Logical Consequence and Entailment

An argument is usually said to be valid iff it is truth-preserving—iff it cannot be that all its premises are true and its conclusion false. But imperatives (it is normally thought) are not truth-apt. They are not in the business of saying how the world is, and therefore cannot either succeed or fail in doing so. To solve this problem, we need to find a new criterion of validity, and I aim to propose such a criterion.

The richer you are, the less equally rich or richer people. The richest is only one (=unique). Maximization of richness or leisure (=classic utility), maximizes the uniqueness (=improbability) that can be maximized also by: extreme sport, suicide, tattoo, count of views... The richest seem unique as the poorest, but the rich can easily become poor, while the poor can hardly get rich. So the aim of maximization reflects IQ and options. Few options increase irrationality, regardless of IQ. I also present (...) the law of values' equality, Utility of Suicide, motivation to support the better, and overview the economics as such. Presented as PhD thesis at CERGE-EI, 2000, exhibited in Holland Park, W8 6LU, 2013, presented at conferences in Santorini 2016, Daejon 2016, Adelaide 2016. (shrink)

Model of intelligence and new methods to assess IQ. MA thesis in 1998 (Comenius University). Art exhibitions "From Animation" London 2013, "Fading Memory" Weißenohe 2015, TAIF Tokyo 2017. Conferences in Santorini, Daejon 2016, Geneva 2017.

Compare two conceptions of validity: under an example of a modal conception, an argument is valid just in case it is impossible for the premises to be true and the conclusion false; under an example of a topic-neutral conception, an argument is valid just in case there are no arguments of the same logical form with true premises and a false conclusion. This taxonomy of positions suggests a project in the philosophy of logic: the reductive analysis of the modal conception (...) of logical consequence to the topic-neutral conception. Such a project would dispel the alleged obscurity of the notion of necessity employed in the modal conception in favour of the clarity of an account of logical consequence given in terms of tractable notions of logical form, universal generalization and truth simpliciter. In a series of publications, John Etchemendy has characterized the model-theoretic definition of logical consequence as truth preservation in all models as intended to provide just such an analysis. In this paper, I will argue that Aristotle intends to provide an account of a modal conception of logical consequence in topic-neutral terms and so is engaged in a project comparable to the one described above. That Aristotle would be engaged in this sort of project is controversial. Under the standard reading of the Prior Analytics, Aristotle does not and cannot provide an account of logical consequence. Rather, he must take the validity of the first figure syllogisms (such as the syllogism known by its medieval mnemonic ‘Barbara’: A belongs to all B; B belongs to all C; so A belongs to all C) as obvious and not needing justification; he then establishes the validity of the other syllogisms by showing that they stand in a suitable relation to the first figure syllogisms. I will argue that Aristotle does attempt to provide an account of logical consequence—namely, by appeal to certain mereological theorems. For example, he defends the status of Barbara as a syllogism by appeal to the transitivity of mereological containment. There are, as I will discuss, reasons to doubt the success of this account. But the attempt is not implausible given certain theses Aristotle holds in semantics, mereology and the theory of relations. (shrink)

David Miller claims that every valid deductive argument begs the question. Other philosophers and logicians have made similar claims. I show that the claim is false. Its appeal depends on the existence of logical terminology, particularly concerning what a proposition 'contains' or its 'logical content,' that is best understood as metaphoric and that, given its aptness to mislead, would be better eschewed. I show how the terminology appears to derive from early modern theories of the nature of mind, ideas and (...) reasoning that have since been rejected. (shrink)

David Miller propounds a theory of objective knowledge from which he mistakenly derives some consequences about question-begging and persuasion that appear to be false. He makes a further claim about persuasion that also seems false. I argue that Miller’s account of objective knowledge is explanatorily weak unless supplemented with an account of subjective knowledge and that the latter enables us to extricate Miller’s theory from the falsehoods he associates with it.

