Quantifiers Are Logical Constants, but Only Ambiguously

2015 ◽  
pp. 51-70
Author(s):  
Sun-Joo Shin
Keyword(s):  
Logical Constants

ON THE NOTION OF CANONICAL DERIVATIONS FROM OPEN ASSUMPTIONS AND ITS ROLE IN PROOF-THEORETIC SEMANTICS

2015 ◽  
Vol 8 (2) ◽  
pp. 296-305 ◽  
Author(s):  
NISSIM FRANCEZ
Keyword(s):  
Natural Deduction ◽  
Proof System ◽  
Logical Constants ◽  
Local Completeness ◽  
Definition Of ◽  
The Impact

AbstractThe paper proposes an extension of the definition of a canonical proof, central to proof-theoretic semantics, to a definition of a canonical derivation from open assumptions. The impact of the extension on the definition of (reified) proof-theoretic meaning of logical constants is discussed. The extended definition also sheds light on a puzzle regarding the definition of local-completeness of a natural-deduction proof-system, underlying its harmony.

AL-FĀRĀBĪ'S KITĀB AL-ḤURŪF AND HIS ANALYSIS OF THE SENSES OF BEING

2008 ◽  
Vol 18 (1) ◽  
pp. 59-97 ◽  
Author(s):  
STEPHEN MENN
Keyword(s):  
Second Order ◽  
First Person ◽  
Logical Syntax ◽  
Logical Constants ◽  
First Order ◽  
Propositional Truth ◽  
Arabic Word ◽  
Ideal Language ◽  
The Senses

Al-Fārābī, in the Kitāb al-Ḥurūf, is apparently the first person to maintain that existence, in one of its senses, is a second-order concept [ma‘qūl thānī]. As he interprets Metaphysics Δ7, ‘‘being'' [mawjūd] has two meanings, second-order ‘‘being as truth'' (including existence as well as propositional truth), and first-order ‘‘being as divided into the categories.'' The paronymous form of the Arabic word ‘‘mawjūd'' suggests that things exist through some existence [wujūd] distinct from their essences: for al-Kindī, God is such a wujūd of all things. Against this, al-Fārābī argues that existence as divided into the categories is real but identical with the essence of the existing thing, and that existence as truth is extrinsic to the essence but non-real (being merely the fact that some concept is instantiated). The Ḥurūf tries to reconstruct the logical syntax of syncategorematic or transcendental concepts such as being, which are often expressed in misleading grammatical forms. Al-Fārābī thinks that Greek more appropriately expressed many such concepts, including being, by particles rather than nouns or verbs; he takes Metaphysics Δ to be discussing the meanings of such particles (comparable to the logical constants of an ideal language), and he takes these concepts to demarcate the domain of metaphysics. This explains how al-Fārābī's title can mean both ‘‘Book of Particles'' and ‘‘Aristotle's Metaphysics.''

Logical constants

Mind ◽  
1999 ◽  
Vol 108 (431) ◽  
pp. 503-538 ◽  
Author(s):  
K Warmbrod
Keyword(s):  
Logical Constants

Normalization theorems for full first order classical natural deduction

1991 ◽  
Vol 56 (1) ◽  
pp. 129-149 ◽  
Author(s):  
Gunnar Stålmarck
Keyword(s):  
Normal Form ◽  
Natural Deduction ◽  
Full System ◽  
Strong Normalization ◽  
Logical Constants ◽  
Restricted Version ◽  
First Order ◽  
Form Theorem ◽  
Normalization Theorem ◽  
Normal Form Theorem

