We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R a... more We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in Brady [7]. We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic. We also contrast the two approaches to classical recapture, again favouring our approach in [7]. We then consider our previous development of Peano arithmetic including primitive recursive functions, finally extending this work to that of general recursion.
This paper uses the relevant logic, MCQ, of meaning containment to explore mathematics without va... more This paper uses the relevant logic, MCQ, of meaning containment to explore mathematics without various classical theses, in particular, without the law of excluded middle.
This paper proves decidability of a range of weak relevant logics using decision procedures based... more This paper proves decidability of a range of weak relevant logics using decision procedures based on the Routley-Meyer semantics. Logics are categorized as F-logics, for those proved decidable using a filtration method, and U-logics, for those proved decidable using a direct (unfiltered) method. Both of these methods are set out as reductio methods, in the style of Hughes and Cresswell. We also examine some extensions of the U-logics where the method fails and infinite sequences of worlds can be generated.
Non-normal relevant systems Multiplying connectives, and multiply intensional logics Operational ... more Non-normal relevant systems Multiplying connectives, and multiply intensional logics Operational semantics The algebraic analysis of relevant affixing systems The more general semantical theory of implication and conditionality Recent developments I Recent developments II On quantified relevant logics.
The paper addresses what we see as the four major issues in logic. The overriding issue is that o... more The paper addresses what we see as the four major issues in logic. The overriding issue is that of the choice of logic. We start with some discussion of the preliminary issue of whether there is such a 'one true logic,' but we reserve the main discussion for the first issue of 'classical logic versus non-classical logic.' Here, we discuss the role of meaning and truth, the relation between classical logic and classical negation, and whether and, if so, how classical logic should reside at the base world. Given the argument in favor of an overall use of non-classical logic, the second issue is that of the choice of non-classical logic. Brady's logic MC of meaning containment is argued for, with some comparison made with other relevant logics. For the remaining two issues, we make a case for relevant deduction, in comparison with classical deduction, and we explore possibilities for the appropriate meta-logic, comparing classical and non-classical approaches.
which corresponds in LR, to the axiom (W) of R, would, in an attempted decision procedure based o... more which corresponds in LR, to the axiom (W) of R, would, in an attempted decision procedure based on LR, , produce unbounded multiplicities of the above structures a. It then seemed likely that the system R without the axiom (W), called now RW, would be decidable. Giambrone, in [8], has shown, by using Dunn's Gentzenization LR, without (WI t) (thus yielding LRW,), that RW, is decidable. Giambrone's decidability argument involves a refinement of Gentzen's original argument for the decidability of sentential calculi in [7], together with original insights appropriate to Dunn's style of Gentzenization, which involves both intensional and extensional sequences in the antecedents of consecutions. In this paper, we apply Giambrone's decidability argument to achieve decidability for RW, after establishing a Cut-free Gentzenization for the logic by extending the work of Dunn in [l]. It is the establishment that the rule Cut: ttA AkB (where 't' is the truth constant in Dunn), is eliminable from the Gentzenization of RW"' (where '0' is fusion, as in Dunn) that requires the careful addition of negation to the Gentzenization LRW,
Skip to content. admin login. La Trobe University Research Online, a digital repository holding w... more Skip to content. admin login. La Trobe University Research Online, a digital repository holding works produced by La Trobe University staff and students La Trobe University Research Online, a digital repository holding works produced by La Trobe University staff and students ...
On Preserving sets out in a very readable style a new approach to paraconsistency, which has been... more On Preserving sets out in a very readable style a new approach to paraconsistency, which has been recently proselytized and researched by a group of Canadians, creating a national viewpoint called 'preservationism'. It consists of a series of essays, written and rewritten for ...
A variety of fineware ceramic vessels are associated with elite and ritual context in and around ... more A variety of fineware ceramic vessels are associated with elite and ritual context in and around the American Bottom. One common example is the beaker, a typically cylindrical form that is highly burnished or slipped and usually decorated with incising or engraving. Recent residue analysis of beakers identifies traces of a plant used to make the highly caffeinated "Black Drink," providing evidence of the beaker's role in ritual contexts. The excavation of portions of the East St. Louis Mound Complex for the new Mississippi River Bridge project has recovered more than 250 beakers from a variety of features. This paper presents preliminary analysis of these and discusses their significance in relation to the consumption of "Black Drink."
