation between empirical propositions and non-empirical propositions. Empirical propositions are synthetic and contingent & falsifiable. Non-empirical propositions are synthetic and contingent & not falsifiable. "Absolute physics of Newton is true" is an empirical propositions, so it is synthetic, contingent, and falsifiable. "There exists a law of nature is true" is a non-empirical proposition, so it is synthetic, contingent, and not falsifiable.

Empirical propositions can't be shown true & empirical propositions can be shown false. Non-empirical propositions can't be shown true & non-empirical propositions can't be shown false. But both propositions are synthetic and contingent, because they are both possibly true and possibly false. Both propositions can't be demonstrated that they are actually true instead of possibly true, but empirical propositions can be demonstrated that they are actually false instead of possibly true and non-empirical propositions can't be demonstrated that they are actually false instead of possibly false. 

"Absolute physics of Newton is true" is possibly true and possibly false; It can't be demonstrated that it is actually true; It can be demonstrated that it is actually false; So "absolute physics of Newton is true" is falsifiable.

"There exists a law of nature is true" is possibly true and possibly false; It can't be demonstrated that it is actually true; It can't be demonstrated that it is actually false; So "there is a law of nature is true" isn't falsifiable.

Falsifiability can be summed up with Modus Tollens:
(1) If p then q :: Cpq or (p-->q)
(2) ~q :: Nq or  ~q
(3) ~p :: Np or ~p 

Here is an example, based on Flew's example, of Modus Tollens:
(1) No Scotsman are a sex maniac.
(2) Mr. Angus MacSporran is a Scotsman and a sex maniac.
(3) Therefore, Not all Scotsman are not sex maniac.

(1) 'No Scotsman are sex maniac' is logically equivalent to 'All Scotsman are not sex maniac'. No p are q is equivalent to All p are not q. One can't be true and the other is false, if one is true then the other is true or if one is false then the other is false. The truth value of one is dependent on the truth value of the other. They must have the same truth value, and it would be contradictory for them to have different truth values.

Falsifiability follows the logical form of Modus Tollens. Falsifiability works with a hypothesis and an observation. Falsifiability works with hypothesis that are possibly false, but works to show that the hypothesis is actually false. Falsifiability shows that a hypothesis is actually false by an observation being accepted as true and the observation contradicts the hypothesis.

(1) For all things, if a thing exists then the thing isn't a law of nature.
(2) There exists a thing, such that the thing exists and the thing is a law of nature.
(3) Not all things, if a thing exists then thing isn't a law of nature.

(1) is the hypothesis and (2) is the observation. Through Modus Tollens, one would have to accept (3). However, if we reject (2), then we wouldn't have to accept (3). However, the hypothesis presented in (1) isn't a falsifiable statement because only (2) can possibly show that (1) is actually false. But, (2) can't be obtained by falsifiability. So, we learn that we don't have to accept (3) by logical implication.

Some would object that you must accept (2) as obviously true or accept that there does exist a law of nature, thus you are committing the "No True Scotsman" fallacy.

But such a statement would be a non-metaphysical statement. We can't show that the statement is not only possibly true but also that it is actually true. If a law of nature exists is true, it can't be shown empirically that it is true. This is because such a thing being true is possible, but empirically impossible to show that actually true. And since it isn't an empirical proposition, it is a non-empirical statement. And non-empirical statements can't be shown false.

This is where the Crux of the No True Scotsman lies. It is between empirical statements & non-empirical statements. Or, to put another way, the crux is between scientific statements & metaphysical statements.

Falsifiability would only accept scientific statements to be placed within the premises of Modus Tollens inference. Premise 1 and Premise 2 must be scientific statements, so that neither premise has a metaphysical statement. If there is at least one premise is a metaphysical statement, then No True Scotsman can't occur.

Now moving back to Flew's example, we show this even further.
(1) No Scotsman are a sex maniac.
(2) Mr. Angus MacSporran is a Scotsman and a sex maniac.
(3) Therefore, Not all Scotsman are not sex maniac.

Suppose that (2) is a metaphysical statement and (1) is an empirical statement. It would follow that (3) doesn't have to be accepted. But some people would complain that (2) is obviously true and so you must accept (3) follows from (1). Thus, the No True Scotsman fallacy would be invoked when faced with a falsification & the individual is performing an Ad hoc rescue to escape the proof of falsity of statement.But no such fallacy has occurred, because the individual proclaiming that the fallacy has failed to recognize that one of the statements accepted is a metaphysical statement.

