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Logic

Logic, from Classical Greek logos, originally meaning the word or what is spoken (but coming to mean thought or reasoning) is generally held to consist of the systematic study of the form of valid arguments. A valid argument form is one whose conclusion cannot be false if the premisses are true. The form of an argument is a schematic way of representing what is common to all arguments of that type.

There is no universal agreement as to the exact scope and subject matter of Logic, but it has traditionally included the classification of arguments, the systematic exposition of the 'logical form' common to all valid arguments, and the study of fallacies and paradoxes.

Division of Logic

Logic has traditionally been divided as follows.

1. Into the study of arguments capable of rigorous demonstration, to which the concept of 'validity' genuinely applies; and the study and exposition of the principles underlying arguments which are not necessarily demonstratively sound, but which are persuasive, or which are likely to give an advantage against an opponent). The latter includes rhetoric. A further discipline known as critical thinking, is taught in many American Universities.

2. Arguments capable of demonstration are further sub-divided into inductive and deductive arguments. The present article deals with deductive arguments only.

3. Deductive reasoning is divided into that which can be formalized by standard predicate logic (especially so-called first-order logic) and that which apparently cannot. The latter is the province of philosophical logic.

4. Types of Predicate logic . Computation.

Logical Form

Logic is formal, in that it aims to analyse and represent the form (or logical form) of any valid argument type. The form of an argument is displayed by representing its sentences using the formal grammar and symbolism of a logical system in such a way as to display its similarity with all other arguments of the same type.

This is known as showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a considerable variety of form and complexity. It requires, first, ignoring those grammatical features which are irrelevant to logic (such as gender, declension (if the argument is in Latin), replacing conjunctions which are not relevant to logic (such as 'but') with logical conjunctions like 'and' and replacing ambiguous or alternative logical expressions ('any', 'every' &c) with expressions of a standard type (such as 'all', or the universal quantifier A).

Second, certain parts of the sentence must be replaced with schematic letters. Thus, for example, the expression 'all A's are B's' shows the logical form which is common to the sentences 'all men are mortals', 'all cats are carnivores', 'all Greeks are philosophers' and so on.

That the concept of form is fundamental to logic was already recognized in ancient times. Aristotle uses variable letters to represent valid inferences the Prior Analytics. (For which reason Łukasiewicz says that the introduction of variables was 'one of Aristotle's greatest inventions'). According to the followers of Aristotle (such as Ammonius), only the logical principles stated in schematic terms belong to logic, and not those given in concrete terms. The concrete terms 'man', 'mortal' &c are analogous to the substitution values of the schematic placeholders 'A', 'B', 'C', which were called the 'matter' (Greek 'hyle') of the inference.

The fundamental difference between modern formal logic and traditional or Aristotelian logic lies in their differing analysis of the logical form of the sentences they treat.

• On the traditional view, the form of the sentence consists of (1) a subject (e.g. 'man') plus a sign of quantity ('all' or 'some' or 'no'); (2) the copula which is of the form 'is' or 'is not'; (3) a predicate (e.g. 'mortal'). Thus: all men are mortal. The logical constants such as 'all', 'no' and so on, plus sentential connectives such as 'and' and 'or' were called 'syncategorematic' terms (from the Greek 'kategorei' – to predicate, and 'syn' – together with. This is a fixed scheme, where each judgement has an identified quantity and copula, determining the logical form of the sentence.
• According to the modern view, the fundamental form of a simple sentence is given by a recursive schema, involving logical connectives, such as a quantifier with its bound variable, which are joined to by juxtaposition to other sentences, which in turn may have logical structure.
• The modern view is more complex, since a single judgement of Aristotle's system will involve two or more logical connectives. For example, the sentence "All men are mortal" involves in term logic two non-logical terms "is a man" (here M) and "is mortal" (here D): the sentence is given by the judgement A(M,D). In predicate logic the sentence involves the same two non-logical concepts, here analysed as $m(x)$ and $d(x)$, and the sentence is given by $\forall x.(m(x)\rightarrow d(x))$, involving the logical connectives for universal quantification and implication.
• But equally, the modern view is more powerful: medieval logicians recognised the problem of multiple generality, where Aristotelean logic is unable to satisfactorily render such sentences as "Some guys have all the luck", because both quantities "all" and "some" may be relevant in an inference, but the fixed scheme that Aristotle used allows only one to govern the inference. Just as linguists recognise recursive structure in natural languages, it appears that logic needs recursive structure.

Semantics

Main article: semantics of logic

The validity of an argument depends upon the meaning or semantics of the sentences that make it up, and so logic cannot completely avoid the need to provide some treatment of semantics.

Aristotle's Organon, especially On Interpretation, gives a cursory outline of semantics which the scholastic logicians, particularly in the thirteenth and fourteenth century, developed into a complex and sophisticated theory, called Supposition Theory. This showed how the truth of simple sentences, expressed schematically, depend on how the terms 'supposit' or stand for certain extra-linguistic items. For example, in book II of his Summa Logicae, William of Ockham presents a comprehensive account of the necessary and sufficient conditions for the truth of simple sentences, in order to show which arguments are valid and which are not. Thus 'every A is B' is true if and only if there is something for which 'A' stands for, and there is nothing for which 'A' stands for, which 'B' does not also stand for.

