The Concept of Logical Consequence: An Introduction to Philosophical Logic
2010th Vol. 7
Author: Matthew W. McKeon
Peter Lang Publishing, Inc., New York
“Truth in a structure” is an intense discussion in the concept of ‘logical model’ through a scheme of ‘non-logical language terminology’.[i] By having introduced these terms (logical model and non-logical language terminology), McKeon commences his explanation to guide his readers to the general characteristic question of the concept of logical consequence. It is notable the term of ‘non-logical language terminology’ and ‘logical model’ genuinely refer to Tarski in “On the Concept of Logical Consequence”.[ii] Tarski uses it for designating the relation dividing the distance between logical and extra-logical domain.
McKeon, as opposed to the Tarski’s opposition, explains the distance existing between logical and extra-logical domain as not arbitrary since the relationship does exist. According to Tarski, the relation lies in the concept of satisfaction.[iii] McKeon justifies the idea by assuming a sentence is satisfied if a non-logical language terminology meets in a simple relation, namely a model; conversely, the model should also be defined in the concept of satisfaction. Nevertheless, the above relation can only be fulfilled at the level of necessary and formal consequences.
Necessary and formal consequences are the key to understanding the concept of logical consequence. Unfortunately, McKeon does not specifically explain both these terms. By contrast, he gives unclear examples regarding a statement cannot be categorized logically necessary if there is an interpretation against the term (1) ‘Hillary Clinton is not the Vice President’ is true; or ‘Barack Obama is President of the United States’ is wrong. He considers the consequences of this statement is necessary if ‘Barack Obama is President of the United States’ is true, and then the ‘Hillary Clinton is not the Vice President’ is true, and then explains this supposition as contradictory since a formal consequence scenario falls into the material consequences. This strategy clearly violates the basic principles of the concept of logical consequence oriented towards formal consequences.
The question arises, how to determine statements which satisfy the conditions of necessary? Considering McKeon does not definitively explain the problem of necessary, I will refer to G. Y. Sher in “Did Tarski Commit ‘Tarski’s Fallacy’?”[iv] to compare. Sher explains that in order to determine the conditions of necessary, a formal condition, or the nature of formal, or formally necessary is required. The formal nature of the concept of logical consequence is reflected in (i) the choice of logical terms, and (II) the construction of models.
Here are the succinct explanations. If a non-logical language terminology can only be definitive in the framework of the model, then a strict logical model can be expressed by the relation functions. X is the logical consequence of K, denoted f (K) R f (X), in which the function (f) (based on the logical schema) selects what the sentences X and theory K say about the object. On the other hand, the R stands for the formal general relation. McKeon, refer to Tarski, says we cannot indicate the logical consequence of K╞ X to be influenced by empirical knowledge since the reduction of material consequence to the logical consequence treating all statements as the logical thing would be problematic. The material consequences do not reflect the logical necessity.[v]
The above statement apparently distinguishes a posteriori and a priori domain, and, undoubtedly, invites numerous rejections among logicians, who one of the them is the persistent opponents of Tarski’s, J. Echenmendy. In 1990, Echenmendy published a book entitled “The Concept of Logical Consequence”, which primarily attacks the Tarski’s definition of logical consequence. He considers that Tarski’s has plunged into a fallacy. The allegations is based on the assumption if Tarski claimes the first adequacy condition [vi](abbreviated as FAC) is able adequately to prove a definition that describe the intuitive notion, it can be compared, if the negation of “all sentences K is correct → X is true” is true, just by simply stating “all sentences K ‘true’ is true and X ‘actually’ is wrong. ” Therefore, if claims of the FAC, such as the definition of K╞ X confronted in such cases, it would produce contradictory consequences.
To that end, how to resolve the weaknesses of the first adequacy condition of the logical model through schemes of non-logical language terminology? This issue is always attributed to the criterion of a prioricity, or an a priori logical consequence proved in a priori way. The criteria of a prioricity is the epistemological foundation of all the underlying of the characteristic of the logical consequences’ relations as necessary and formal relational consequences. Consequently, when we reject the building of the epistemology of a prioricity, relation FAC claims to describe the intuitive notion will again stuck in the ontological grounding.[vii]
However, the question with respect to the Tarski’s fallacy actually has been repudiated by Sher (1996). He retains the concept of necessary consequences and formal consequences as an adequate basis to proof the intuitive notion. He then concludes that the FAC actually does not contain contradictions. Different solutions come from McKeon. First of all, Mckeon receives a knowledge only based on the characteristics of empirical knowledge. From the empirical, he grounds his epistemic building in two steps as follows: firstly the a priori knowledge is knowledge that has the value of truth, providing an understanding of the interrelated terms without having to refer to empirical experience. Vice versa, secondly, the a posteriori knowledge should require the direct validity of empirical experience.