Conclusion:

S1: An individual proposes that (1) is true. p.
S2: Another individual proposes that (2) is true. Np.
S3: Discussion between both individuals contain (3). KpNp.
S4: Discussions can't contain (3) because discussion follows (4). NKpNp.
S5: Discussions must contain either (2) or (3) because discussions follows (5). ApNp
S6: Examine if both (1) and (2) are Falsifiable.

If (1) & (2) are falsifiable then not accepting (2) & (3) violates (4), so No True Scotsman Fallacy has occurred.

If either (1) or (2) aren't falsifiable then not accepting (2) & (3) doesn't violate (4), so No True Scotsman Fallacy hasn't occurred.

If neither (1) nor (2) are falsifiable then not accepting (2) & (3) possibly does violate (4), so No True Scotsman Fallacy might have occurred.

One Value Logic

Unvalence Logic, would only contain one value. But, most other logics would contain two values or more. A typical example would be the two value logic of Aristotle. Take the Aristotelian syllogism of Disamis: If all golden retrievers are dogs & some beloved household pets are golden retrievers, then some dogs are beloved household pets. This syllogism is always true under all two valued logics.

Two valued logic is usually called Bivalent. This is because two valued logic has the fundamental thesis that all propositions have two, and only two, values that can possibly be attributed to them. The two values that every proposition will have is True or False. The Bivalent logic will be able to tell you which propositions are True and which propositions are False. The only classes for propositions are Truth class and False class. Truth class is everything and False class is nothing.

Bivalent logic would only work if have only those two classes. From having these two classes it will always give you true propositions. It will be able to answer only two types of questions.

Suppose a student asks Bivalent logic a question, specifically a question on mathematics. The student asks, "What isn't true in mathematics?". The response is "two plus two isn't five". The student asks, "What is true in mathematics?". The response is "two plus two is four". So it can only give answers to the questions of form of "What isn't true" & "What is true". 

Bivalue logic deals with propositions, but these propositions must be of a certain sort. The propositions must take on a certain form for Bivalue logic to be applied. The mathematical logic text "Mathematical Logic and Computability", gives some indication of mathematical logic being a bivalent logic.

"A vocabulary for propositional logic is a non-empty set P(o) of symbols; the elements of the set P(o) are called proposition symbols and denoted by lower case letters p, q, r, s, p,q, ... In the standard semantics of propositional logic the proposition symbols will denote propositions such as 2+2 = 4 or 2+2 = 5. Propositional logic is not concerned with any internal structure these propositions may have; indeed, for us the only meaning a proposition symbol may take is a truth value – either true or false in the standard semantics." - (Italics my emphasis)

The Bivalent logic machine will only answer questions that have meaning. The only questions that concern the machine are propositions that are either true or false, and nothing more and nothing less. It will always give true answers to questions that meet that requirement of meaningful questions, i.e. meaningful propositions.

As the authors move further on to show of mathematical logic that:

"The primitive symbols of the propositional logic are the following: (1) propositional symbols p,q,r...from P(o); (2) negation sign (~); (3) conjunction sign (&); (4) disjunction sign (V); (5) implication sign (-->); (6) equivalence sign (<-->); (8) left bracket [ ; (9) right bracket ]."

We can represent all these primitive symbols of Bivalent logic, or at least mathematical logic, like this: Primitive Symbols = (p, ~, &, V, -->, <-->, [,and  ]).

"Any finite sequence of these symbols is called a string. Or first task is to specify the syntax of propositional logic; i.e. which strings are grammatically correct. These strings are called well-formed formulas. The phrase well-formed formula is often abbreviated to wff (or a wff built using the vocabulary P(o) if we wish to be specific about exactly which proposition symbols may appear in the formula). The set of wffs of propositional logic is then inductively defined by the following rules: (W:P ) Any proposition symbol is a wff; (W:~ If A is a wff, then ~A is a wff; (W:∧, ∨, -->, <-->) If A and B are wffs, then [A ∧ B], [A ∨ B], [A --> B], and 0 [A <--> B] are wffs." - (Italics are authors)

The primitive symbols will now be able to be joined together. This joining together will be based on some rules, and these rules determine what is a well formed formula. So if well formed formula is presented based on some primitive symbols, then Bivalent logic may deal with them.

We can represent these rules of Bivalent logic, at least mathematical logic, like this:
WFF of Primitive Symbols= [(p, ~, &, V, -->, <-->, [,and  ]) P, ~P, P&Q, PvQ, P-->Q, A<-->Q]

Another way to put this would be that (i) all propositional symbols are wff, (ii) all wff that are prefixed with ~ are wff, (iii) all two wff joined by either (&,V,-->, or <-->) and  enclosed by brackets are wff. Bivalent logic can only deal with propositions that are either of these criterion and either true or false.