David Miller argues that there are no good reasons, either sufficient or insufficient. I show that most of his arguments are invalid or unsound. Several of his arguments depend upon the false claim that every deductively valid argument is circular. I accept one of Miller's arguments for the conclusion that there are no good reasons which are less-than-sufficient. I accept one of his arguments to the conclusion that there are no probative sufficient reasons. But I explain how there are epistemic (...) sufficient reasons which are relative to the game of science and how their existence is consistent with Miller's main contentions while avoiding Miller's apparently anarchic account of our knowledge. (shrink)

I give an account of proof terms for derivations in a sequent calculus for classical propositional logic. The term for a derivation δ of a sequent Σ≻Δ encodes how the premises Σ and conclusions Δ are related in δ. This encoding is many–to–one in the sense that different derivations can have the same proof term, since different derivations may be different ways of representing the same underlying connection between premises and conclusions. However, not all proof terms for a sequent Σ≻Δ (...) are the same. There may be different ways to connect those premises and conclusions. -/- Proof terms can be simplified in a process corresponding to the elimination of cut inferences in sequent derivations. However, unlike cut elimination in the sequent calculus, each proof term has a unique normal form (from which all cuts have been eliminated) and it is straightforward to show that term reduction is strongly normalising—every reduction process terminates in that unique normal form. Further- more, proof terms are invariants for sequent derivations in a strong sense—two derivations δ1 and δ2 have the same proof term if and only if some permutation of derivation steps sends δ1 to δ2 (given a relatively natural class of permutations of derivations in the sequent calculus). Since not every derivation of a sequent can be permuted into every other derivation of that sequent, proof terms provide a non-trivial account of the identity of proofs, independent of the syntactic representation of those proofs. (shrink)

Inductive logic would be the logic of arguments that are not valid, but nevertheless justify belief in something like the way in which valid arguments would. Maybe we could describe it as the logic of “almost valid” arguments. There is a sort of transitivity to valid arguments. Valid arguments can be chained together to form arguments and such arguments are themselves valid. One wants to distinguish the “almost valid” arguments by noting that chains of “almost valid” arguments are weaker than (...) the links that form them. But it is not clear that this is so. I have an apparent counterexample the claim. Though: as is typical in these sorts of situations, it is hard to tell where the problem lies. (shrink)

The paper begins with a conceptual discussion of Michael Dummett's proof-theoretic justification of deduction or proof-theoretic semantics, which is based on what we might call Gentzen's thesis: 'the introductions constitute, so to speak, the "definitions" of the symbols concerned, and the eliminations are in the end only consequences thereof, which could be expressed thus: In the elimination of a symbol, the formula in question, whose outer symbol it concerns, may only "be used as that which it means on the basis (...) of the introduction of this symbol".' The intuitive philosophical content of Dummett's notions of harmony and stability is that harmony obtains if the grounds for asserting a proposition match the consequences of accepting it, and stability obtains if the converse also holds. Rules of inference define the meanings of a logical constant they govern if and only if they are stable. Gentzen observed that 'it should be possible to establish on the basis of certain requirements that the elimination rules are functions of the corresponding introduction rules.' One of the objectives of this paper is to specify such a function: I will specify a process by which it is possible to determine the elimination rules of logical constants from their introduction rules, and conversely, to determine the introduction rules from the elimination rules. I'll give the general forms of rules of inference and generalised reduction procedures for the normalisation of deduction. I'll give a formally precise characterisations of harmony and stability and show that deductions in logics that contain only constants governed by stable rules always normalise. (shrink)

