User:Dcresti

Hello. I am a formal semanticist by training, but I'm not doing much of that these days (see my current home page).

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Notes to self
I thought I would contribute to the page on Counterfactual conditionals. In the process of thinking up a possible edit to the introduction, I saw the page for the Indicative conditional, and then decided I should make a distinction between the natural language if-then conditionals and the

Head scratchers
While the philosophy section is often in a state of anarchy, some words (or terms) have been appropriated by people who seem competent at math and logic but not natural language. For instance:

Propositional logic link redirects to Propositional calculus.

Assertion link is interpreted as Logical assertion and redirects to Sequent.

Notes for Logic page
Logic is formal in the sense that it analyzes the ‘form’ of statements, as opposed to how such statements relate to facts about the world. In a basic logic, we assume that a statement can be True or False (1 or 0) in a given context of use, but we are not concerned with how this truth or falsity comes about. There are, however, logics that admit more than two truth values; these multi-valued logics reflect a common intuition that there are systematic cases where a statement (in a given context of use) cannot be evaluated as either True or False.

At the sentence level, a basic logic typically has truthfunctional operators such as $$\lnot$$ for not or it-it-not-the-case-that, $$\land$$ for and, $$\lor$$ for what we call 'inclusive' or, and $$\rightarrow$$ for statements of the form if...then. These are defined in terms of their truth tables:

This type of abstraction entails a loss of some complexities and ambiguities of natural language. At the same time, it provides a systematic way of describing many different kinds of procedures, and a variety of forms of human reasoning. The tension and interplay between the expressiveness and subtlety of natural language, and the functionality of formal languages such as the logic we are describing here, is at the core of several areas of philosophy, linguistics, computer science etc. See Partee et. al. (1990) for an introduction to this area of inquiry.

Example (I)

 * How do we know that the statement It is raining and it is not raining is always False, independently of whether or not it is actually raining in a given context? Let’s “tease out” the sentence It is raining and call it p; rephrase the sentence It is not raining as It is not the case that p. Our example can then be written as:


 * "p and it-is-not-the-case-that p"


 * Or, in symbols:
 * "p$ \land \lnot$p"


 * where '$$\land$$' substitutes for natural language ‘and’, and '$$\lnot$$' substitutes for ‘it-is-not-the-case-that’.


 * We define $$\land$$ and $$\lnot$$ in terms of truth tables, from which we can compute p$$ \land \lnot$$p as follows:
 * {| border="1" cellpadding="2" cellspacing="0" style="text-align:center;"

! style="width:35px;background:#eeeeee;" | p ! style="width:35px;background:#ffffff;" | $$\lnot$$ p ! style="width:65px;background:#ffffff;" | p $$\land \lnot$$p
 * T || F || F
 * F || T || F
 * }
 * From the last column in this table, we see that no matter whether p is True or False, a statement of the form p-and-not-p is always interpreted as False in this system. A statement of this kind (one that is always False) is called a contradiction. Conversely, a statement that is always True, such as It is raining or it is not raining, is called a tautology.
 * }
 * From the last column in this table, we see that no matter whether p is True or False, a statement of the form p-and-not-p is always interpreted as False in this system. A statement of this kind (one that is always False) is called a contradiction. Conversely, a statement that is always True, such as It is raining or it is not raining, is called a tautology.

Example (II)

 * If we know that (a) all cats are mammals, and (b) all mammals have a heart, how do we deduce that all cats have a heart?
 * Intuitively, we can think in terms of relations between sets of entities in the universe (U - the universe of discourse):
 * [[Image:cat-mammal-heart.png]]


 * If CAT is a subset of MAMMAL, and MAMMAL is a subset of HAS-HEART, then CAT is a subset of HAS-HEART. Note that it is merely a coincidence that these sets are realistic—i.e., that they reflect what we agree are facts about the world. We easily take our premises (a) and (b) as true because of this, but the deduction itself does not rely on truth in the actual world. Suppose we hear of the discovery of a new species called fwox, and we are told that (d) all fwoxes are mammals. If we believe (d) and thus take it as a premise ("fact"), we are entitled to deduce from (d) and (b) that all fwoxes have a heart.
 * The sets in the picture above have isomorphic predicates via their characteristic functions. Let's call $$C$$ the predicate for CAT, $$M$$ the one for MAMMAL, and $$H$$ the one for HAS-HEART. Following a tradition that goes back to Aristotle, we could paraphrase our statement (a) as All C are M. In addition to the predicates C and M, we introduce the universal quantifier $$\forall$$, generally paraphrased as for-all.
 * We are now "inside" our sentences; this is an extension of our statement logic commonly known as predicate logic. In this system, we interpret (a) as follows:
 * {| style="border:1px solid #bbbbbb;text-align:left;" cellpadding="8" cellspacing="0"


 * (a) || All cats are mammals. || $$\forall x (Cx \rightarrow Mx)$$ || For all x: if x is-a-cat then x is-a-mammal
 * }
 * }


 * {| style="border:1px solid #bbbbbb;" cellpadding="6" cellspacing="0"


 * + Example of a formal proof
 * 1 || $$\forall x (Cx \rightarrow Mx)$$ || Premise
 * 2 || $$\forall x (Mx \rightarrow Hx)$$ || Premise
 * 3 || $$\forall x (Cx \rightarrow Mx) \land  \forall x (Mx \rightarrow Hx)$$ || 1, 2,  Conjunction
 * 4 || $$(Ca \rightarrow Ma) \land  (Ma \rightarrow Ha)$$ || 3,  Universal Instantiation (twice)
 * 5 || $$Ca \rightarrow Ha$$ || 4, Hypothetical Syllogism
 * 6 || $$\forall x (Cx \rightarrow Hx)$$ || 5, Universal Generalization
 * }
 * 4 || $$(Ca \rightarrow Ma) \land  (Ma \rightarrow Ha)$$ || 3,  Universal Instantiation (twice)
 * 5 || $$Ca \rightarrow Ha$$ || 4, Hypothetical Syllogism
 * 6 || $$\forall x (Cx \rightarrow Hx)$$ || 5, Universal Generalization
 * }
 * 6 || $$\forall x (Cx \rightarrow Hx)$$ || 5, Universal Generalization
 * }


 * Let's now add to our knowledge the premise ("fact") that some mammals are erbivores; can we deduce from this, plus our pre-existing statements, that some cats are erbivores? Can we deduce that some cows are erbivores?