An Introduction to Non-Classical Logic: From If to Is, Second Edition

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• Author: ndl.ethernet.edu.et
• Title: An Introduction to Non-Classical Logic: From If to Is, Second Edition

Page Notes#to-process

Errata

Highlights

to-process

• If everyproof-theoretically valid inference is semantically valid (so thatentails|=)the proof-theory is said to besound. If every semantically valid inference isproof-theoretically valid (so that|=entails) the proof-theory is said to becomplete. — Updated on 2024-05-03 12:15:24
  • Semantic Validity: $tack.double$ Proof-theoretic/Syntactic Validity: $tack$

If $tack$ entails $tack.double$, the system is sound, since the syntax corresponds to our semantics. If $tack.double$ entails $tack$, the system is complete, since our semantics are all syntactically valid.

• Aninterpretationof the language is a function,ν, which assigns to eachpropositional parameter either 1 (true), or 0 (false). — Updated on 2024-05-03 12:17:45

• Ais alogical truth(tautology)(|=A)iff it is a semantic consequence ofthe empty set of premises(φ|=A), that is, every interpretation makesAtrue — Updated on 2024-05-03 12:36:45

  • $Sigmatack.doubleA$ iff there is no interpretation ($v$) where all $Sigma$ are true and $A$ false.

Tautologies then are a special case, where $tack.doubleA$ is shorthand for $phi.alttack.doubleA$.

• A tableau iscompleteiff every rule that can be applied has been applied — Updated on 2024-05-07 12:24:47

• A is a proof-theoretic consequence of the set of formulas(A)iffthere is a complete tree whose initial list comprises the members ofandthe negation ofA, and which is closed. — Updated on 2024-05-07 12:28:17

  • Earlier, he has said that a complete tree/tableau can always be made.

Analytic tableau show syntactic entailment/proof-theoretic consequences.

• These are sometimes called the ‘paradoxes of material implication’. — Updated on 2024-05-07 12:40:31

  • Referring to $Btack.doubleA->B$ and $notAtack.doubleA->B$, which does not seem to make sense in terms of “conditional.”

• Communication between people is governed by many pragmaticrules of conversation, for example ‘be relevant’, ‘assert the strongest claimyou are in a position to make’. We often use the fact that these rules are inplace to draw conclusions. Consider, for example, what you would inferfrom the following questions and replies: ‘How do you use this drill?’,‘There’s a book over there.’ (It is a drill manual.Relevance.) ‘Who won the3.30 at Ascot?’, ‘It was a horse named either Blue Grass or Red Grass.’ (Thespeaker does not know which.Assert the strongest information.) These infer-ences are inferences, not from thecontentof what has been said, but from thefact thatit has been said. The process is often dubbed ‘conversational impli-cature’. — Updated on 2024-05-07 12:44:14

  • Material conditionals do not always adhere to these, which is a reason why it seems strange at times.

• distinguish between two sorts of conditionals: conditionals inwhich the consequent is expressed using the word ‘would’ (called ‘sub-junctive’ or ‘counterfactual’), and others (called ‘indicative’). — Updated on 2024-05-14 23:44:27

• If you close switchxand switchythe light will go on. Hence, it is thecase either that if you close switchxthe light will go on, or that if youclose switchythe light will go on. — Updated on 2024-05-18 01:23:06

  • The material conditional does not fit in to the English conditional, if, then. The situation stated is clearly invalid, but its representation with the material conditional isn’t:

$(AandB)->Ctack(A->C)or(B->C)$

• It is not the case that if there is a good god the prayers of evil peoplewill be answered. Hence, there is a god — Updated on 2024-05-18 01:27:49
  • This one is an even clearer example of the issues of the material conditional:

$not(A->B)tackA$

• First, suppose that ‘IfAthenB’ is true. Either¬Ais true orAis. Inthis first case,¬A∨Bis true. In the second case,Bis true bymodus ponens.Hence, again,¬A∨Bis true. Thus, in either case,¬A∨Bis true. — Updated on 2024-05-18 01:30:51
  • This also appears logically sound, but this assumes that all statements of the English “if” can be treated in this way.