Sorites (Σωρίτης), ISSN 11351349 http://www.sorites.org Issue # 20 — March 2008. Pp. 141156 Hypothesis Testing Analysis Copyright © by Mikael Eriksson and Sorites 
Hypothesis Testing Analysis
by Mikael Eriksson
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1. Introduction
Looking directly into some technicalities of the philosophical empirical tradition, the essence of hypothesis confirmation and hypothesis falsification is viewed such that hypotheses are based in, and perhaps are stronger than, the evidence that experience in turn provides. Confirmation singles out those hypotheses that base in, and perhaps go beyond, the evidence. Falsification, on the other hand, singles out those hypotheses being inconsistently based in evidence. The very same traditions recognise the wellknown consequences that confirmation and falsification involve induction and correlation problems. Confirmation tends to include more information than the evidence provides, while falsification is not strong enough to generally tell what is false in the falsified hypothesis. As if this would not be problem enough, it is also generally viewed that confirmation and falsification clash with each other. These problems seem to put confirmation and falsification in a pretty awkward place of fulfilling the conditions of being a set of logical constants validating the natural phenomenon of empirical knowledge gaining activities.
For starters, look at the notion of falsification. Popper talked about a hypothesis h being better corroborated than a hypothesis h' on some evidence e, in the sense that h is capable of being tested for falsification by more observational statements than h'. (Popper,1972) Corroboration includes this falsification, which in turn has the consequence that some given observation, testing a hypothesis, falsifies it without being able to show which part of the hypothesis is false. The accustomed logical understanding of this is as follows. The predicate «e falsifies h» boils down to the formula «h implicates e». A wellknown logical theorem says that in denying a formula consequent, it does not follow which part of the antecedent is false. Therefore, when e falsifies h and h is a complex formula, it is not clear which part of h is false. However, looking at the natural idea of falsification, it clearly indicates the idea of proper evidence and hypothesis correlation, demanding more of «e falsifies h» than the implication analysis shown. Following this, it seems to me that the suggested Popper logical understanding of falsification needs revision. Rather, how does h and e correlate?
Now, what has the empiricists to offer? The vast empiricist tradition searches for methods of gaining empirical knowledge along the notion of induction, taking off in the Hume tradition. (Hume,1777) In the early 20^{th} century the empirical ideas took further steps, by understanding confirmation as testing, which by that time was viewed as an inductive and nondemonstrative inference from evidence to hypotheses. Later on, empiricists developed the basic confirmation idea understanding into demonstrative probability, still emulating induction. There, Carnap stipulated that the formula «evidence confirms a hypothesis» means a hypothesis becoming accepted scientific knowledge and nearly (increasingly) certain in the light of one (several) bodies of evidence — absolute (relevance) confirmation. (Carnap,1962) This induction idea was governed by the assumptions that hypotheses are based in evidence, and might be stronger than that evidence. For those cases when the proper hypotheses do not go beyond the evidence, the hypotheses and evidence are indeed identical. I claim that this seemingly fair view is false and leads to paradoxes as section 5.1 will show.
With this outline I have tried to show the problems involved in both empirical accounts — each aiming at the trophy of validating the natural phenomenon of empirical knowledge gaining. The rising question is whether there might be another way to define this empirical set of constants. We know that the falsification account claims formulas on the form «hypothesis h being better corroborated than hypothesis h' on evidence e». Empiricists, on the other hand, claim formulas on the form «evidence e confirms hypothesis h». Putting it this way, it seems to me that both accounts put their efforts on analysing the relation between e and h, while I think the emphasis first need to be on e and h in separate. Developing a system along this view give rise to a new terminology which I hope will bring some light to both the Popper notion of falsification modus tollens and the Carnap notion of confirmation probability.
2. Natural language analysis
2.1 Idea
In pursuing empirical knowledge reasoning, I search for valid argument schemata characterising the scientific everyday language. A usual sentence is «I am interested in this phenomenon or this hypothesis». Out of such ordinary scientific language, it could be possible to single out argument schemata. One way of unveiling these suggested schemata is to put the following question. What makes scientists to accept and be nearly certain of a hypothesis? It seems clear that scientists, after making their conjectures, use evidence to test their hypotheses, aiming at confirming or falsifying the hypotheses. However, the clearness is not preserved when it comes to understand the involved phenomenon of testing. Aiming at this seemingly opacity, I believe that this notion should be understood in terms of correlation of hypothesis and evidence as well as understanding the meaning of hypothesis and evidence. Consider this ancient example. In my view, when Galilei claimed that there is a halo around the moon (not the face of God turned to the observer as claimed by contemporaries), he used his perception to collect evidence. He also made a hypothesis claiming the idea of «halo around the moon». Finally, he used his human ability to relate his hypothesis with his evidence. In a language view, I refer to these as evidence formula, hypothesis formula, and procedure formula. I put these three formulas together in the following way. The procedure formula tests the hypothesisformula «halo around the moon» with the evidenceformula «halo around the moon».
With this view of testing established, I go further by analysing what these three formulas are. By evidenceformula I mean a logical formula describing the evidence claim (for instance the perception of some entity). By hypothesisformula I mean a logical formula describing the hypothesis claim (for instance an idea claimed by someone). I mean that the evidence and hypothesis formulas are distinct in character but similar to each other, having the same simple and logical form. Finally, the procedure formula I mentioned is a logical construction of these evidence and hypothesis formulas. This logical construction will depend on the logical form of the evidence and hypothesis formulas, as section 3 shows. Summing this up, I view and analyse testing as including procedures of hypothesis and evidence formulas.