This paper describes a decision procedure for disjunctions of conjunctions of anti-prenex normal forms of pure first-order logic (FOLDNFs) that do not contain V within the scope of quantifiers. The disjuncts of these FOLDNFs are equivalent to prenex normal forms whose quantifier-free parts are conjunctions of atomic and negated atomic formulae (= Herbrand formulae). In contrast to the usual algorithms for Herbrand formulae, neither skolemization nor unification algorithms with function symbols are applied. Instead, a procedure is described that rests on (...) nothing but equivalence transformations within pure first-order logic (FOL). This procedure involves the application of a calculus for negative normal forms (the NNF-calculus) with A -||- A & A (= &I) as the sole rule that increases the complexity of given FOLDNFs. The described algorithm illustrates how, in the case of Herbrand formulae, decision problems can be solved through a systematic search for proofs that reduce the number of applications of the rule &I to a minimum in the NNF-calculus. In the case of Herbrand formulae, it is even possible to entirely abstain from applying &I. Finally, it is shown how the described procedure can be used within an optimized general search for proofs of contradiction and what kind of questions arise for a &I-minimal proof strategy in the case of a general search for proofs of contradiction. (shrink)

Substructural logics and their application to logical and semantic paradoxes have been extensively studied, but non-reexive systems have been somewhat neglected. Here, we aim to (at least partly) ll this lacuna, by presenting a non-reexive logic and theory of naïve consequence (and truth). We also investigate the semantics and the proof-theory of the system. Finally, we develop a compositional theory of truth (and consequence) in our non-reexive framework.

The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic value of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide the valid deductive inference. Sound deductive conclusions are the result of these finite string transformation rules.

Both Tarski and Gödel “prove” that provability can diverge from Truth. When we boil their claim down to its simplest possible essence it is really claiming that valid inference from true premises might not always derive a true consequence. This is obviously impossible.

This sentence G ↔ ¬(F ⊢ G) and its negation G ↔ ~(F ⊢ ¬G) are shown to meet the conventional definition of incompleteness: Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ)). They meet conventional definition of incompleteness because neither the sentence nor its negation is provable in F (or any other formal system).

We can simply define Gödel 1931 Incompleteness away by redefining the meaning of the standard definition of Incompleteness: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ). This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system. Since self-contradictory expressions are neither provable nor disprovable (...) only because they are self-contradictory we could define them as unsound instead of defining the formal system as incomplete. (shrink)

Minimal Type Theory (MTT) is based on type theory in that it is agnostic about Predicate Logic level and expressly disallows the evaluation of incompatible types. It is called Minimal because it has the fewest possible number of fundamental types, and has all of its syntax expressed entirely as the connections in a directed acyclic graph.

By extending the notion of a Well Formed Formula to include syntactically formalized rules for rejecting semantically incorrect expressions we recognize and reject expressions that have the semantic error of Pathological self-reference(Olcott 2004). The foundation of this system requires the notion of a BaseFact that anchors the semantic notions of True and False. When-so-ever a formal proof from BaseFacts of language L to a closed WFF X or ~X of language L does not exist X is decided to be semantically (...) incorrect. (shrink)

Minimal Type Theory (MTT) shows exactly how all of the constituent parts of an expression relate to each other (in 2D space) when this expression is formalized using a directed acyclic graph (DAG). This provides substantially greater expressiveness than the 1D space of FOPL syntax. -/- The increase in expressiveness over other formal systems of logic shows the Pathological Self-Reference Error of expressions previously considered to be sentences of formal systems. MTT shows that these expressions were never truth bearers, thus (...) never sentences of any formal logic system. (shrink)

What are the probabilities that this universe is repeated exactly the same with you in it again? Is God invented by human imagination or is the result of human intuition? The intuition that the same laws/mechanisms (evolution, stability winning probability) that have created something like the human being capable of self-awareness and controlling its surroundings, could create a being capable of controlling all what it exists? Will be the characteristics of the next universes random or tend to something? All these (...) ques-tions that with different shapes (but the same essence) have been asked by human be-ings from the beginning of times will be developed in this paper. (shrink)

