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Location: http://www.cs.mun.ca/~ulf/gloss/logic.html. By Ulf Schünemann since 2001. Please mail any comments.

1. Divisions of Logic
2. Philosophical Logic: Knowledge-(Expression)-Structure and -Inference
3. Aristotelean Logic: Comprehensive Study of the Forms of Knowledge
4. The Terms of Formal Logic
5. Inferential Reasoning

1. Formal logic studies which rules have to be used to get from true statements to true statements.
2. Methodology studies the reasoning according to logical laws, i.e., the use of those laws in different areas. Methodology studies the following methods:
   • direct methods:
     • the phenomenological method - used in (continental European) philosophy and humanities. It was mainly developed by Husserl (1859-1938): (1) Eidetic reduction is used to eliminate from intuition about some object everything subjective, all pre-knowledge (theories etc.), and all tradition. (2) Phenomenologic reduction is used to further abstract from the object's existence and from everything unessential.
   • indirect methods:
     • linguistic analysis (semantic method) - used by Russell, Gödel, Lukasiewicz, Tarski, et.al., in reasoning about languages, e.g., about the liar's paradox, etc.

       See	levels of semantics

     • inference:
       • The deductive method is used in the mathematics.
       • The reductive method is used in the empirical sciences:
         • inductive reduction - used in the `inductive sciences', esp., the natural sciences.
         • non-inductive reduction - used in historical sciences, and (branches of) geology, geography, astronomy, ...

       See	inferential reasoning.

3. Philosophy of logic studies the ontological status of logical laws (Are they linguistic, psychologic, or objective entities? Do they apply universally? Are they true?), what do ``for all'' and ``exists'' mean? ...

A possible division of the subject-matter of philosophical logic would be:

1. «Theories of reference[x] are concerned with the relationships between subpropositional or subsentential parts of thought or speech and their extra-mental or extra-linguistic objects - for instance, with the relationship between names[x] and things named, and with the relationship between predicates and the items to which they apply» [x].

   -> denotation of

   names

   ,

   predicates

   (in: logical parts of speech)

2. Theories of truth

   -> denotation of

   prepositions

   (in: logical parts of speech)

3. Analysis of complex propositions

   ->

   complex propositions

   (in: logical parts of speech)

4. Theories of modality[x], that is, accounts of such notions as necessity[x], possibility, and contingency, along with associated concepts such as that of analyticity.

   ->	de-re, de-dicto

5. «Finally, we come to questions concerning relations between propositions or sentences - relations such as those of entailment[x], presupposition, and confirmation[x] (or probabilistic support). Such relations are the subject-matter of the general theory of rational argument[x] or inference[x]»

   ->

   inference

1. Definition [in Aristotle's Categories]

«[T]he first operation begins with grasping the unity in things ... and culminates in the art of definition. Once we have grasped the definitum (the thing to be defined) in a unified way, we must see what differentiates it from other similar things. ... In differentiating the definitum from other similar things we will have to divide.»

«Good definition is the foundation for clear thinking since it tells us not only the kind of thing something is, but also how it is so in a distinct way.» «For the sake of defining well, ... we must know what kind of thing each thing is, hence we must see what distinguishes, separates, or divides, one species from all others within the same genus. The necessary tool in order to achive this is division. Division is first developed by Plato.»

->

Meta-models of things

->

categories of definitions

2. Predication or statement [in Aristotle's De Interpretatione]

«Once we have grasped adequately the notion of the subject we are speaking about, we can now begin to make a statement about it - an activity called predicating in the Aristotelian tradition.» «[T]he second operation creates statements in which a predicate is either affirmed or denied of a subject (i.e., composed with or divided from). The predicate may be said of the whole nature of the subject, or of some individuals in which that nature is found.»

3. Inference or argument [in the rest of Aristotle's logical works]

«The final operation of reason is concerned with the valid ordering of our statements for the sake of a conclusion. Validity and invalidity are properties of an argument as a wole, whereas truth and falsity are properties of individual statements. Of course, since the conclusion of an argument is also a statement, it too is either true or false. In the third operation Aristotelian logic focuses on deductive argument in the form of the syllogism.»

