Logic and Reasoning
«Logic is pure form, and ontology is the content.
Without ontology, logic says nothing about anything.
Without logic, ontology can only be analyzed, represented, and discussed in
vague generalities» [TOC 669].|
-> see ontological commitment.
[Löf] Per Martin-Löf: On the Meaning of the Logical Constants and the Justifications of the Logical Laws;
3-10 in Nordic Journal of Philosophical Logic 1/1; 1996.
- Divisions of Logic
- Philosophical Logic: Knowledge-(Expression)-Structure and -Inference
- Aristotelean Logic: Comprehensive Study of the Forms of Knowledge
- The Terms of Formal Logic
- Inferential Reasoning
[DvR] Walter Robert Fuchs: Denkspiele vom Reißbrett - eine Einführung in die moderne Philosophie; Droemer Knaur 1972.
[ZDM] I M Boche'nski: Die zeitgenössischen Denkmethoden (6th ed.); UTB/Francke 1975. (first edition 1954).
Logic, as the study of reasoning
(cf. the traditional broader philosophical logic),
has three parts [ZDM]:
- Formal logic studies which rules have to be used
to get from true statements to true statements.
- 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.
- 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, ...
- 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? ...
Loosely described, philosophical logic is
«the philosophical elucidation of those notions
that are indispensable for the proper characterization of
rational thought and its contents - notions like those of reference,
predication, truth, negation, necessity, definition, and entailment.
These and related notions are needed in order to give adequate
accounts of the structure of thoughts - particularly as expressed in
language - and of the relationships in which thoughts stand both to
one another and to objects and states of affairs in the world»
«But it must be emphasized that philosophical logic is not concerned with
thought inasmuch as the latter is a psychological process, but only in
so far as thoughts have contents which are assessable as true or
false. To conflate these concerns is to fall into the error of
much decried by Frege.»
A possible division of the subject-matter of philosophical logic would be:
«The order of topics in the list reflects a general progression
from the study of parts of propositions[x]
through the study of whole and compound propositions,
to the study of relations between propositions»
- «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»
- Theories of truth
- Analysis of complex propositions
- Theories of modality[x],
that is, accounts of such notions as
possibility, and contingency,
along with associated concepts such as that of analyticity.
- «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]
«Aristotle's logical works are commonly referred to as the Organon ...
The Organon is divided by Thomas Aquinas according to the three operations
of our reason in its activity of coming to know about things.» [AUOOP]
- 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.»
- 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.»
In logic, certain distinctions are important to make
(which are not normally made in everyday life / ordinay language):
proposition vs. judgement, axiom vs. theorem,
implication vs. entailment/consequence vs. inference.
explained the use of the terms `proposition', `judgement', `assertion'
«The word `proposition' ... comes from Aristiotle
and has dominated the logical tradition, wheras the word `theorem' ...
is in Euclid, I believe, and has dominated the mathematical tradition.
With Kant, something important happened, namely that the term `judgement'
... came to be used instead of `proposition'.
The term `judgement' also has a long history. ...
The traditional way of relating the notions of `judgement' and `proposition'
was by saying that `proposition is the verbal expression of a judgement'.
... Now this means that [with Kant] the term `judgement' acquired a double meaning.»
- judgements (Frege), aka assertions (Russell)
= the affirmation [meta language] of some proposition A [object language]
= the premises and conclusions of logical inference
- can be evident/proven/known, or not
|| A is true || (e.g.
|| A & B is true ||) ,
|| A is false ||,
|| A is proposition
|| |- A ||
|| |- A & B ||
|| ? ||
|| |- A <>
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'»
[Löf]: «It seems to have been Bolzano who took the crucial step of replacing
the Aristotelean forms of judgement
by the single form
In this, he was followed by Brentano, who also introduced the opposite form
Martin-Löf additionaly considers
| A is,
|| A is true,
||or|| A holds.
«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.
| A prop.,
||or|| A is a proposition.
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]:
See also hypothetical judgements below.
- the act of judging/asserting (`judgement' in the traditional sense),
- that which is judged/asserted, an ``object of knowledge'':
- 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'.