We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R a... more We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in Brady [7]. We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic. We also contrast the two approaches to classical recapture, again favouring our approach in [7]. We then consider our previous development of Peano arithmetic including primitive recursive functions, finally extending this work to that of general recursion.
This paper uses the relevant logic, MCQ, of meaning containment to explore mathematics without va... more This paper uses the relevant logic, MCQ, of meaning containment to explore mathematics without various classical theses, in particular, without the law of excluded middle.
This paper proves decidability of a range of weak relevant logics using decision procedures based... more This paper proves decidability of a range of weak relevant logics using decision procedures based on the Routley-Meyer semantics. Logics are categorized as F-logics, for those proved decidable using a filtration method, and U-logics, for those proved decidable using a direct (unfiltered) method. Both of these methods are set out as reductio methods, in the style of Hughes and Cresswell. We also examine some extensions of the U-logics where the method fails and infinite sequences of worlds can be generated.
Non-normal relevant systems Multiplying connectives, and multiply intensional logics Operational ... more Non-normal relevant systems Multiplying connectives, and multiply intensional logics Operational semantics The algebraic analysis of relevant affixing systems The more general semantical theory of implication and conditionality Recent developments I Recent developments II On quantified relevant logics.
The paper addresses what we see as the four major issues in logic. The overriding issue is that o... more The paper addresses what we see as the four major issues in logic. The overriding issue is that of the choice of logic. We start with some discussion of the preliminary issue of whether there is such a 'one true logic,' but we reserve the main discussion for the first issue of 'classical logic versus non-classical logic.' Here, we discuss the role of meaning and truth, the relation between classical logic and classical negation, and whether and, if so, how classical logic should reside at the base world. Given the argument in favor of an overall use of non-classical logic, the second issue is that of the choice of non-classical logic. Brady's logic MC of meaning containment is argued for, with some comparison made with other relevant logics. For the remaining two issues, we make a case for relevant deduction, in comparison with classical deduction, and we explore possibilities for the appropriate meta-logic, comparing classical and non-classical approaches.
which corresponds in LR, to the axiom (W) of R, would, in an attempted decision procedure based o... more which corresponds in LR, to the axiom (W) of R, would, in an attempted decision procedure based on LR, , produce unbounded multiplicities of the above structures a. It then seemed likely that the system R without the axiom (W), called now RW, would be decidable. Giambrone, in [8], has shown, by using Dunn's Gentzenization LR, without (WI t) (thus yielding LRW,), that RW, is decidable. Giambrone's decidability argument involves a refinement of Gentzen's original argument for the decidability of sentential calculi in [7], together with original insights appropriate to Dunn's style of Gentzenization, which involves both intensional and extensional sequences in the antecedents of consecutions. In this paper, we apply Giambrone's decidability argument to achieve decidability for RW, after establishing a Cut-free Gentzenization for the logic by extending the work of Dunn in [l]. It is the establishment that the rule Cut: ttA AkB (where 't' is the truth constant in Dunn), is eliminable from the Gentzenization of RW"' (where '0' is fusion, as in Dunn) that requires the careful addition of negation to the Gentzenization LRW,
Skip to content. admin login. La Trobe University Research Online, a digital repository holding w... more Skip to content. admin login. La Trobe University Research Online, a digital repository holding works produced by La Trobe University staff and students La Trobe University Research Online, a digital repository holding works produced by La Trobe University staff and students ...
On Preserving sets out in a very readable style a new approach to paraconsistency, which has been... more On Preserving sets out in a very readable style a new approach to paraconsistency, which has been recently proselytized and researched by a group of Canadians, creating a national viewpoint called 'preservationism'. It consists of a series of essays, written and rewritten for ...
A variety of fineware ceramic vessels are associated with elite and ritual context in and around ... more A variety of fineware ceramic vessels are associated with elite and ritual context in and around the American Bottom. One common example is the beaker, a typically cylindrical form that is highly burnished or slipped and usually decorated with incising or engraving. Recent residue analysis of beakers identifies traces of a plant used to make the highly caffeinated "Black Drink," providing evidence of the beaker's role in ritual contexts. The excavation of portions of the East St. Louis Mound Complex for the new Mississippi River Bridge project has recovered more than 250 beakers from a variety of features. This paper presents preliminary analysis of these and discusses their significance in relation to the consumption of "Black Drink."
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