The empirical division McKeon made, a priori and a posteriori, basically remains on the empirical thesis. So, in the case we are allowed to reject a prioricity criteria, and only recognize the characteristics of the a posteriori one, do not we cancel the status of satisfaction itself? If not, what is the consequence of necessary which may be maintained in the status of satisfaction providing that it does not rely on the formal consequences? Can necessary consequences guarantee logical consequence? Could it reconsider its relation to the substitutional definition? Or is it re-definition present form of the proof-theoretic definition? How to treat a logical model if individual identities do not have a formal structure, even the formal identity of the object?
The questions can be expanded further to the rules of deductive system, the structure of U, M domain, neutral topic, individual constant, constant logical, ZFC, function objects, relationships, and so on. In short, McKeon, in his book, gives us a brief summaries as regards the basis for the concept related to the implementation of logical consequence (although some caveats need to be added). That is to say, at least, we can keep track down the standard process of logical consequence.
(1) Relationship between logical and extra-logical proposition.
(2) Topics neutral
(3) Logical Constant
(4) Logical Model scheme, also called Schema-T.
It means, supposedly, there is a relation between logical and extra-logical in the neutral topic. Since the topic neutral play a significant role in the language as the vaguest devices, then its task is to provide definitive clues. The definitive requirement is needed to be justified to the logical sentential form, and then rigidly arranged to be logical constants. With this in mind, we might obtain a model in the form of a logical scheme, i.e. schema-T which becomes the basic reference for the concept of logical consequence.[viii]
[i] McKeon conteds that the model in the logical schema is decomposition’s structure of the sentence that accepts a truth-value. Because if a sentence (through non-logical language terminology) can explain the object variable, then the interpretation of the truth value is depicted in the model structure of the sentence. Therefore, the ability of a structure to load the value of truth is the basis for clarifying how a sentence can accept this form of truth-value (s).
[ii] Tarski, A., “On the Concept of Logical Consequence” (1963) 409-420. Also, translated as “On the Concept of Following logically”, trans. by M. Stroińska and D. Hitchcock (2002), History and Philosophy of Logic 23, 155-196.
[iii] The concept of satisfactions is the concept of truth in formalized language. Formalized language must meet the logical standard, as the standard of modern logic which recognizes when X is the logical consequence of the class K, then it is not possible to interpret the language of the non-logical terminology that sentence is true and X K wrong. Consistency is needed that, X is the logical consequence of K if and only if X is able to be deduced from K.
[iv] GY Sher, “Did Tarski Commit” Tarski’s Fallacy “?”, in The Journal of Symbolic Logic 61, no. 2, (1996), 653-686.
[v] Tarski explained that the consequence relation cannot be affected by replacing designations of the objects Referred to in these sentences by the designations of any other objects. Tarski, “On the Concept…”, 414-415.
[vi] The concept of logical consequence of necessary and formal plays an important role in the definition of the first adequacy conditions. A definition of logical consequence is said to be adequate only if it produces a necessary and formal consequences, means K╞ X.
[vii] Ontological grounding is the main problem of logic in determining the condition of the truth value of an reffered object and the specific term referring to them. This debate was triggered because of differences in strategy or scheme to find a logical structure, namely the relationship between language as a logical schema and the actual world as non-logical language terminology. Two major schemes can be mentioned here, the first metaphysical conception including the study of representational and inflationary, and the second conception of linguistics including the study of interpretational and deflationary.
[viii] Tarski shows an insufficiencycy in proof-theoretic definition, and proceeds to put two conditions on the adequate definition of logical intuitive. Tarski then considers the substitutional definition of logical consequence, but in fact the definition of substitution is not sufficient. At this point Tarski states that we should find new ways (non-syntax) to express the intuitive condition; suggesting that “every means provided by semantics.”