This analysis of testing makes the first face of empirical knowledgegaining (besides conjecturing). Note that this face of testing involves the principle of similar evidence and hypothesis formulas, which later will be shown to explains the rationalist's evidence and hypothesis correlation problem. I believe there is a second face of empirical knowledgegaining, which I claim to be hypothesis inclusion. Now, this means that my viewed twofaced formulation of empirical knowledgegaining divides a scientist hypothesis claim into two parts. The first part is a sub hypothesis, which is tested by evidence. The second part is the rest of the hypothesis, which belongs to the intuition of the scientist, but is not present in the evidence. This way of understanding empirical knowledgegaining explains the empiricist's induction inference problems, as I will show.
Summing the idea up, I view confirmation as constructive empirical knowledgegaining and I analyse it in terms of testing and inclusion. This can be expressed by the conjunction of the two formulas «this evidence tests this hypothesispart» and «the full hypothesis compares with that hypothesispart». Along this view, I fit in falsification as reductive empirical knowledgegaining and I analyse it in terms of reductive testing, as below will show. In this way, both confirmation and falsification is defined in terms of testing, seemingly opening a common ground for both the Carnap and Popper terminologies.
2.2 Analysis
2.2.1 Distinctiveness
The above show that I understand testing as including the condition of evidence and hypothesis formulas being distinct from each other. Going into this in detail, I will say that by formulas I mean logical formulas, where the evidenceformula simple parts refer to simple evidenceclaims, and the hypothesisformula simple parts refer to simple hypothesisclaims. By the term distinct, I mean that the simple parts of evidenceformulas are distinct in kind from the simple parts of hypothesisformulas.
This view of distinctiveness bases in my intuition of what hypothesis and evidence are. To me, evidenceclaims and hypothesesclaims are human activities about the natural world. A natural entity differs from both the human perception of it and the human understanding of it. Evidenceclaims (related to science) are claims focusing upon natural entities from a perceptual point of view. Analogously, hypothesisclaims (related to science) are claims focusing upon natural entities from an idea point of view. These different points of views of the natural entity include that the evidence have at least some quality different from the hypothesis. This makes evidence distinct from hypotheses.
Let's see how this distinctiveness applies to the Galilei example. When Galilei made a claim that there is a halo around the moon (not the face of God turned to the observer), he focused upon the natural situation of the moon from his perceptual point of view. His perception included biochemical properties. The very same properties were not included in his hypothesisclaim that there is a halo around the moon. This makes his evidence claim distinct from his hypothesis claim. Relating to Carnap, he also points out on page 12 that neurological factors determine inductive reasoning. (Carnap,1971) Still, he claims that «e implicates h», if e and h refer to statedescriptions having inclusion relation. That is, h is included in e. Understanding the Carnap mentioned neurological factors as I do, makes his claim contradictory, as e and h has some neurological properties distinct from each other. However, it would be possible to rewrite the Carnap claim in the following manner. My claim of hypothesis and evidence distinctiveness is consistent with worldstates including the two kinds of evidence and hypotheses referring to the same situation in the worldstate. The evidence, having its properties, is part of a twofaced worldstate alongside with the hypothesis, having some distinct properties. However, contrary to Carnap the evidence and hypothesis relation, due to the distinctiveness, is rather conjunctive than implicative. To formalise this evidence E and hypothesis H distinctiveness, I use the following terminology. (A ╞ B means B is true in A.)
E,H ╞ F,G iff E ╞ F , H ╞ G
E,H ╞ F ∧ G iff E ╞ F and H ╞ G
E,H ╞ ¬F iff not Ε,Η ╞ F
E,H ╞ F → G iff Ε,Η ╞ ¬(F ∧ ¬G)
2.2.2 Similarity
After analysing the distinctiveness of e and h I go on investigating the e and h relation. Rather, I understand testing as including the condition of an evidenceformula F and a hypothesisformula G being similar in simple form and the same in logical form. By similar simple form I mean that the logically simple evidence claim F of a natural entity and the logically simple hypothesis claim G of that natural entity is similar. Same logical form means that F has the same logical structure as G. I set a terminology for this, as follows.
The indexed formulas F_{i} and G_{i} mean that F has similar simple form and same logical form as G.
2.2.3 Matching, procedure and equivalence
I also understand the notion of testing as including the matching condition. Matching is the combination of the condition of distinct evidence and hypothesis formulas with the condition of similar evidence and hypothesis formulas. Matching M is the conjunction of the distinct evidence and hypothesis formulas, having similar simple and same complex logical form.
M(F_{i},G_{i}) iff F_{i} ∧ G_{i}
To me, the notion of testing also includes a procedure showing how the test is about to be done — its procedure. In language, the procedure is defined as a logical construction of the involved matching predicates. That is, the logical construction depend upon which arguments the matching predicates have. For instance, the procedure of testing a conjunction formula is to first test one of the conjuncts and then to test the second conjunct.
Procedure of testing G_{i} ∧ G_{j} is the procedure of testing G_{i} and the procedure of testing G_{j}
Procedure of testing G_{i} ∨ G_{j} is either the procedure of testing G_{i} or the procedure of testing G_{j}
My final understanding of testing is that it involves an expression condition, inspired by the elegant natural formulation «evidence tests hypothesis». This natural expression form denotes any test procedure, indifferent of its logical construction. In formal language, this behaves much like a Frege predicate showing free variables within parentheses. Here, the predicate variables are instead exchanged with formulas. I believe there is a brilliant natural language focusing feature involved here which I extract for this article purpose as follows. The testpredicate T, having the simple argument F_{i} and G_{i}, defines as the simple test procedure M(F_{i},G_{i}).
T(F_{i},G_{i}) iff M(F_{i},G_{i})
Test procedures differ with the hypotheses being tested because the logical constructions of the hypotheses differ. This means that the logical properties of the test predicate T differ with the logical form of the arguments, as section 3.3 will show. This has the unexpected consequence that a system built on these premises include cases where two test predicates have logically equivalent formulas as arguments without the test predicates having the same logical properties. This phenomenon has special consequences for the wellknown raven paradox, as section 6 will show.
3. Formal suggestion
My basic aim in this article is to view the idea of empirical knowledge gaining in terms of confirmation and falsification, but give both those notions a base in the notion of testing. I will now try to fully formalise the notion of testing into a system and then derive the notion of confirmation and falsification from that system.
3.1 Simple test predicate rule
For starters, using natural deduction style, the section 2.2 formally includes below deduction.