Many recent theories of epistemic discourse exploit an informational notion of consequence, i.e. a notion that defines entailment as preservation of support by an information state. This paper investigates how informational consequence fits with probabilistic reasoning. I raise two problems. First, all informational inferences that are not also classical inferences are, intuitively, probabilistically invalid. Second, all these inferences can be exploited, in a systematic way, to generate triviality results. The informational theorist is left with two options, both of them radical: (...) they can either deny that epistemic modal claims have probability at all, or they can move to a nonstandard probability theory. (shrink)

Lycan (2010: 617) maintained that if we know about a deductive argument P1, P2…, ∴C that it is valid, this shows that a set containing the premises and the negation of conclusion is internally inconsistent {P1, P2,…, ~C}. Using a similar reasoning, we should conclude that an invalid deductive argument shows that a set containing the premises and the negation of conclusion is internally consistent. This implies that premises and conclusion of any deductive argument are truth-apt. Since it is uncontroversial (...) that the argument that motivates the liar paradox is deductive, it follows that the liar sentences are truth-apt. The reason is obvious: if a sentence can’t be true, it can’t be a part of a consistent or inconsistent set, by definition. Consequently, the liar paradox cannot be solved by arguing that liar sentences are truth-valueless. (shrink)

The conventional wisdom about conditionals claims that (1) conditionals that have non-assertive acts in their consequents, such as commands and promises, are not plausibly interpreted as material implications; (2) the most promising hypothesis about these sentences is conditional-assertion theory, which explains a conditional as a conditional speech act, i.e., a performance of a speech act given the assumption of the antecedent. This hypothesis has far-reaching and revisionist consequences, because conditional speech acts are not synonymous with a proposition with truth conditions. (...) This paper argues against this prevalent view in two steps. First, it presents a battery of objections against conditional-assertion theory. Second, it argues that these examples can be convincingly interpreted as categorical assertions of material implications. (shrink)

Russell’s role in the controversy about the paradoxes of material implication is usually presented as a tale of how even the greatest minds can fall prey of basic conceptual confusions. Quine accused him of making a silly mistake in Principia Mathematica. He interpreted ‘if-then’ as a version of ‘implies’ and called it material implication. Quine’s accusation is that this decision involved a use-mention fallacy because the antecedent and consequent of ‘if-then’ are used instead of being mentioned as the premise and (...) the conclusion of an implication relation. It was his opinion that the criticisms and alternatives to the material implication presented by C. I. Lewis and others would never be made in the first place if Russell simply called the Philonian construction ‘material conditional’ instead of ‘material implication’. Quine’s interpretation on the topic became hugely influential, if not universally accepted. This paper will present the following criticisms against this interpretation: (1) the notion of material implication does not involve a use-mention fallacy, since the components of ‘if-then’ are mentioned and not used; (2) Quine’s belief that the components of ‘if-then’ are used was motivated by a conditional-assertion view of conditionals that is widely controversial and faces numerous difficulties; (3) the Philonian construction remains paradoxical even if it is called ‘material conditional’; (4) the Philonian construction is more plausible when it is interpreted as a material implication. (shrink)