During the decline of scholasticism in the fifteenth and sixteenth centuries, these insights were lost. Early modern logic defined semantics purely as a relation between ideas. Antoine Arnauld in the Port Royal Logic, says that 'after conceiving things by our ideas, we compare these ideas, and, finding that some belong together and some do not, we unite or separate them. This is called affirming or denying, and in general judging. [Buroker p. 82]. Thus truth and falsity are no more than the agreement or disagreement of ideas. This suggests obvious difficulties, leading Locke to distinguish between 'real' truth, when our ideas have 'real existence' and 'imaginary' or 'verbal' truth, where ideas like harpies or centaurs exist only in the mind (Essay IV. v. 1-8). This view (psychologism) was taken to the extreme in the nineteenth century, and is generally held by modern logicians to signify a low point in the decline of logic before the twentieth century.

Modern semantics is in some ways closer to the medieval view, in rejecting such psychological truth-conditions. However, the introduction of quantification, needed to solve the problem of multiple generality, rendered impossible the kind of subject-predicate analysis that underlies medieval semantics. The main modern approach is model-theoretic semantics, based on Alfred Tarski's semantic theory of truth. The approach assumes that the meaning of the various parts of the propositions are given by the possible ways we can give a recursively specified group of interpretation functions from them to some predefined mathematical domains: an interpretation of first-order predicate logic is given by a mapping from terms to a universe of individuals, and a mapping from propositions to the truth values "true" and "false". Model-theoretic semantics is one of the fundamental concepts of model theory.

Inference

Inference is not to be confused with implication. An implication is a sentence of the form 'If p then q', and can be true or false. The Stoic logician Philo of Megara was the first to define the truth conditions of such an implication: false only when the antecedent p is true and the consequent q is false, in all other cases true. An inference, on the other hand, consists of two separately asserted propositions of the form 'p therefore q'. An inference is not true or false, but valid or invalid. However, there is a connection between implication and inference, as follows: if the implication 'if p then q' is true, the inference 'p therefore q' is valid. This was given an apparently paradoxical formulation by Philo, who said that the implication 'if it is day, it is night' is true only at night, so the inference 'it is day, therefore it is night' is valid in the night, but not in the day.

The theory of inference (or 'consequences') was systematically developed in medieval times by logicians such as William of Ockham and Walter Burley. It is uniquely medieval, though it has its origins in Aristotle's Topics and Boethius' De Syllogismis hypotheticis. This is why many terms in logic are Latin. For example, the rule that licenses the move from the implication 'if p then q' plus the assertion of its antecedent p, to the assertion of the consequent q is known as modus ponens (or 'mode of positing'). Its Latin formulation is 'Posito antecedente ponitur consequens'. The Latin formulations of many other rules such as 'ex falso quodlibet' (anything follows from a falsehood), 'reductio ad absurdum' (disproof by showing the consequence is absurd) also date from this period.

However, the theory of consequences, or of the so-called 'hypothetical syllogism' was never fully integrated into the theory of the 'categorical syllogism'. This was partly because of the resistance to reducing the categorical judgment 'Every S is P' to the so-called hypothetical judgment 'if anything is S, it is P'. The first was thought to imply 'some S is P', the second was not, and as late as 1911 in the Encyclopedia Britannica article on Logic, we find the Oxford logician T.H. Case arguing against Sigwart's and Brentano's modern analysis of the universal proposition. Cf. problem of existential import

[... Charles can you take over from the year 1910?]

Rival Conceptions of Logic

In the periodic of scholastic philosophy, logic was predominantly Aristotelian. Following the decline of scholasticism, logic was thought of as an affair of ideas by early modern philosophers such as Locke and Hume . Immanuel Kant took this one step further. He begins with the assumption of the empiricist philosophers, that all knowledge whatsoever is internal to the mind, and that we have no genuine knowledge of 'things in themselves'. Furthermore, (an idea he seemed to have got from Hume) the material of knowledge is a succession of separate ideas which have no intrinsic connection and thus no real unity. In order that these disparate sensations be brought into some sort of order and coherence, there must be an internal mechanism in the mind which provides the forms by which we think, perceive and reason.

Kant calls these forms Categories (in a somewhat different sense than employed by the Aristotelian logicians), of which he claims there are twelve:

• Quantity (Singular, Particular, Universal)
• Quality (Affirmative, Negative, Infinite)
• Relation (Categorical, Hypothetical, Disjunctive)
• Modality (Problematic, Assertoric, Apodictic)

However, this seems to be an arbitrary arrangement, driven by the desire to present a harmonious appearance than from any underlying method or system. For example, the triple nature of each division forced him to add artificial categories such as the infinite judgment.

This conception of logic eventually developed into an extreme form of psychologism espoused in the nineteenth by Benno Erdmann and others. The view of historians of logic is that Kant's influence was negative.

Another view of logic espoused by Hegel and others of his school (such as Lotze, Bradley, Bosanquet and others, was the 'Logic of the Pure Idea'. The central feature of this view is the identification of Logic and Metaphysics. The Universe has its origin in the categories of thought. Thought in its fullest development becomes the Absolute Idea, a divine mind evolving itself in the development of the Universe.