In this paper we prove the strong normalization theorem for full first order classical N.D. (natural deduction)—full in the sense that all logical constants are taken as primitive. We also give a syntactic proof of the normal form theorem and (weak) normalization for the same system.The theorem has been stated several times, and some proofs appear in the literature. The first proof occurs in Statman [1], where full first order classical N.D. (with the elimination rules for ∨ and ∃ restricted to atomic conclusions) is embedded in a system for second order (propositional) intuitionistic N.D., for which a strong normalization theorem is proved using strongly impredicative methods.A proof of the normal form theorem and (weak) normalization theorem occurs in Seldin [1] as an extension of a proof of the same theorem for an N.D.-system for the intermediate logic called MH.The proof of the strong normalization theorem presented in this paper is obtained by proving that a certain kind of validity applies to all derivations in the system considered.The notion “validity” is adopted from Prawitz [2], where it is used to prove the strong normalization theorem for a restricted version of first order classical N.D., and is extended to cover the full system. Notions similar to “validity” have been used earlier by Tait (convertability), Girard (réducibilité) and Martin-Löf (computability).In Prawitz [2] the N.D. system is restricted in the sense that ∨ and ∃ are not treated as primitive logical constants, and hence the deductions can only be seen to be “natural” with respect to the other logical constants. To spell it out, the strong normalization theorem for the restricted version of first order classical N.D. together with the well-known results on the definability of the rules for ∨ and ∃ in the restricted system does not imply the normalization theorem for the full system.

Logical Constants

2014 ◽  
pp. 163-185
Author(s):  
Jaroslav Peregrin
Keyword(s):  
Logical Constants

Logical constants

2018 ◽  
Author(s):  
Timothy McCarthy
Keyword(s):  
Fundamental Problem ◽  
Logical Consequence ◽  
Logical Form ◽  
Logical Truth ◽  
Semantic Property ◽  
Truth Conditions ◽  
Property A ◽  
Logical Constants ◽  
Language L

A fundamental problem in the philosophy of logic is to characterize the concepts of ‘logical consequence’ and ‘logical truth’ in such a way as to explain what is semantically, metaphysically or epistemologically distinctive about them. One traditionally says that a sentence p is a logical consequence of a set S of sentences in a language L if and only if (1) the truth of the sentences of S in L guarantees the truth of p and (2) this guarantee is due to the ‘logical form’ of the sentences of S and the sentence p. A sentence is said to be logically true if its truth is guaranteed by its logical form (for example, ‘2 is even or 2 is not even’). There are three problems presented by this picture: to explicate the notion of logical form or structure; to explain how the logical forms of sentences give rise to the fact that the truth of certain sentences guarantees the truth of others; and to explain what such a guarantee consists in. The logical form of a sentence may be exhibited by replacing nonlogical expressions with a schematic letter. Two sentences have the same logical form when they can be mapped onto the same schema using this procedure (‘2 is even or 2 is not even’ and ‘3 is prime or 3 is not prime’ have the same logical form: ‘p or not-p’). If a sentence is logically true then each sentence sharing its logical form is true. Any characterization of logical consequence, then, presupposes a conception of logical form, which in turn assumes a prior demarcation of the logical constants. Such a demarcation yields an answer to the first problem above; the goal is to generate the demarcation in such a way as to enable a solution of the remaining two. Approaches to the characterization of logical constants and logical consequence are affected by developments in mathematical logic. One way of viewing logical constanthood is as a semantic property; a property that an expression possesses by virtue of the sort of contribution it makes to determining the truth conditions of sentences containing it. Another way is proof-theoretical: appealing to aspects of cognitive or operational role as the defining characteristics of logical expressions. Broadly, proof-theoretic accounts go naturally with the conception of logic as a theory of formal deductive inference; model-theoretic accounts complement a conception of logic as an instrument for the characterization of structure.

scholarly journals On a Formalism Which Makes any Sequence of Symbols Well-Formed

1968 ◽  
Vol 32 ◽  
pp. 1-4
Author(s):  
Shigeo Ōhama
Keyword(s):  
Formal System ◽  
Finite Sequence ◽  
Universal Quantifier ◽  
Logical Constants ◽  
System A

Any finite sequence of primitive symbols is not always well-formed in the usual formalisms. But in a certain formal system, we can normalize any sequence of symbols uniquely so that it becomes well-formed. An example of this kind has been introduced by Ono [2]. While we were drawing up a practical programming along Ono’s line, we attained another system, a modification of his system. The purpose of the present paper is to introduce this modified system and its application. In 1, we will describe a method of normalizing sentences in LO having only two logical constants, implication and universal quantifier, so that any finite sequence of symbols becomes well-formed. In 2, we will show an application of 1 to proof. I wish to express my appreciation to Prof. K. Ono for his significant suggestions and advices.

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