E ├ F_{i} and H ├ G_{i} 
I∧ 
E,H ├ F_{i} ∧ G_{i} 
The formation rule of matching is as follows.
FM 
E,H ├ F_{i} ∧ G_{i} 

E,H ├ M(F_{i},G_{i}) 
Let the indexed formulas F_{i} and G_{i} be «the evidence claim F has similar simple form and same logical form as the hypothesis claim G». Let F_{i} be a simple formula in system E and let G_{i} be a simple formula in system H. Let M be matching. Let M(F_{i},G_{i}) be «matching M of F_{i} and G_{i} denotes conjunction formula F_{i} ∧ G_{i} focusing upon F_{i} and G_{i}». Read M(F_{i},G_{i}) as F_{i} matches G_{i}.
IM 
E,H ├ F_{i} ∧ G_{i} 

E,H ├ M(F_{i},G_{i}) 
EM 
E,H ├ M(F_{i},G_{i}) 

E,H ├ F_{i} ∧ G_{i} 
The section 2.2 also includes a way of expressing matching, formally expressed as the test predicate T(G_{i}). The test predicate T takes hypothesis formula G_{i}, the outstanding part of the formula M(F_{i},G_{i}), as argument.
FT 
E,H ├ M(F_{i},G_{i}) 

E,H ├ T(F_{i},G_{i}) 
F_{i} and G_{i} are simple formulas. M(F_{i},G_{i}) is a formula in the systems E,H. Let T be testing. Let T(F_{i},G_{i}) be «testing T of F_{i} and G_{i} denotes M(F_{i},G_{i}) focusing upon F_{i} and G_{i}». Read T(F_{i},G_{i}) as F_{i} tests G_{i}.
IT' 
E,H ├ M(F_{i},G_{i}) 

E,H ├ T `(G_{i}) 
Read T `(G_{i}) as test of G_{i}
I will now reformulate the rule IT ` to below rule for more natural reading of testing.
IT 
E,H ├ M(F_{i},G_{i}) 

E,H ├ T(F_{i},G_{i}) 
Read T(F_{i},G_{i}) as F_{i} tests G_{i}
ET 
E,H ├ T(F_{i},G_{i}) 

E,H ├ M(F_{i},G_{i}) 
3.2 Probability rule
Carnap viewed confirmation as being defined as probability, which can be seen in his quantitative concept «evidence supports hypothesis to some degree». (Carnap,1962) In this article I surely claim that probability is part of the quantitative empirical knowledge gaining intuition, but it does not coincide with the qualitative notion of confirmation. Along this line, I suggest the probability notion «the probability of evidence testing hypotheses», distinguishing testing from probability. Here, the quantitative empirical knowledge intuition passes over to the notion of probability, leaving testing a purely qualitative empirical knowledge intuition. In this way, I can nest probability formulas inside test formulas and vice versa as the following two examples will show. In the first example, I test T that at least one individual being a swan G_{i} and white G_{j}; T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}). (Section 3.3 shows this T conjunction case.) I also test precisely one thousand individuals being white; T(F_{k},G_{k}). This means two tested hypotheses. Now, the probability P of there being tested white individuals that also are tested as swans, is at least one of one thousand; P( T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j})  T(F_{k},G_{k}) ). (Below IP applied to 3.3 explains this formulation.) In this way, I talk about tested hypotheses, and probability applied to these. In the second example, I test the following probability hypothesis. At least one of one thousand whites is a swan. That is, P( G_{i} ∧ G_{j}  G_{k} ). To test this I need an evidence formula similar to the hypothesis. I use the evidence arguments of above T predicates, and apply probability to these. That is, P( F_{i} ∧ F_{j}  F_{k} ). Now, the evidence formula tests the similar hypothesis formula — T( P( F_{i} ∧ F_{j}  F_{k} ) , P( G_{i} ∧ G_{j}  G_{k} ) ). This shows the relationships between the two notions of testing and probability.
IP 
S ├ F 

S ├ P(F) 
, where P(F_{i} ∨ F_{j}) = P(F_{i}) + P(F_{j})  P(F_{i} ∧ F_{j}) and P(F_{i} ∧ F_{j}) = P(F_{i}  F_{j}) * P(F_{j}) and the conditional P(F_{i}  F_{j}) = P(F_{i} ∧ F_{j}) / P(F_{j}), as well as its converse that if P(F_{i}  F_{j}) then P(F_{j}  F_{i}) = P(F_{j} ∧ F_{i}) / P(F_{j} ∧ F_{i}) + P(¬F_{j} ∧ F_{i}).
3.3 Complex test predicates
The section 2.2 makes clear that the test predicates emulate the test part of the natural phenomenon of confirmation. The section 3.1 IT rule singles out the test predicates having simple arguments. Below will define the remaining test predicates, having logically complex evidence and hypotheses.
Τ∧ __Ε,Η ├ T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j})
__________⇔