- A judgement which is actually known, an `evident' judgement
(`proposition' in the traditional sense in logic):
- A judgement which is immediately evident;
i.e., evident by itself, not through other judgements;
i.e., evident by `intuitive' proof (`axiom').
- A judgement which is made evident
mediately through some previously made evident judgements
(`theorem' in mathematics);
i.e., evident by `discursive' proof.
- propositions (in the modern sense), aka statements
= expressions/formulae combinable by logical operations (the connectives and the quantifiers)
- can be true, or false (in bi-valued logic)
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»
|| A & B
|| A -> B
- a mode, aka rule of argument/inference/derivation in a proof
= 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.»
| || ||modus ponens
|| ||modus tollens
|| ||ex falso quodlibet
||- A |- B
|- A & B
||- A -> B |- A
||- A -> B |- ¬B
||- the unprovable
- 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).
- 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
|| : |
|| step k+1:
|| : |
|- A -> B
|| : |
- A regressive step constructs the proof from the thesis to axioms (`top-down'):
| step k:
|| step k+1:
|| ? |
|- A -> B
|| ? |
= a `tree' of truth-preserving arguments whose final conclusion, the thesis |- A, is the proved judgement
and whose initial premisses (|- A1, ..., |- An)
= that which makes a (hypothetical) judgement evident, IOW,
turns an enunciation into a theorem (or `proposition' in the traditional sense)
«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
- immedeately evident truths, i.e., axioms of the logical system (a categorial proof), or
- 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].
«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»
| |- A1
|| |- An
| |- A
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»
- hypothetical judgement, logical consequence, sequent:
= the assertion/affirmation/judgement that
a consequent (A true) is entailed
by certain antecedents (A1 true, ..., An true)
«It is the relation of logical consequence,
which must be careful distinguished from implication» [Löf].
| IF A1 true, ..., An true THEN A true
|| A1, ..., An |- A
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.»
= necessity OR entailment + relevance = necessity?
«Some theorists [of inference] regard entailment as analysable in
terms of the modal notion of logical necessity - holding that a
proposition p entails a proposition q
just in case the conjunction of p and the negation of
q is logically impossible.
This view, however, has the queer consequence
that a contradiction entails any proposition whatever, whence it is
rejected by philosophers who insist that there must be a 'relevant
connection' between a proposition and any proposition which it can be
said to entail. (Relevance logic[x].)»
«The notion of presupposition, though widely appealed to by philosophers,
is difficult to distinguish precisely from that of entailment,
but according to one line of thought a statement S presupposes a
statement T just in case S fails to be either true or false
unless T is true. For instance, the statement that the present
King of France is bald might be said to presuppose, in this sense,
that France currently has a male monarch. (Such an approach obviously
requires some restriction to be placed on the principle of bivalence.)»
«As for the notion of confirmation, understood as a relation between
propositions licensing some form of non-demonstrative inference (such
as an inference to the truth of an empirical generalization[x]
from the truth of observation statements in agreement with it), this is
widely supposed to be explicable in terms of the theory of
- though precisely how the notion of probability should itself be
interpreted is still a matter of widespread controversy»
Inference comes in three modes of inference [x]:
Inference = «upgrading or adjustment of belief in the light
of the play of new information upon current beliefs»
- Deduction uses a rule, called `modus ponens',
of the following general form in proofs:
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.
| ||(S1) If A is true then B is true
|And: ||(S2) A is true
|Therefore: ||(S3) B is true
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].
«Inductive inferences are those that project beyond the known data,
as in the paradigm of generalizing that all emeralds are green»
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.
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.
| ||(S1) If A is true then B is true
|And: ||(S2) B is true
|Therefore: ||(S3) A is true
- (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.)
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.
| ||(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]
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.
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
Abduction is recognized in two varieties:
- Inference to the best explanation
(the more common definition both in philosophy and computing
«justifying the postulation of unobservable phenomena on the strength of
explanations they afford of observable phenomena»
«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»
«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.»
- Generic inference (as opposed to general, as in induction),
or ``the generation of hypotheses to explain observations or conclusion''
«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»
Ulf Schünemann 210901-171201