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5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic

[logic when interpreted, rather than mere formal systems]

23 ideas
Semantics is the concepts of connections of language to reality, such as denotation, definition and truth [Tarski]
     Full Idea: Semantics is the totality of considerations concerning concepts which express connections between expressions of a language and objects and states of affairs referred to by these expressions. Examples are denotation, satisfaction, definition and truth.
     From: Alfred Tarski (The Establishment of Scientific Semantics [1936], p.401)
     A reaction: Interestingly, he notes that it 'is not commonly recognised' that truth is part of semantics. Nowadays truth seems to be the central concept in most semantics.
A language containing its own semantics is inconsistent - but we can use a second language [Tarski]
     Full Idea: People have not been aware that the language about which we speak need by no means coincide with the language in which we speak. ..But the language which contains its own semantics must inevitably be inconsistent.
     From: Alfred Tarski (The Establishment of Scientific Semantics [1936], p.402)
     A reaction: It seems that Tarski was driven to propose the metalanguage approach mainly by the Liar Paradox.
Tarski built a compositional semantics for predicate logic, from dependent satisfactions [Tarski, by McGee]
     Full Idea: Tarski discovered how to give a compositional semantics for predicate calculus, defining truth in terms of satisfaction, and showing how satisfaction for a complicated formula depends on satisfaction of the simple subformulas.
     From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Vann McGee - Logical Consequence 4
     A reaction: The problem was that the subformulas may contain free variables, and thus not be sentences with truth values. 'Satisfaction' can handle this, where 'truth' cannot (I think).
Tarksi invented the first semantics for predicate logic, using this conception of truth [Tarski, by Kirkham]
     Full Idea: Tarski invented a formal semantics for quantified predicate logic, the logic of reasoning about mathematics. The heart of this great accomplishment is his theory of truth. It has been called semantic 'theory' of truth, but Tarski preferred 'conception'.
     From: report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Richard L. Kirkham - Theories of Truth: a Critical Introduction 5.1
In standard views you could replace 'true' and 'false' with mere 0 and 1 [Dummett]
     Full Idea: Nothing is lost, on this view, if in the standard semantic treatment of classical sentential logic, we replace the standard truth-values 'true' and 'false' by the numbers 0 and 1.
     From: Michael Dummett (The Justification of Deduction [1973], p.294)
     A reaction: [A long context will explain 'on this view'] He is discussing the relationship of syntactic and semantic consequence, and goes on to criticise simple binary truth-table accounts of connectives. Semantics on a computer would just be 0 and 1.
Classical two-valued semantics implies that meaning is grasped through truth-conditions [Dummett]
     Full Idea: The standard two-valued semantics for classical logic involves a conception under which to grasp the meaning of a sentence is to apprehend the conditions under which it is, or is not, true.
     From: Michael Dummett (The Justification of Deduction [1973], p.305)
     A reaction: The idea is that you only have to grasp the truth tables for sentential logic, and that needs nothing more than knowing whether a sentence is true or false. I'm not sure where the 'conditions' creep in, though.
Beth trees show semantics for intuitionistic logic, in terms of how truth has been established [Dummett]
     Full Idea: Beth trees give a semantics for intuitionistic logic, by representing sentence meaning in terms of conditions under which it is recognised to have been established as true.
     From: Michael Dummett (The Justification of Deduction [1973], p.305)
In real reasoning semantics gives validity, not syntax [Searle]
     Full Idea: In real-life reasoning it is the semantic content that guarantees the validity of the inference, not the syntactical rule.
     From: John Searle (Rationality in Action [2001], Ch.1.II)
Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG]
     Full Idea: There are two approaches to an 'interpretation' of a logic: the first method assigns objects to names, and then defines connectives and quantifiers, focusing on truth; the second assigns objects to variables, then variables to names, using satisfaction.
     From: report of David Bostock (Intermediate Logic [1997], 3.4) by PG - Db (lexicon)
     A reaction: [a summary of nine elusive pages in Bostock] He says he prefers the first method, but the second method is more popular because it handles open formulas, by treating free variables as if they were names.
There are three different standard presentations of semantics [Hodges,W]
     Full Idea: Semantic rules can be presented in 'Tarski style', where the interpretation-plus-valuation is reduced to the same question for simpler formulas, or the 'Henkin-Hintikka style' in terms of games, or the 'Barwise-Etchemendy style' for computers.
     From: Wilfrid Hodges (First-Order Logic [2001], 1.3)
     A reaction: I haven't yet got the hang of the latter two, but I note them to map the territory.
A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables [Hodges,W]
     Full Idea: To have a truth-value, a first-order formula needs an 'interpretation' (I) of its constants, and a 'valuation' (ν) of its variables. Something in the world is attached to the constants; objects are attached to variables.
     From: Wilfrid Hodges (First-Order Logic [2001], 1.3)
I |= φ means that the formula φ is true in the interpretation I [Hodges,W]
     Full Idea: I |= φ means that the formula φ is true in the interpretation I.