Logic textbooks advertise that they can teach how to spot a valid argument by its logical form alone. They also boast having collections with the most basic valid argumentative forms people of flesh and blood can use in deductive matters. Think about this for a moment. These are bold statements. If they were accurate, philosophers would be in higher demand than software engineers and no one would be able to make contributions to theoretical physics without ever taking a logic class. (...) Of course, none of that is true. Their basic argumentative forms are nowhere to be seen in real-life argumentation and people will carry on their cognitive business without ever putting their hands on a logic textbook. I believe philosophers overpromise and underdeliver due to a subtle, but profound confusion they make about their own object of study. They take metalogical principles such as hypothetical syllogism for real forms of the arguments that are used in natural language. I try to connect these principles with real-life argumentation by interpreting their ‘premises’ as background assumptions and their ‘conclusions’ as either arguments or categorical assertions. This is not enough to save face for two reasons. First, the arguer’s assumptions should count as additional premises, but they will involve numerous arguments and the resulting formal interpretation will be too cumbersome to be practical. Second, we can’t determine the validity of an argument by formal means except when the conclusion repeats the premise, the premise is contradictory or the conclusion is tautological. These limitations are usually ignored due to the prevalent but questionable assumption that conditionals are connectives and not claims to implication. But conditionals can be elegantly explained as claims to implication in a variety of modal ranges that are all classical in nature. Consequently, the substantial validity of a deductive argument can only be determined in an informal manner and requires an extensive investigation of the nature of the subject. I conclude the article with some observations about the need to rethink the role of logic, its cognitive importance, and its teaching. Logic is important because it allows us to understand the nature of logical consequence and not because it is supposed to be useful for other deductive endeavours. I also make some observations about the need to rethink informal logic. Similarly to baby logic conventions involving formal logic, the textbook fallacies have no uncontroversial corresponding examples in real-life argumentation and are better interpreted as a list of epistemic vices that any honest epistemic agent should avoid. In this sense, informal logic has been misunderstood as a poor cousin of formal logic for decades, when it should be viewed as a speciality of virtue epistemology. (shrink)

The modus ponens can be interpreted as an answer to a circularity charge, but this strategy is only feasible if the additional conditional premise is interpreted as a claim to a material implication. Here’s how it works.

It is usually accepted that conditional sentences are sui generis and enigmatic. In this paper I try to make them more accessible by interpreting them as claims to relations of implication restricted to a parameter world. This interpretation revives an old idea that fell into disuse, but in its improved version leads to refreshing solutions to known problems in conditional theory. The many benefits of this approach are evidenced by its insightful explanation of some apparent counter-examples to classical argumentative forms (...) (the paradoxes of material implication, antecedent strengthening, contraposition, hypothetical syllogism, conditional negation), conditional standoffs, the problem of counterfactuals, the referential/inferential nature of conditionals, the Apartheid thesis, the triviality results and conditionals embedding. (shrink)

There is a profound, but frequently ignored, relationship between the classical conception of logical consequence and the material implication. The first repeats the patterns of the latter, but with a wider modal reach. This relationship suggests that there should be also a connection between the notion of logical consequence and the conditional connective of any given logical system. This implies, among other things, that it is incoherent to propose alternatives to the material implication while maintaining the classical conception of logical (...) consequence. The other important implication is that we need to posit different conceptions of logical consequence that are consistent with different theories of conditionals in order to evaluate their relative merits from new and unexplored angles. As a pilot study of this research program, we evaluate two new notions of logical consequence motivated by conditional-assertion theory and possible world theories. Those alternatives are compared unfavourably with the classical conception of logical consequence. (shrink)

An argument is deductive when the arguer believes the truth of the premises necessitates the truth of the conclusion. A deductive argument is valid when the arguer’s claim is true, i.e., when there are no possible worlds whether the premises are true and the conclusion is false. But in order to evaluate this claim in an accurate manner we need to consider three qualifications that have been repeatedly ignored in the literature, namely, consistency in the attribution of truth values, world (...) consistency, and the inclusion of the reasons offered by the arguer in support of the premises. These apparently innocuous restrictions have significant and far-reaching implications. (shrink)

The material account of indicative conditionals states that indicative conditional sentences and the material implication have the same truth conditions. Many conditional logics are motivated by attempts to fix the counter-intuitive aspects associated with the material account. Some counter-intuitive instances of classical argumentative forms, e.g., strengthening of the antecedent, contraposition, and conditional negation, are regarded as evidence that the material account is wrong and that classical logic should be rejected in favour of a new logic system in which these argumentative (...) forms are invalid. It is argued that these logical revisions are ad hoc, because those controversial argumentative forms are implied by other argumentative forms we want to keep. It is impossible to remove a counter-intuitive argumentative form from a logical system without getting entangled in an intricate logical web, since these revisions imply the removal of other parts of logic we want to maintain. Consequently, these revisions are incoherent and unwarranted. At the very least, the usual approach in the analysis of putative counter-examples of argumentative forms must be seriously reconsidered. (shrink)