Ε ├ F_{i} ∧ F_{j} , H ├ G_{i} ∧ G_{j} 
I∧ 
Ε,Η ├ (F_{i} ∧ F_{j}) ∧ (G_{i} ∧ G_{j}) 
IM 
Ε,Η ├ M(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) 
Τ→ Ε,Η ├ T(F_{i} → F_{j} , G_{i} → G_{j})
__________⇔

E ├ F_{i} , E ├ F_{i} → F_{j} 
H ├ G_{i} , H ├ G_{i} → G_{j} 
I∧ 
E,H ├ (F_{i} → F_{j}) ∧ (G_{i} → G_{j}) 
IM 
Ε,Η ├ M(F_{i} → F_{j} , G_{i} → G_{j}) 
Τ∨ ____Ε,Η ├ T(F_{i} ∨ F_{j} , G_{i} ∨ G_{j})
___________⇔

Either_Ε ├ F_{i} or Ε ├ F_{j} , either H ├ G_{i} or H ├ G_{j} 
I∨ 
Ε,Η ├ (F_{i} ∨ F_{j}) , (G_{i} ∨ G_{j}) 
I∧ 
Ε,Η ├ (F_{i} ∨ F_{j}) ∧ (G_{i} ∨ G_{j}) 
IM 
Ε,Η ├ M(F_{i} ∨ F_{j} , G_{i} ∨ G_{j}) 
Τ¬ _____Ε,Η ├ T(¬F_{i} , ¬G_{i})
_________⇔

Ε ├ F_{i} , H ├ G_{i} 

: , : 

⊥ , ⊥ 
I¬ 
Ε ├ ¬F_{i} , H ├ ¬G_{i} 
I∧ 
Ε,Η ├ ¬F_{i} ∧ ¬G_{i} 
IM 
Ε,Η ├ M(¬F_{i} , ¬G_{i}) 
Τ∀______E,H ├ T( ∀vF_{i}(v) , ∀uG_{i}(u) )
___________⇔

Ε ├ F_{i}(c) , H ├ G_{i}(d) 
I∀ 
Ε ├ ∀vF_{i}(v) , H ├ ∀uG_{i}(u) 
I∧ 
Ε,Η ├ ∀vF_{i}(v) ∧ ∀uG_{i}(u) 
IM 
E,H ├ M( ∀vF_{i}(v) , ∀uG_{i}(u) ) 
, for arbitrary c in E and arbitrary ├ in H.
Τ∃ _______E,H ├ T( ∃vF_{i}(v) , ∃uG_{i}(u) )
___________⇔

Ε ├ F_{i}(c) , H ├ G_{i}(d) 
I∃ 
Ε ├ ∃vF_{i}(v) , H ├ ∃uG_{i}(u) 
I∧ 
Ε,Η ├ ∃vF_{i}(v) ∧ ∃uG_{i}(u) 
IM 
E,H ├ M( ∃vF_{i}(v) , ∃uG_{i}(u) ) 
, for some c in E and some d in H.
The T disjunction case is exclusive disjunction. In the T quantification case, the variables are relative their E and H domains. The conjunction case for understanding above test predicates is as follows. Suppose the hypothesis conjunction and find an evidence conjunction matching the hypothesis. Introduce T to denote the performed logical steps and read it as «evidence tests hypothesis».
The system developed above has a special effect on the wellknown substitution principle. In that principle, any formula including A → B means the same as the formula instead including ¬B → ¬A. Now, check this in the system above. Suppose that the formula including A → B is above T implication case. The T implication case definition shows that T(A → B) includes A in separate. It is easy to see by the same T implication definition that T(¬B → ¬A) includes ¬B in separate. Now, take the three formulas A, A → B, and ¬B included in above two T predicates. These formulas show that a contradiction follows from claiming that ¬B → ¬A substitutes A → B in T. Therefore, the substitution principle does not hold for extensional logic with T predicates added.
3.4 Test predicate logical properties
I will now show some logical consequences of the 3.1 and 3.3 test predicates.
Simple F_{i} and G_{i} T(F_{i},G_{i}) ↔ (F_{i} ∧ G_{i})
Conjunction T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) ↔ ( T(F_{i},G_{i}) ∧ T(F_{j},G_{j}) )
Implication T(F_{i} → F_{j} , G_{i} → G_{j}) ↔ ( T(F_{i},G_{i}) ∧ G_{i} → G_{j} )
Disjunction T( F_{i} ∨ F_{j} , G_{i} ∨ G_{j} ) ↔ (either T(F_{i},G_{i}) or T(F_{j},G_{j}))
Contradiction ( T(¬F_{i},¬G_{i}) → T(⊥) ) → T(F_{i},G_{i})
Negation ( T(F_{i},G_{i}) → T(⊥) ) → T(¬F_{i},¬G_{i})
( T(F_{i},G_{i}) ∧ T(¬F_{i},¬G_{i}) ) → T(⊥)
All quantification T( ∀vF(v) , ∀uG(u) ) ↔ T( F(c) , G(d) ), for arbitrary c and d.
Existence quantification T( F_{i}(c) , G_{i}(d) ) → T( ∃vF_{i}(v) , ∃uG_{i}(u) ), for some c and d.
( ( ( T(F_{i}(c) , G_{i}(d)) → I ) → (T(∃vF_{i}(v) , ∃uG_{i}(u)) → I) ) ∧
( T(F_{i}(c) , G_{i}(d)) → T(∃vF_{i}(v) , ∃uG_{i}(u)) ) ) → I
Probability ( P( T(F_{i},G_{i}) ) ∧ (T(F_{i},G_{i}) ↔ T(F_{j},G_{j})) ) → P( T(F_{j},G_{j}) )
( T( P(A_{i}) ) ∧ P(A_{i}) = P(B_{i}) ) → T( P(B_{i}) )
With this formalism hanging over our heads, here is an intuitive example of understanding above formulas. T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) means that the evidence formula tests the hypothesis formula, following the conjunction test procedure (logical steps). The natural language reading of above conjunction formula is «testing a conjunction formula means that each conjunct tests in separate». I will not prove all these theorems due to the length of this paper. I will only prove the conjunction case.
Suppose 
Ε,Η ├ T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) 
(i) 
By 3.3 T∧ def. below (ii)(iii) 
Ε ├ F_{i} ∧ F_{j} , H ├ G_{i} ∧ G_{j} 
(ii) 
I∧ 
Ε,Η ├ (F_{i} ∧ F_{j}) ∧ (G_{i} ∧ G_{j}) 

