Coppercock - Invitation into Formal Semantics

Introduction

Entailment

When A entails B -> when A happens, B happens

To reason correct, it is not enough if the presmises and conclusion is true, it needs to also be true if things happen differently.

Arguments whose premises entail the conclusion are \(\texttt{valid}\), others \(\texttt{invalid}\).

An argument is valid iff: under all circumstances in which the presmises would be true, the conclusion is also true.

An argument that is valid whose premises are true is called \(\texttt{sound}\).

Quantifiers

Example: {{< hint danger >}} a. Florida is north of no US state

b. No US state is north of Alaska

c. \(\cancel{\therefore}\) Florida is north of Alaka {{< /hint >}}

The quantifier here of \(\texttt{no US state}\) is a shorthand ambiguity

Sentences are ambigious in nautral language

Two types of amibguity:

Scope amibguity - How you read the argument
Lexical amibguity - When a word has multiple meanings

Entailment vs Implicate

Conversational implicate

Inferences that the listener can derive, assuming the speaker is adhereing to some axioms
Maxim of Quantity -> The speaker provides as much information as needed

\(\texttt{Question under discussion}\) -> Subject matter at hand

Conversational implicative can be cancelled without producing a contradiction.

\(\texttt{Defeasbility test}\) - Check if youy can construct an example where A is followed by the negation of B, and see if that still makes sense

Contrary opposition & Contradictory Opposition

{{< columns >}} A is contrary oppposition to B iff A & B cannot be true in the same circumstance but both can be false <—> A & B cannot be true together and cannot be false together {{< /columns >}}

{{< hint info >}} Note that negation is alwatys contradictory opposition. {{< /hint >}}

Tests

{{< columns >}} Defeasbility Test - Attempts to form A && \(\neg\) B <—> Reinforcement Test - Attempts to form A && B {{< /columns >}}

Entailment vs Presupposition

Most negations of \(\neg\) true are True
Some negations are false

Definition of presupposition {{< columns >}} Pragmatic -> Something the speaker of a sentence does <—> Semantic -> When a sentence presupposes something, the presupposed must be true in order for the sentence to be true. {{< /columns >}}

If you presuppose something false, the sentence is not true NOR false, but rather nonsense

Presupposition projects from the antecedent of the conditional

Projection Test

Asses whether the inference in question projects over the negation Presuppositions thrive where entailments die

Truth Conditional Semantics

Truth conditions are the situations in which the sentence is true, not whether the sentence is in fact true.