Lee Archie argued that if any truth values are consistently assigned to a natural language conditional to which Modus Ponens and Modus Tollens are valid argumentative forms, and affirming the consequent is an invalid argumentative form, this conditional would have the same truth conditions than a material implication. This argument is simple and it requires few assumptions that are relatively uncontroversial. We show that it is possible to extend Archie’s argument to three-valued logics and five-valued logics and vindicate a slightly (...) weaker conclusion, but that is still important: even if we would not believe in bivalence and in the classical negation operator, we would still have good reasons to accept that natural language conditionals and the material implication share truth conditions. (shrink)

In a typical modus ponens the reasoner will (a) assert that a premise materially implies a conclusion in a given world; (b) assert this premise and (c) infer the conclusion. But this restricted modus ponens has little in common with the unrestricted textbook modus ponens, since the latter claims that there are no possible worlds where: (a’) a premise materially implies a conclusion, (b’) that premise is true and (c’) the conclusion is false. It is clear that this textbook modus (...) ponens cannot be a real argumentative form since the same inference cannot be repeated across all possible worlds by one reasoner. What is wrong? (shrink)

Intensional evidence is any reason to accept a proposition that is not the truth values of the proposition accepted or, if it is a complex proposition, its propositional contents. Extensional evidence is non-intensional evidence. Someone can accept a complex proposition, but deny its logical consequences in two circumstances: (1) when her acceptance is based on intensional evidence, while the logical consequences of the proposition presuppose the acceptance of extensional evidence, e.g., she can refuse the logical consequence of a proposition she (...) accepts because she doesn’t know what the truth-values of its propositional contents; (2) when she accepts a proposition based on extensional evidence, but thinks that this evidence is insufficient to establish its logical consequences, which would require intensional evidence. It is argued that this tension is responsible for the counter-intuitive aspects of the material account of conditionals involving the negation of conditionals, hypothetical syllogism, contraposition and the inferential passage from disjunctions to conditionals (or-to-if). This tension is also behind some known puzzles involving conditionals, namely, conditional stand-offs, Adam pairs, the cheating partner example, jump-out conditionals, the problem of counterfactuals and the burglar’s puzzle. It is shown that this tension is always dissolved in favour of extensional evidence, since intensional evidence is defeasible, while extensional evidence is not. Thus, it is irrational to deny the logical consequences of an accepted proposition due to its reliance on intensional evidence and ignorance of its extensional evidence. (shrink)

The material account claims that indicative conditionals are material. However, the conventional wisdom even among material account enthusiasts is that the material account cannot be extended to subjunctive conditionals. There are mainly three reasons that motivate this consensus: (1) the belief that if subjunctives were material, most subjunctive conditionals would be vacuously true, which is implausible; (2) its inconsistency with Adam pairs, which suggest that indicative and subjunctive conditionals have different truth conditions; and (3) the belief that it is an (...) inferior hypothesis compared to the possible world theories. I will argue against (1) that the counterintuitive aspects of vacuously true conditionals can be explained away in a uniform fashion, regardless of whether they are indicatives or subjunctives. I reinforce this assumption by showing that the positive arguments for the material account of indicatives are also intuitively valid for subjunctives. The point mentioned in (2) is resisted by explaining Adam pairs as logically equivalent conditionals that can be appropriate at different times, depending of the speaker’s epistemic situation. Finally, (3) is criticised by making the case that the possible world account faces insurmountable problems and that a full-blown material account of indicatives and subjunctives is overall a more elegant solution. (shrink)