IM 
Ε,Η ├ M(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) 
(iii) 
EM 
Ε,Η ├ (F_{i} ∧ F_{j}) ∧ (G_{i} ∧ G_{j}) 

E∧ 
Ε ├ F_{i} ∧ F_{j} , H ├ G_{i} ∧ G_{j} 

E∧ 
Ε ├ F_{i} , Ε ├ F_{j} , H ├ G_{i} , H ├ G_{j} 

Permutation 
Ε ├ F_{i} , H ├ G_{i} ,_ Ε ├ F_{j} , H ├ G_{j} 
(iv) 
I∧ 
Ε,Η ├ F_{i} ∧ G_{i} , E,H ├ G_{j} ∧ G_{j} 

IM 
Ε,Η ├ M(F_{i},G_{i}) , Ε,Η ├ M(F_{j},G_{j}) 
(v) 
By T def. above (iv)(v) 
Ε,Η ├ T(F_{i},G_{i}) , Ε,Η ├ T(F_{j},G_{j}) 

I∧ 
Ε,Η ├ T(F_{i},G_{i}) ∧ T(F_{j},G_{j}) 
(vi) 
I→_(i),(vi) 
Ε,Η ├ T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) → ( T(F_{i},G_{i}) ∧ T(F_{j},G_{j}) ) 

Suppose 
Ε,Η ├ T(F_{i},G_{i}) ∧ T(F_{j},G_{j}) 
(i) 
E∧ 
Ε,Η ├ T(F_{i},G_{i}) , Ε,Η ├ T(F_{j},G_{j}) 

By T∧ def. (ii)(iii) 
Ε ├ F_{i} , H ├ G_{i} , E ├ F_{j} , H ├ G_{j} 
(ii) 
I∧ 
Ε,Η ├ F_{i} ∧ G_{i} , E,H ├ F_{j} ∧ G_{j} 

IM 
Ε,Η ├ M(F_{i},G_{j}) , Ε,Η ├ M(F_{j},G_{j}) 
(iii) 
EM 
Ε,Η ├ F_{i} ∧ G_{i} , E,H ├ F_{j} ∧ G_{j} 

E∧ 
Ε ├ F_{i} , H ├ G_{i} , E ├ F_{j} , H ├ G_{j} 

Permutation 
Ε ├ F_{i} , E ├ F_{j} , H ├ G_{i} , H ├ G_{j} 
(iv) 
I∧ 
Ε,Η ├ F_{i} ∧ F_{j} , E,H ├ G_{i} ∧ G_{j} 

IM 
Ε,Η ├ M(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) 
(v) 
By T∧ def. (iv)(v) 
Ε,Η ├ T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) 
(vi) 
I→_(i),(vi) 
Ε,Η ├ ( T(F_{i},G_{i}) ∧ T(F_{j},G_{j}) ) → T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) 

3.5 Complex test predicate rules
Based on 3.3 and 3.4 it is now possible to form natural deduction rules for the complex T predicate. I show this for the conjunction case.
FT∧ 
Ε,Η ├ T(F_{i},G_{i}) ∧ T(F_{j},G_{j}) 

Ε,Η ├ T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) 
F_{i} is a formula in E and G_{i} is a formula in H. T(F_{i},G_{i}) is a formula in the systems E,H. Let T be testing. Read T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) as F_{i} ∧ F_{j} tests G_{i} ∧ G_{j}.
IT∧ 
Ε,Η ├ T(F_{i},G_{i}) ∧ T(F_{j},G_{j}) 

Ε,Η ├ T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) 
ET∧ 
Ε,Η ├ T(F_{i} ∧ F_{j} , G_{i} ∧ G_{j}) 

Ε,Η ├ T(F_{i},G_{i}) ∧ T(F_{j},G_{j}) 
3.6 Confirmation
Until now I have definied the notion of testing in a formal system. The remaining task is to derive the notion of confirmation and falsification in this system. I start with the notion of confirmation. In my view, confirmation includes more than the above test notion. Scientists often claim hypotheses that after some consideration are shown to partly base in evidence and partly base in their own intuition. Therefore, I view testing as the confirmation base step and complement it with an induction step that will deal with comparing the tested hypothesis part with the full hypothesis. In natural language this will sound like «the evidence F_{i} tests T the hypothesispart G_{i}« and «the full hypothesis G_{j} compares with G_{i}». This conjunction is then expressed by «F_{i} confirms G_{i}» using the natural language feature of focusing mentioned in section 2.2.3. Formally, the confirmation C base step is testing T the hypothesis part G_{i} using the evidence part F_{i}. The confirmation C induction step is a measure function P' of G_{i} and G_{j}.
IC 
Ε,Η ├ T(F_{i},G_{i}) ∧ P'(G_{i}G_{j}) 