1. judgements (Frege), aka assertions (Russell)
   = the affirmation [meta language] of some proposition A [object language]

   Cf.	the meta level.

   
   = the premises and conclusions of logical inference
   - can be evident/proven/known, or not

   basic forms:	A is true	(e.g.	A & B is true	) ,	A is false	,	A is proposition
   formally:	|- A		|- A & B		?		|- A <>

   footnote: Ramsey's redundance theory of truth [x] claims that the truth-predicate 'is true' is superfluous, based on the (apparent) equivalence between asserting a proposition p and asserting p is true. «Difficulties appear to arise for the theory from cases in which propositions are said to be true even though the speaker may not know which propositions they are, and so cannot assert them himself, or when there are too many such propositions for each to be asserted individually, for example when someone claims 'Something that John said yesterday is true' or 'Everything asserted by a Cretan is true'» [x]

   [Löf]: «It seems to have been Bolzano who took the crucial step of replacing the Aristotelean forms of judgement by the single form

   A is,

   A is true,

   or

   A holds.

   In this, he was followed by Brentano, who also introduced the opposite form

   A is not,

   or

   A is false.

   and Frege.» Martin-Löf additionaly considers

   A prop.,

   or

   A is a proposition.

   «And through Frege's influence, the whole of modern logic has come to be based on the single form of `judgement', or `assertion', A is true.
   Once this step was taken, the question arose, What sort thing is it that is affirmed in an `affirmation' and denied in a `denial'? that is, What sort of thing is the A here? The isolation of this concept ... was a step which was entirely necessary for the development of modern logic. Modern logic simply would not work unless we had this concept, because it is on the things that fall under it that the logical operations operate.
   This new concept, which simply did not exist before the last century, was variously called. And since it was something that one had not met before. one had difficulties with what one should call it. Among the terms that were used, I think that the least committing one is ... `content of a judgement' ... Bolzano, who was the first to introduce this concept, called it `proposition in itself' ... Frege also grappled with this terminological problem. In Begriffsschrift, he called it `judgeable content' ...
   Now, Russell used the term `proposition' for this new notion, which has become the standard term in Anglo-Saxon philosophy and in modern logic. ... And [he chose] the word `assertion' rather than translate [Frege's] Urteil literally by `judgement'.»

   The terms `judgement'/`assertion' can refer to many things [Löf]:

   1. the act of judging/asserting (`judgement' in the traditional sense), or
   2. that which is judged/asserted, an ``object of knowledge'':
      1. A judgement before it is known / proven (traditionally called `enunciation'). E.g. |- "every even number is the sum of two primes" is not a `proposition'.
      2. A judgement which is actually known, an `evident' judgement (`proposition' in the traditional sense in logic):
         1. A judgement which is immediately evident; i.e., evident by itself, not through other judgements; i.e., evident by `intuitive' proof (`axiom').
         2. A judgement which is made evident mediately through some previously made evident judgements (`theorem' in mathematics); i.e., evident by `discursive' proof.
   See also hypothetical judgements below.

2. propositions (in the modern sense), aka statements
   = expressions/formulae combinable by logical operations (the connectives and the quantifiers) [Löf].
   - can be true, or false (in bi-valued logic)

   A

   or

   A & B

   or

   A -> B

   etc.

   Material implication: Statement A (materially) implies statement B, written A -> B, unless A is true and B is false. «One 'paradox' of material implication is that this relation holds between statements wholly unrelated in subject-matter: 'If Oxford is a city, then Italy is sunny'. Another 'paradox' is that the relation holds merely if p is false ('If pigs can fly, then ...') or merely if q is true ('If ... then Plato was a philosopher'). These are all ways in which material implication diverges from 'if ... then ...' as ordinarily used» [x].

3. a mode, aka rule of argument/inference/derivation in a proof [x]
   = a complex consisting of a set of premisses, i.e., the affirmations of some propositions A and of B, and a conclusion, i.e., the affirmation of some proposition like A & B.
   - can be good, or valid/truth preserving, or bad: «An argument is valid when its conclusion follows from its premisses (other descriptions are 'is deducible from' or 'is entailed by'). It can be a good argument even when not valid, if its premisses support its conclusion in some non-deductive way, for example inductively.» [x].