     From: Wilfrid Hodges (First-Order Logic [2001], 1.5)
     A reaction: [There should be no space between the vertical and the two horizontals!] This contrasts with |-, which means 'is proved in'. That is a syntactic or proof-theoretic symbol, whereas |= is a semantic symbol (involving truth).
When an 'interpretation' creates a model based on truth, this doesn't include Fregean 'sense' [Hodes]
     Full Idea: A model is created when a language is 'interpreted', by assigning non-logical terms to objects in a set, according to a 'true-in' relation, but we must bear in mind that this 'interpretation' does not associate anything like Fregean senses with terms.
     From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131)
     A reaction: This seems like a key point (also made by Hofweber) that formal accounts of numbers, as required by logic, will not give an adequate account of the semantics of number-terms in natural languages.
Assigning an entity to each predicate in semantics is largely a technical convenience [Fine,K]
     Full Idea: In doing semantics we normally assign some appropriate entity to each predicate, but this is largely for technical convenience.
     From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
Classical semantics has referents for names, extensions for predicates, and T or F for sentences [Fine,K]
     Full Idea: A precise language is often assigned a classical semantics, in which the semantic value of a name is its referent, the semantic value of a predicate is its extension (the objects of which it is true), and the value of a sentence is True or False.
     From: Kit Fine (Vagueness: a global approach [2020], 1)
     A reaction: Helpful to have this clear statement of how predicates are treated. This extensionalism in logic causes trouble when it creeps into philosophy, and people say that 'red' just means all the red things. No it doesn't.
Syntactical methods of proof need only structure, where semantic methods (truth-tables) need truth [Lowe]
     Full Idea: Syntactical methods of proof (e.g.'natural deduction') have regard only to the formal structure of premises and conclusions, whereas semantic methods (e.g. truth-tables) consider their possible interpretations as expressing true or false propositions.
     From: E.J. Lowe (Introduction to the Philosophy of Mind [2000], Ch. 8)
     A reaction: This is highly significant, because the first method of reasoning could be mechanical, whereas the second requires truth, and hence meaning, and hence (presumably) consciousness. Is full rationality possible with 'natural deduction'?
The semantics shows how truth values depend on instantiations of properties and relations [Zalabardo]
     Full Idea: The semantic pattern of a first-order language is the ways in which truth values depend on which individuals instantiate the properties and relations which figure in them. ..So we pair a truth value with each combination of individuals, sets etc.
     From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.3)
     A reaction: So truth reduces to a combination of 'instantiations', which is rather like 'satisfaction'.
We can do semantics by looking at given propositions, or by building new ones [Zalabardo]
     Full Idea: We can look at semantics from the point of view of how truth values are determined by instantiations of properties and relations, or by asking how we can build, using the resources of the language, a proposition corresponding to a given semantic pattern.
     From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6)
     A reaction: The second version of semantics is model theory.
Valuations in PC assign truth values to formulas relative to variable assignments [Sider]
     Full Idea: A valuation function in predicate logic will assign truth values to formulas relative to variable assignments.
     From: Theodore Sider (Logic for Philosophy [2010], 4.2)
     A reaction: Sider observes that this is a 'double' relativisation (due to Tarski), since propositional logic truth was already relative to an interpretation. Now we are relative to variable assignments as well.
Situation semantics for logics: not possible worlds, but information in situations [Mares]
     Full Idea: Situation semantics for logics consider not what is true in worlds, but what information is contained in situations.
     From: Edwin D. Mares (Negation [2014], 6.2)
     A reaction: Since many theoretical physicists seem to think that 'information' might be the most basic concept of a natural ontology, this proposal is obviously rather appealing. Barwise and Perry are the authors of the theory.
An ontologically secure semantics for predicate calculus relies on sets [McGee]
     Full Idea: We can get a less ontologically perilous presentation of the semantics of the predicate calculus by using sets instead of concepts.
     From: Vann McGee (Logical Consequence [2014], 4)
     A reaction: The perilous versions rely on Fregean concepts, and notably Russell's 'concept that does not fall under itself'. The sets, of course, have to be ontologically secure, and so will involve the iterative conception, rather than naive set theory.
In classical semantics singular terms refer, and quantifiers range over domains [Linnebo]
     Full Idea: In classical semantics the function of singular terms is to refer, and that of quantifiers, to range over appropriate domains of entities.
     From: Řystein Linnebo (Philosophy of Mathematics [2017], 7.1)
It is easier to imagine truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F) [Horsten]
     Full Idea: It is easier to imagine what it is like for a sentence to lack a truth value than what it is like for a sentence to be both truth and false. So I am grudgingly willing to entertain the possibility that certain sentences (like the Liar) lack a truth value.
     From: Leon Horsten (The Tarskian Turn [2011], 02.5)
     A reaction: Fans of truth value gluts are dialethists like Graham Priest. I'm with Horsten on this one. But in what way can a sentence be meaningful if it lacks a truth-value? He mentions unfulfilled presuppositions and indicative conditionals as gappy.