Ε,Η ├ C(F_{i},G_{i}) 
, where P' is a measure tool used to measure the difference between G_{i} and G_{j}.
Read C(F_{i},G_{i}) as F_{i} confirms G_{i}
3.7 Falsification
Traditionally, confirmation is an empiricist term about hypothesissupport, while falsification is a rival rationalist term about hypothesisdenial. However, in my analysis both terms have testing T in common. In consequence, it seems that it is possible to define both these concepts without rival views emerging. Rather, they go side by side forming a threeunity with testing as base.
I view the natural notion of falsification as reductive testing. That is, falsification is a way of focus upon the evidence and hypothesis parts of the test procedure, where the evidence denies the hypothesis. Formally, this is formulated as follows as section 3.4 testpredicate negationintroduction.
IC 
E,H ├ ( T(F_{i},G_{i}) → T(⊥) )_ → T(¬F_{i},¬G_{i}) 

E,H ├ F(¬F_{i},G_{i}) 
, where F focuses upon ¬F_{i} and G_{i} in the first row formula
Read F(¬F_{i},G_{i}) as ¬F_{i} falsifies G_{i}
The first row antecedent includes that the test argument F_{i} implicates contradiction ⊥. Considering the T indexcondition, the same applies to G_{i}. By negationintroduction and testintroduction, the consequent T(¬F_{i},¬G_{i}) follows. (Accordingly, E and H revise to exclude F_{i} and G_{i}.) In the falsificationintroduction, the second row focuses upon the evidence and hypothesis parts in the first row. That is, the rule IF starts with a test procedure going wrong and ends up with focusing upon the evidence and hypothesis parts of that procedure. Falsification application is not used for hypothesis choice, but for evidence and hypothesis revision. Here is an example. If a claimed individual has certain properties, and some is falsified. Then, that individual is falsified.
4. Semantics
The language shown above has a reference as follows. Language L includes symbols for constants c, variables v, functions f, assignment function a, satisfication s, and logical operators ∧, ∨, →, ↔, ¬, ∀, ∃. U is the set of terms, that is constants C, variables V, and functions. S is semantics. M is model set.
4.1 Notation convention
Read the formalism f : A → B [X]
as function f from set A to set B such that condition(s) X
4.2 Assignment function
a : V → C [a(v_{i}) = c_{i}]
4.3 Interpretation function
i : C → S [(a),(b),(c)]
(a) i(c_{j}) = s_{j}
(b) i( f(c_{1},c_{2},…,c_{n}) ) = i( f(i(c_{1}),i(c_{2}),…,i(c_{n})) )
(c) i( f(v_{1},v_{2},…,v_{n}) ) = i( f(a(v_{1}),a(v_{2}),…,a(v_{n})) )
4.4 Satisfication
M ╞ c_{j} = c_{k} ⇔ i(c_{j}) = i(c_{k})
M ╞ v_{j} = v_{k} ⇔ a(v_{j}) = a(v_{k})
M ╞ f(t_{1},t_{2},…,t_{n}) ⇔ i( f(t_{1},t_{2},…,t_{n}) )
4.5 Logical operators
Logical definition over formula A. A is a Frege predicate, a test predicate, or a logical construction of these.
M ╞ A_{i} ∧ A_{j} iff M ╞ A_{i} and M ╞ A_{j}
M ╞ ¬A iff not M ╞ A
M ╞ A_{i} ∨ A_{j} iff either M ╞ A_{i} or M ╞ A_{j}
M ╞ A_{i} → A_{j} ⇔ M ╞ ¬(A_{i} ∧ ¬A_{j})
M ╞ ∀vA(v) iff for any v if v ∈ U then M ╞ A(v)[a(v)=c]
M ╞ ∃vA(v) ⇔ M ╞ ¬∀v¬A(v)
M ╞ T(A_{i},A_{j}) ⇔ M ╞ A, where T focuses upon A_{i} and A_{j} in A.
5. Application
The basic aim in this article is to contribute to understand the notion of empirical knowledge gaining. I have suggested that the traditional notions of confirmation and falsification can be somewhat modified and derived in an ordinary logical system complemented with the test rule. It is now time to show the strength of this system by defining stronger notions.
5.1 Evidencedevices
Scientists use instruments to get information out of the natural world by quantifying properties. I emulate these instruments as sets of evidence formulas, and name the set evidencedevices E.
Let E be the set of evidenceformulas
Let E_{j} ∈ ℘(Ε), where ℘ is poweroperator and E_{j} is evidencedevice.
So E ├ F_{i}(v) ⇒___∃E_{j}( E_{j} ├ F_{i}(v) )
Evidencedevices are language descriptions of the instruments used by scientist. Each such device is a language expression of what the instrument has registered. If there should be any use with the hypothesis testing theory in this article, a central task would be to paraphrase the scientific everyday instruments as evidencedevices. This is a vast task.
5.2 Hypothesisdevices
Scientists claim ideas in order to suggest naturalworld views. I emulate these ideas as sets of hypothesis formulas, and name the set hypothesisdevices H.
Let H be the set of hypothesisformulas
Let H_{j} ∈ ℘(Η), where ℘ is poweroperator and H_{j} is hypothesisdevice.
So H ├ G_{i}(u) ⇒___∃H_{j}( H_{j} ├ G_{i}(u) )
Hypothesisdevices would usually be associated with humans being smart enough to come up with something scientifically interesting. However, it would also be possible for machines to be hypothesisdevices.
5.3 Testdevices.
Testdevices T(E_{j},H_{j}) are evidence devices testing hypothesis devices. The testdevices show the proper test procedure for any hypothesis and evidence possible. This shows the strength of the theory, which would be able to guide any kind of empirical knowledge gaining activity. I call the following theorem hypothesistesting completeness.
Def. E_{j},H_{j} ├ T(E_{j},H_{j}) ⇔____ E_{j},H_{j} d_∀F_{i},∃G_{i} Τ(F_{i},G_{i})__∧__∀G_{i},∃F_{i} Τ(F_{i},G_{i})