   For more see

   reasoning.

   		modus ponens		modus tollens		ex falso quodlibet
   |- A   |- B |- A & B		|- A -> B   |- A |- B		|- A -> B   |- ¬B |- ¬A		|- the unprovable |- A

   
   - ex falso quodlibet is valid because it can never be used in a proper proof (since the unprovable cannot be proven in a proper proof).

4. a step of argument/inference/derivation is the application of a mode/rule in the search for/construction of a proof.
   - A mode/rule can be used several ways to make a step. E.g., modus ponens can be used in either way of the implication for the tentative construction of a proof:
   • A progressive step constructs the proof from axioms to a thesis (`bottom-up'):

     step k:		: |- A -> B		: |- A		step k+1:		: |- A -> B		: |- A
     								|- B

   • A regressive step constructs the proof from the thesis to axioms (`top-down'):

     step k:			?			step k+1:		? |- A -> B		? |- A
     		|- B						|- B
     	:						:

5. a proof = an argument/inference/derivation ->

   Inferential Reasoning

   = a `tree' of truth-preserving arguments whose final conclusion, the thesis |- A, is the proved judgement and whose initial premisses (|- A₁, ..., |- A_n) are
   1. immedeately evident truths, i.e., axioms of the logical system (a categorial proof), or
   2. judgements (hypotheses) which are assumed to be evident/proven for the sake of argument (a hypothetical proof). «The notion of a hypothetical proof ... is explained by saying that it is a proof which, when supplemented by proofs of the hypotheses ... becomes a [categorial] proof of the thesis ...» [Löf].
   = that which makes a (hypothetical) judgement evident, IOW, turns an enunciation into a theorem (or `proposition' in the traditional sense) [Löf]. «For proving a conclusion you need more than a good argument to it. The premisses from which the proof starts must also be true (the word 'sound' is sometimes reserved for valid arguments with true premisses) and must be already 'given' - i.e. accepted or acceptable at a stage when the conclusion is not (you cannot, for example, prove a true conclusion from itself, even though you would be arguing soundly).» [x].

   |- A1		...		|- An
   :		:		:
   |- A

   Entailment: «A set of propositions (or statements, or sentences) entails a proposition (etc.) when the latter follows necessarily (logically, deductively) from the former, i.e. when an argument consisting of the former as premisses and the latter as conclusion is a valid deduction» [x].
   Proving with axioms vs. proving with hypotheses is the dividing line between axiomatic systems and natural deduction systems: «A natural deduction system has rules of inference, but no logical truths assumed axiomatically. A formula may be introduced into a derivation as a hypothesis at any stage» [x]. «In order to keep track of the hypotheses on which each line of a natural deduction derivation depends, these lines may be shown as sequents» [x]. See next:

   hypothetical judgement, logical consequence, sequent:
   = the assertion/affirmation/judgement that a consequent (A true) is entailed by certain antecedents (A₁ true, ..., A_n true)

   IF A1 true, ..., An true THEN A true

   formally,

   A1, ..., An |- A

   «It is the relation of logical consequence, which must be careful distinguished from implication» [Löf].

   The proof for a hypothetical judgement is the hypothetical proof whose conclusion (thesis) is the consequent and whose premisses (hypotheses) are the antecedents. «The difference between an inference and a logical consequence, or hypothetical judgement, is that an inference is a proof of a logical consequence ... [T]he difference is precisely that it is the presence of a proof of a logical consequence that turns its antecedents into presises and consequent into conclusion of the proof in question» [Löf]. If G |- A «occurs as a line of a correct derivation, the formula A will be a logical consequence of the formulae G.» [x].

1. Deduction uses a rule, called `modus ponens', of the following general form in proofs:

   	(S1)   If A is true then B is true
   And:	(S2)   A is true
   Therefore:	(S3)   B is true

   Deduction is independent of the meaning of statements S1, S2, S3. Deduction works for all statements of the corresponding form. Deductive proofs can be constructed by progressive or regressive use of modus ponens, or by a combination of both.