Moti Mizrahi (2013) presented a putative counterexample to Hypothetical Syllogism (HS) for indicative conditionals aiming to succeed where previous attempts to refute HS have failed. Lee Walters (2014a) objected that Mizrahi’s putative counterexample results from an inadequate analysis of conditionals with embedded modals, but advanced new putative counterexamples to HS for subjunctive conditionals that are supposed to bypass this issue (Walters, 2014a; 2014b). It is argued that Walter’s analysis of embedded modals is unnecessary to prevent Mizrahi’s putative counterexample, since the (...) example can be disarmed either as a fallacy of equivocation or as a contextual fallacy. Walters’s putative counterexamples to HS are also criticized as contextual fallacies. The conclusion is that neither Mizrahi nor Walters provided compelling reasons to reject HS. (shrink)

The material account of indicative conditionals faces a legion of counterexamples that are the bread and butter in any entry about the subject. For this reason, the material account is widely unpopular among conditional experts. I will argue that this consensus was not built on solid foundations, since these counterexamples are contextual fallacies. They ignore a basic tenet of semantics according to which when evaluating arguments for validity we need to maintain the context constant, otherwise any argumentative form can be (...) rendered invalid. If we maintain the context fixed, the counterexamples to the material account are disarmed. Throughout the paper I also consider the ramifications of this defence, make suggestions to prevent contextual fallacies, and anticipate some possible misunderstandings and objections. (shrink)

Four intuitions are recurrent and influential in theories about conditionals: the Ramsey’s test, the Adams’ Thesis, the Equation, and the robustness requirement. For simplicity’s sake, I call these intuitions ‘the big four’. My aim is to show that: (1) the big four are interdependent; (2) they express our inferential dispositions to employ a conditional on a modus ponens; (3) the disposition to employ conditionals on a modus ponens doesn’t have the epistemic significance that is usually attributed to it, since the (...) acceptability or truth conditions of a conditional is not necessarily associated with its employability on a modus ponens. (shrink)

The conclusion of a valid deductive argument is contained in its premises, but the deductive argument can still be informative if the arguer is trying to find the missing premises that together with the accepted premises will ensure the truth of the conclusion. Necessarily true conditionals have a deductive-like character so their consequents are contained in their antecedents. These conditionals can be informative, so we need to find the missing antecedent that makes the connection between the antecedent and the consequent. (...) So where is it? (shrink)

It is usually accepted that one of the properties of classical logic is monotonicity, which states that the validity of implication is not affected by the addition of new premises. In this piece, I will argue that this common notion is unjustified since it is motivated by a category mistake. The notion of monotonicity is primarily epistemic in character and can’t be meaningfully attributed to a system. This is acutely clear in the contrast of monotonicity with non-monotonicity, which we tend (...) to associate with defeasible inferences, where reasoners can abandon a previous inference based on new information. So non-monotonicity is actually the reasoners’ willingness to abandon a previous inference based on new information. Therefore, it can be a property of a system only in a manner of speaking. But since non-monotonicity is the reasoners’ willingness to abandon a previous inference based on new information, monotonicity should be understood in a similar epistemic fashion, namely, as the reasoners’ willingness to maintain a previous inference after newly discovered information. By analyzing the problem from this perspective, a classical inference can be considered non-monotonic if the reasoner is willing to retract her previous conclusions based on new information. (shrink)

In a material implication the relation of logical consequence is restricted to a world parameter, whereas in a formal implication the relation of logical consequence extends over many worlds. One could infer from this pattern that the material implication should be reduced to formal implication since it is just a restricted version of it, or, inversely, that formal implication should be reduced to material implication since it is an unrestricted version of it. But both reductionist claims would betray a superficial (...) understanding of this connection. The obvious kinship between material and formal implication just means that they are versions of the same type of implication. The only difference is that in each case the same type of implication is presented in two degrees. (shrink)

The material account of indicative conditionals states that indicative conditional sentences and the material implication have the same truth conditions. Recently, Adam Rieger has carried out a survey of arguments in favour of the material account. In this paper, I extend this survey by presenting yet more arguments for the material account. On top of presenting more arguments, I also want to argue that it is plausible to extend the material account to subjunctive conditionals. For that reason, the arguments here (...) presented contain principles that hold for both indicative and subjunctive conditionals. Towards the end, I defend why there are good reasons to accept these new arguments and try to explain why similar arguments have been somewhat unpopular. (shrink)