The testdevices claim that every hypothesisdevice can be tested by some evidencedevice and every evidencedevice can test some hypothesisdevice. The proof is as follows. By the definition T(E_{j},Η_{j}) is the case. Let H_{j} be a simple formula. Then by the definition, there is a simple formula E_{j} testing H_{j}. Let E_{j} be a simple formula. Then by the definition, there is a simple formula H_{j} tested by E_{j}. Let H_{j} be a conjunction formula. Then by the definition, there is a conjunction formula E_{j} testing H_{j}. Let E_{j} be a conjunction formula. Then by the definition, there is a conjunction formula H_{j} tested by E_{j}. Let H_{j} be an implication formula. Then by the definition, there is an implication formula E_{j} testing H_{j}. Let E_{j} be an implication formula. Then by the definition, there is an implication formula H_{j} tested by E_{j}. Let H_{j} be an exclusive disjunction formula. Then by the definition, there is an exclusive disjunction formula E_{j} testing H_{j}. Let E_{j} be an exclusive disjunction formula. Then by the definition, there is an exclusive disjunction formula H_{j} tested by E_{j}. Let H_{j} be a negation formula. Then by the definition, there is a negation formula E_{j} testing H_{j}. Let E_{j} be a negation formula. Then by the definition, there is a negation formula H_{j} tested by E_{j}. Let H_{j} be an all quantification formula. Then by the definition, there is an all quantification formula E_{j} testing H_{j}. Let E_{j} be an all quantification formula. Then by the definition, there is an all quantification formula H_{j} tested by E_{j}. Let H_{j} be an existence quantification formula. Then by the definition, there is an existence quantification formula E_{j} testing H_{j}. Let E_{j} be an existence quantification formula. Then by the definition, there is an existence quantification formula H_{j} tested by E_{j}. So for any H_{j} there is an E_{j} testing H_{j} and for any E_{j} there is an H_{j} tested by E_{j}.
5.4 Empirical knowledgegaining programs
Scientists build up theories by putting ideas and information about the world together, perhaps letting the ideas going beyond the actual information at hand. I understand this phenomenon as confirmation, in terms of section 3.6. Applying the above notion of devices to this definition will result in the following formula. Example. Let H_{k} be Newton mechanics. Then there is a hypothesis part H_{j}, of H_{k}, being tested by its proper evidence E_{j}.
Ε,Η ├ C(E_{j},H_{j}) ⇔____Ε,Η ├ T(E_{j},Η_{j}) ∧ P'(Η_{j}H_{k})
Another knowledgegaining part is when scientists revise their theories according to hypothesis contradiction or falsification. Section 3.7 shows this. In the device manner, it is possible to formulate.
E,H ├ F(¬E_{j},H_{j})
Besides confirmation and falsification, scientists use probability. Section 3.2 shows how this works with the notion of testing, and therefore how it works with confirmation and falsification.
E,H ├ P( T(E_{j},H_{j})  T(E_{k},H_{k}) )
Finally, I define the strongest notion in this theory, called empirical knowledgegaining program. T(Q,M,A) focuses upon the reasoning characteristics of at least one of above three knowledgegaining parts. This focusing uses the same natural language principle claimed in above 2.2.3. Q is question, M is method, and A is answer. Example. Let T(Q,M,A) be T(E_{j},H_{j}). Q is the supposition of T(E_{j},H_{j}), M is the deduction, and A is the conclusion. I think this theorem shows the procedure of confirming any imaginable hypothesis.
6. Some brief ontologicalconsequential notes
6.1 Distinctiveness
Empirical knowledgegaining programs T(Q,M,A)_include the condition of evidence being distinct in kind from hypotheses. This condition challenges the traditional confirmation view that «hypotheses are based upon evidence and go beyond the evidence». In this view, «based upon» means referring to, and «going beyond» means that the hypothesis includes the evidence. In my view, «based upon» means that evidence tests a distinct hypothesis part, and «going beyond» means that the full hypothesis includes this hypothesis part. To me it seems that natural entities differ from those creatures observing those. Intelligent creatures make notes both by perception and understanding and forms two distinct evidence and hypothesis entities.
6.2 Similarity
T(Q,M,A) also includes the condition of evidence being similar to hypotheses. I use indexes like F_{i} and G_{i} to denote that F and G have similar simple and the same logical form. To me it seems that there is no point in confirming or falsifying a hypothesisclaim using an evidenceclaim not similar to it. The hypotheticaldeductional principle, (Popper,1972) includes the problem of identifying the hypothesis part being false out of a given falsification of the hypothesis. This problem partly includes the problem of correlating evidence with hypotheses. The T(Q,M,A) similarity condition directs evidence to its proper hypothesis, avoiding this part of the hypotheticaldeductional problem. However, the correspondence of evidence and hypotheses is not definite. Someone might suggest that evidence F is similar to hypothesis G, someone else might instead suggest that F' is similar to G. One such way of defining similarity for evidence and hypotheses results in one set of testpredicates. That is, one empirical knowledgegaining program. Using another similarity defined set of testpredicates ends up with some other empirical knowledgegaining program. These programs cannot compare properly, due to the different conventions or paradigms of similarity. Therefore, such empirical knowledge gaining programs are incommensurable. This is an application of the notion of paradigm. (Kuhn,1970) The consequence for the empirical knowledge gaining programs is that programs are true in their paradigms. A supported hypothesis is true if and only if it is confirmed in the section 3.6 formal sense, where the involved simple predicates are true in the supported paradigm. There are also other consequences of the notion of similarity. Relative verisimilitude is a relative way of approximating to truth presupposing observational nesting, using the idea that increasing theory truthcontents entails increasing observational success, predictive power. (NewtonSmith,1990) In my analysis, the natural relation of evidence and hypotheses makes an empirical request out of this thesis. This follows by its presupposition of truthvalues relative a way of defining test predicates, having its set of suggested evidence and hypothesis similarities. Going on; suggestions that scientific methods are no better than other knowledge gaining methods, (Feyerabend,1975) could be interpreted such that the formulas F and G cannot fulfil the index condition suggested in F_{i} and G_{i}.