   Safe rule of logic =/= good rule of inference? «Some critics doubt whether, even if justified, the rules of logic are good rules of deductive inference. Modus ponens is a case in point. Asserting that it is always permissible to infer B from A and 'If A then B', Harman points out that although B is here implied, it would not be correct to accept B for any reasoner who came to notice that B was false» [x].

   Reduction/Induction. «Inductive inferences are those that project beyond the known data, as in the paradigm of generalizing that all emeralds are green» [x].
   Bertrand Russel about Isaac Newton's "Mathematical Foundations of Natural Sciences": «It embodies the scientific method in ideal form: From the observation of single facts one obtains by induction a general law, and by deduction one obtains other specific facts from the general law» [DvR].

   - The general Reduction (by Lukasiewicz?) is the derivation from B to A ``against'' the direction of the implication.

   	(S1)   If A is true then B is true
   And:	(S2)   B is true
   Therefore:	(S3)   A is true

   Analgously to modus ponens, this rule can be used progressively (`verification') to infer (S2) from (S3) via (S1), or regressively (`explanation') to infer (S3) from (S2) via (S1) [ZDM]. `Explanation' is not unfailable, it yields (S3) as a hypothesis. Hypothesis (S2) is confirmed/falsified by checking the consequence of (S3) (derived by progressive reduction, among them (S2)) whether in within the logical system (consistency), or empirically.

   - (Scientific) Induction (not mathematical = complete ``induction'', since that does not go beyond known data) is the special case of reduction where A is a generalization of B [ZDM]. This is used in natural sciences all the time. (Historical sciences do not strive for generalizations, but for explaining historic events - they use non-inductive reduction.) For example,

   	(S1)   If all ravens are black [is true] then all ravens we see are black [is true]
   And:	(S2)   all ravens we see are black [is true]
   Therefore:	(S3)   all ravens are black [is true]

   Natural laws are not verified (proven, deduced) but only shown to be true for more and more specific cases. Which makes them more and more certain. But a single counter-example (a white raven) can falsify the ``law''. What is required from a natural law (instead of verification) is that it explains (allows to deduce) correctly all the already known single cases and that makes (allows to deduce) a prediction on yet unknown cases.

   So by which rule can one select among similarly licenced inductions from available data to `all emeralds are green' and to `they are all green if observed before 1 January 2050, and blue thereafter'? By inference to the best explanation -> see abduction below.

   - Probabilistic Reasoning. Inductive reasoning is also thought to include probabilistic reasoning. It is said that an inference is justified if it conforms to the theorems of the probability calculus.

   Abduction «accepts a conclusion on the grounds that it explains the available evidence. The term was introduced by Charles Peirce to describe an inference pattern sometimes called 'hypothesis' or 'inference to the best explanation'. He used the example of arriving at a Turkish seaport and observing a man on horseback surrounded by horsemen holding a canopy over his head. He inferred that this was the governor of the province since he could think of no other figure who would be so greatly honoured. In his later work, Peirce used the word more widely» [x]. Abduction is recognized in two varieties:

   1. Inference to the best explanation (the more common definition both in philosophy and computing [Wikipedia]), ie. «justifying the postulation of unobservable phenomena on the strength of explanations they afford of observable phenomena» [x] «Accepting a statement because it is the best available explanation of one's evidence; deriving the conclusion that best explains one's premisses. ... [A]cceptable inductive inferences are all inferences to the best explanation» [x].
      «The semantics and the implementation of abduction cannot be reduced to those for deduction, as explanation cannot be reduced to implication.
      Applications include fault diagnosis, plan formation and default reasoning.
      Negation as failure in logic programming can both be given an abductive interpretation and also can be used to implement abduction. The abductive semantics of negation as failure leads naturally to an argumentation-theoretic interpretation of default reasoning in general.» [Wikipedia]
   2. Generic inference (as opposed to general, as in induction), or ``the generation of hypotheses to explain observations or conclusion'' [Wikipedia], ie. «the process of forming generic beliefs from known data. Observations incline us to think that tigers are four-legged, a proposition we hold true even upon discovery of a three-legged tiger. Generic sentences differ from general (i.e. universally quantified) sentences by their accommodation of negative instances, that is, of instances which would falsify general sentences» [x]