Here is a truth table of two propositions, both of which are wff:
P ~P
T   F
F   T

This truth table exhausts all logically possible outcomes of either well formed formula P or well formed formula ~P. These are the only possible outcomes for Bivalent logic. The truth table shows that each well formed formula is consistent with each proposition being either true or false.

Unvalent Logic will have some slight alternations. Unvalent logic will only deal with propositions that take on one truth value and only one truth value. This truth value would be truth itself, perhaps might be false value. So Unvalent logic will not deal with any propositions that take on two truth values or more than two truth values.

This alteration leads to an automatic change. It would immediately lead to a well formed formula not having the primitive symbols of (2) through (9). So it will not accept the symbols (~, &,V,-->, <-->). It will also not accept as well formed formula like (ii) and (iii). This also leads to an immediate change in the laws of logic.

For example, the Bivalent logic has at least three laws. It is the Law of Identity, Law of Excluded Middle, and Law of Non-Contradiction. None of these laws would hold within Unvalent logic. All of this becomes obvious from recognition that the logical operator of negation, i.e. ~, is rejected from Unvalent logic.

Here are all three laws represented in symbolic notation:
(LoI): P-->P or ~P-->~P
(LEM): Pv~P or ~PvP
(LNC): ~(P&~P) or ~(~P&P)

Excluded Middle has a negation within it, and Unvalent rejects negation, thus Excluded Middle is rejected. Likewise, Non-Contradiction has two negations in it, so Non-Contradiction is rejected. Because Identity in Bivalent logic equally applies to negation and non-negation, and negation is rejected in Unvalent logic, the identity is also rejected.

With two valued logic, each proposition with a propositional symbol (either p,q, or r) is implicitly saying that "it is true that...". For example, take the propositional symbol P. P is either true or false. P states that "it is true that two plus two is four". The negation operator is implicitly saying that "it isn't true that...". So given symbol P and applying ~ operator to it, then we obtain ~P, which means that "it isn't true that two plus two is four".

So LoI says, If "it is true that two plus two is four" then "it is true that two plus two is four" & If "it isn't true that two plus two is four" then "it isn't true that two plus two is four". LEM says, either "it is true that two plus two is four" or "it isn't true that two plus two is four". LNC says, not both "it is true that two plus two is four" & "it isn't true that two plus two is four". So Unvalue logic drops all propositions with "it isn't true that..." from being well formed formula. It would drop two valued logics well formed formula.

If all golden retrievers are dogs & some beloved household pets are golden retrievers, then some dogs are beloved household pets, is a true proposition in two valued logic, but isn't true in Uncvalue logic. This is because it has the operators of --> and &. Implication is dropped as a logical operator in this single value logic because implication is defined by material implication.

Implication: P-->Q
Material Implication: ~PvQ

As becomes obvious, material implication contains negation operator within it. This means that whenever a conditional statement is made in two value logic, there is a negation contained within it. If  P-->Q is true, then ~PvQ is true. If P-->Q is true, then a negation is also true. Substitute the wff X for the wff (P-->Q) and suppose the proposition X is true, then a negation is also true. Single value can't have a negation true, so P-->Q can't be true. Furthermore, suppose that P-->Q, then the contraposition would be true as well, which is ~Q-->~P. So P-->Q being true implies a negation is true as well.

In two value logic, there are two basic logical operators. They are negation and one of the other four operators of -->, &, V, or <-->. Usually, the two basic logical operators are implication and negation, or ~ or -->. Because --> can be replaced with material implication, which is just a combination of negation and disjunction, or ~ and V, the disjunction can't be the second operator.

If we only assume propositional symbols P and Q & logical operators of negation and implication, then we can define the other logical operators.

Conjunction (&): ~(P-->~Q)
Disjunction (V): (~P-->Q)
Equivalence (<-->): (P-->Q)&(Q-->P)

The law of identity is an implication, and implications contain negations within them, so implications aren't valid in single value logic. The law of excluded middle is a disjunction, and we notice that disjunction contains negation within it, so disjunction isn't valid in single value logic. The law of non-contradiction is a conjunction, and we notice that conjunction contains negation within it, so conjunction isn't valid in single value logic. Equivalence contains both implication and conjunction, and so they aren't valid in single value logic.

It becomes obvious that single value logic will not have negations, conjunctions, disjunctions, equivalence, or implications. It won't answer "What isn't true?" questions. It will only answer "What is true?" questions. It will give you nothing but true statements, and only applies to true statements.

The primitive symbols of the Uncal logic are (1) propositional symbols p,q,r...from P(o). The set of well formed formula of single value logic is then inductively defined by the rule (i) (W:P ) that any proposition symbol is a well formed formula.