6.3 Procedure
Evidence being distinct from, and similar to, hypotheses defines evidence matching hypotheses. Logical constructions of matching define the test procedures, as section 3.3 shows. Now, consider the raven paradox. (Hempel,1985) The premise of the paradox is «confirmation of all ravens are black» and paraphrases in my analysis as the formula T( ∀x(F_{i}(x) → F_{j}(x)) , ∀yG_{i}((y) → G_{j}(y)) ), where F_{i}(x) is the evidence that any individual is raven and F_{j}(x) is the evidence that any individual is black. G_{i}(x) is the hypothesis that any individual is raven and G_{j}(x) is the hypothesis that any individual is black. The conclusion of the paradox is «confirmation of all nonblack are nonravens» and is analogously paraphrased as above T formula, but with the two arguments in logical counter position. (My analysis of confirmation also includes probability, but this is not needed to comment the Hempel paradox. The second part P' of confirmation is not used in the paradox either.) The sections 3.3 and 3.6 show that the two C formulas define as two different test procedures. 3.3 shows that in this case it is not possible to substitute the arguments in T with logically equivalent arguments, due to the non equivalent logical constructions of matching predicates defining the T predicates. Therefore, the raven paradox is explained and avoided in this theory. Going on; the Carnap concept of confirmation includes the relevance concept, (Carnap,1962) involving terminology as positive, negative, and irrelevant confirmation. The T(Q,M,A) makes this concept pointless.
Empirical knowledgegaining programs T(Q,M,A)_view the test part of confirmation as focusing upon evidence and hypotheses being parts of test procedures. This view is the result of stressing the meaning and relation of evidence and hypotheses. (This might be somewhat related to the Keynes idea of explicating the confirmation evidence part.) One consequence of this ontological view is that it puts new challenging questions to notions like induction, confirmation and falsification. For instance, I see it as Carnap that confirmation involves a classificatory, a comparative, and a quantitative part. With the first part I mean «evidence confirms hypothesis» T(E,H). With the second and third part I mean «this evidence confirms this hypothesis better than that evidence confirms that hypothesis» T(E,H) ∧ T(E',H') ∧ P'(HH'), and «evidence confirms hypothesis to some degree» P( T(E_{j},H_{j})  T(E_{k},H_{k}) ). However in my account, the classificatory part does neither define the comparative nor the quantitative part. Another consequence example is induction, where there might be more to it than logical and psychological issues. A third effect is to question whether the rationalistic falsification problem can be solved. To me it seems that the traditional clash between empiricists and rationalists do not hold in the aspect of testing, as the empiricist hypothesissupport view and the rational hypothesisdenial rival view seem to converge in the same principle of testing T.
Finally, in my view empirical knowledgegaining does not aim at theory choice, but at the cultural use of tested hypotheses. I view T(Q,M,A) as a possible template of the empirical knowledgegaining method of hypothesis testing, complementing the traditional knowledgegaining method of theory proof.
7. Conclusion
Empirical knowledge gaining is traditionally viewed as being about confirmation and falsification. Those natural notions, in turn, come with suggested analyses, known to include both insights and paradoxes. I try to show that there is a third terminology about testing available. The essence of this testing terminology is to stress the ontology of confirmation and falsification and to clarify the meaning and relation of evidence and hypotheses. With this settled, the testing terminology tries to wedge itself into the traditional terminologies of confirmation and falsification. With this done, the testing terminology shows the ability to define strong theorems like empirical knowledge gaining programs. These programs are noninductive and correlative and aim to template some logical aspect of insightful everyday scientific empirical knowledgegaining in perhaps a less nonparadoxical way.
8. References
Carnap, R., Logical Foundations of Probability 2^{nd} ed., 1962, University of Chicago Press.
Carnap, R., Studies in Inductive Logic and Probability vol. I, 1971, University of Chicago Press.
Feyerabend, P.K., Against Method, 1975, London: New Left Books.
Frege, G., Begriffsschrift, 1879, Halle: Verlag Louis Nebert.
Goodman, N., Fact, Fiction, and Forecast, 1955, Cambridge Mass.
Hempel, C.G., Epist., Methodology, and Phil. of Science, 1985, Repr. from Erkenntniss, Vol. 22, Nos. 1, 2 and 3.
Hume, D., Treatise on Human Nature ed. by L.A. SelbyBigge, 1958, Clarendon Press.
Hume, D., Enquiry Concerning Human Understanding ed. by L.A. SelbyBigge, 1927, Oxford.
Keynes, J.M., A treatise on probability 2^{nd} ed., 1929, London and New York.
Kuhn, T.S., The Structure of Scientific Revolutions 2^{nd} ed., 1970, University of Chicago Press.
NewtonSmith, W.H., The Rationality of Science, 1990, Routledge.
Popper, K., Objective Knowledge, 1972, London: Oxford University Press.
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Mikael Eriksson
Karolinska Institutet
Stockholm
mikael.eriksson@ki.se