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

In this paper we provide a proof theoretical investigation of logical argumentation, where arguments are represented by sequents, conflicts between arguments are represented by sequent elimination rules, and deductions are made by dynamic proof systems extending standard sequent calculi. The idea is to imitate argumentative movements in which certain claims are introduced or withdrawn in the presence of counter-claims. This is done by a dynamic evaluation of sequences of sequents, in which the latter are considered ‘derived’ or ‘not derived’ according (...) to the content of the sequence. We show that decisive conclusions of such a process correspond to well-accepted consequences of the underlying argumentation framework. The outcome is therefore a general and modular proof-theoretical approach for paraconsistent and non-monotonic reasoning with argumentation systems. (shrink)

When discussing Logical Pluralism several critics argue that such an open-minded position is untenable. The key to this conclusion is that, given a number of widely accepted assumptions, the pluralist view collapses into Logical Monism. In this paper we show that the arguments usually employed to arrive at this conclusion do not work. The main reason for this is the existence of certain substructural logics which have the same set of valid inferences as Classical Logic—although they are, in a clear (...) sense, non-identical to it. We argue that this phenomenon can be generalized, given the existence of logics which coincide with Classical Logic regarding a number of metainferential levels—although they are, again, clearly different systems. We claim this highlights the need to arrive at a more refined version of the Collapse Argument, which we discuss at the end of the paper. (shrink)

In this paper we discuss the extent to which the very existence of substructural logics puts the Tarskian conception of logical systems in jeopardy. In order to do this, we highlight the importance of the presence of different levels of entailment in a given logic, looking not only at inferences between collections of formulae but also at inferences between collections of inferences—and more. We discuss appropriate refinements or modifications of the usual Tarskian identity criterion for logical systems, and propose an (...) alternative of our own. After that, we consider a number of objections to our account and evaluate a substantially different approach to the same problem. (shrink)

This paper scrutinizes the debate over logical pluralism. I hope to make this debate more tractable by addressing the question of motivating data: what would count as strong evidence in favor of logical pluralism? Any research program should be able to answer this question, but when faced with this task, many logical pluralists fall back on brute intuitions. This sets logical pluralism on a weak foundation and makes it seem as if nothing pressing is at stake in the debate. The (...) present paper aims to improve this situation by looking at a promising case study and drawing general lessons about the kind of evidence that would support logical pluralism. I argue that the best motivation for logical pluralism will ultimately be rooted in certain kinds of performative data. (shrink)

This paper is about the underlying logical principles of scientific theories. In particular, it concerns ex contradictione quodlibet the principle that anything follows from a contradiction. ECQ is valid according to classical logic, but invalid according to paraconsistent logics. Some advocates of paraconsistency claim that there are ‘real’ inconsistent theories that do not erupt with completely indiscriminate, absurd commitments. They take this as evidence in favor of paraconsistency. Michael calls this the non-triviality strategy. He argues that this strategy fails in (...) its purpose. I will show that Michael's criticism significantly over-reaches. The fundamental problem is that he places more of a burden on the advocate of paraconsistency than on the advocate of classical logic. The weaknesses in Michael's argument are symptomatic of this preferential treatment of one viewpoint in the debate over another. He does, however, make important observations that allow us to clarify some of the complexities involved in giving a logical reconstruction of a theory. I will argue that there are abductive arguments deserving of further consideration for the claim that paraconsistent logic offers the best explanation of the practice of inconsistent science. In this sense, the debate is still very much open. (shrink)

This paper is a guide to the main ideas and innovations in Robert Stalnaker's "Indicative Conditionals". The paper is for a volume of essays on twenty-one classics of formal semantics edited by Louise McNally, Yael Sharvit and Zoltàn Gendler Szabò.