Supports

Formal definition

Support is a binary relation $E \succ H$ (read: $E$ supports $H$):

$$(E : \mathrm{Evidence},\; H : \mathrm{Hypothesis})$$

where:

• $E$ is the evidence — the information, observation, argument, or fact that bears on $H$; $E$ may be a data point, an experimental result, a theoretical consideration, or a background assumption; $E$ is the supporting item
• $H$ is the hypothesis — the claim, proposition, or theory that $E$ supports; $H$ is the supported item; $H$ may be a specific empirical prediction, a general theory, or a principle

The support relation $E \succ H$ holds when $E$ raises the credibility of $H$ — $H$ is more likely true, better grounded, or more warranted given $E$ than without it. Support is a graded, partial relation: stronger evidence gives stronger support; support is not binary.

One invariant. $E \succ H$ iff:

1. Positive relevance: $E$ makes $H$ more credible than $H$ is in the absence of $E$. Formally in the probabilistic account: $P(H \mid E) > P(H)$. $E$ is irrelevant to $H$ if $P(H \mid E) = P(H)$. $E$ is negative evidence for $H$ (undermines $H$) if $P(H \mid E) < P(H)$. Positive relevance distinguishes genuine support from neutral information.

Support is not grounding: grounding provides the complete mathematical foundation for $H$ — the unique sufficient reason. Support provides partial evidence — $H$ can have multiple independent supports, and $H$ could be true without any of them (through other evidence). The relation $E$ grounds $H$ implies $E \succ H$, but $E \succ H$ does not imply $E$ grounds $H$.

Support is not implication: $E \Rightarrow H$ (logical implication) means $H$ is necessarily true whenever $E$ is true — the support is deductive and complete. $E \succ H$ (support) means $E$ raises $H$’s probability but does not necessitate $H$ — the relation is inductive and partial.

Carnap: inductive logic and degrees of confirmation

Rudolf Carnap (Logical Foundations of Probability, 1950; The Continuum of Inductive Methods, 1952) developed inductive logic as a formal theory of the support relation:

Carnap distinguishes two notions of probability:

• Probability$_1$ (logical or epistemic probability): the degree of support — the degree to which evidence $E$ confirms hypothesis $H$, denoted $c(H, E)$ (the confirmation function)
• Probability$_2$ (empirical probability): the relative frequency of an event in the long run

The confirmation function $c(H, E)$ is a function from hypothesis-evidence pairs to real numbers in $[0, 1]$, satisfying:

• $c(\neg H, E) = 1 - c(H, E)$ (negation complement)
• $c(H_1 \vee H_2, E) = c(H_1, E) + c(H_2, E)$ when $H_1$ and $H_2$ are logically exclusive
• $c(H, E_1 \wedge E_2) = c(H, E_1) \cdot c(H \mid E_1, E_2)$ (product rule)

$E$ confirms $H$ relative to background $K$ iff $c(H, E \wedge K) > c(H, K)$ — the evidence raises $H$’s confirmation above its background level. This is Carnap’s formal definition of support.

Carnap’s lambda continuum: a parameterized family of confirmation functions $c_\lambda$ indexed by $\lambda \in [0, \infty)$, capturing the degree of reliance on prior (inductive bias) vs evidence. At $\lambda = 0$: pure frequency-based (no prior); at $\lambda \to \infty$: pure prior-based (ignores evidence). The continuum of inductive methods is the space of possible evidential support relations, not just one fixed relation.

Hempel: qualitative confirmation theory

Carl Hempel (Studies in the Logic of Confirmation, 1945, Mind) gave the first systematic qualitative theory of support/confirmation:

Hempel’s satisfaction criterion: evidence $E$ confirms hypothesis $H$ iff $E$ satisfies $H$ in the sense that $E$ includes all instances that $H$ predicts (the universal hypothesis “all ravens are black” is confirmed by evidence “this black raven”).

Hempel identifies three adequacy conditions for a confirmation relation:

1. Entailment condition: if $E \Rightarrow H$, then $E$ confirms $H$ — logical implication is a limit case of support
2. Consistency condition: a consistent body of evidence cannot confirm two mutually exclusive hypotheses
3. Equivalence condition: if $E$ confirms $H$ and $H \equiv H'$, then $E$ confirms $H'$ — support respects logical equivalence of hypotheses

The paradox of confirmation (Hempel 1945): “All ravens are black” ($H$) is logically equivalent to “All non-black things are non-ravens” ($H'$). By the equivalence condition, if “this white shoe” confirms $H'$, it also confirms $H$. This is the raven paradox: a white shoe appears to confirm the raven hypothesis, which seems absurd. Hempel accepts the result: in principle, any non-black non-raven is confirming evidence for the raven hypothesis, though very weak evidence.

The raven paradox shows that support is sensitive to the reference class: the white shoe confirms the hypothesis within the reference class of “non-black things” but not within the narrower class of “ravens.” Different reference classes yield different support judgments.

Toulmin: argument structure and backing

Stephen Toulmin (The Uses of Argument, 1958) developed a model of argument structure that distinguishes several roles for supporting evidence:

The Toulmin argument model has six components:

1. Claim ($C$): the conclusion being argued for
2. Data/Grounds ($D$): the evidence (facts, data) supporting the claim; $D \succ C$
3. Warrant ($W$): the rule or principle that licenses the move from $D$ to $C$; “given $D$, so $C$ because $W$”
4. Backing ($B$): support for the warrant $W$ itself; $B \succ W$
5. Qualifier ($Q$): the degree of confidence in the claim (necessarily, probably, presumably)
6. Rebuttal ($R$): conditions under which the warrant does not apply; exceptions to $W$

The support hierarchy: $B$ supports $W$; $W$ licenses the move from $D$ to $C$; $D$ supports $C$. Support is transitive through the warrant: if $B \succ W$ and $W$ licenses $D \Rightarrow C$, then $B$ transitively supports $C$.

Backing vs grounding: Toulmin’s backing $B$ provides support for the warrant — it makes the warrant credible, but does not logically entail the warrant. If $B$ were the complete logical foundation for $W$ (making $B \Rightarrow W$ analytically), then $B$ would ground $W$; otherwise $B$ merely supports $W$.

The Toulmin model makes explicit that in real argumentation, support is layered: we support conclusions with data, warrants license the inference, and backings support the warrants. Each layer involves a distinct support relation — none is deductively complete.

Bayesian confirmation theory

Bayesian confirmation theory (Good 1950; Howson & Urbach, Scientific Reasoning: The Bayesian Approach, 1989) formalizes support using conditional probability:

$E$ confirms $H$ iff $P(H \mid E) > P(H)$ — the probability of $H$ given $E$ exceeds $H$’s prior probability. This is the positive relevance account of confirmation.

Measures of degree of support:

• Difference measure: $c(H, E) = P(H \mid E) - P(H)$
• Ratio measure: $c(H, E) = P(H \mid E) / P(H)$
• Log-likelihood ratio: $c(H, E) = \log P(E \mid H) / P(E \mid \neg H)$ (Bayes factor)

Bayesian updating: when evidence $E$ is received, beliefs update by Bayes’ theorem:

$$P(H \mid E) = \frac{P(E \mid H) \cdot P(H)}{P(E)}$$

The support relation is dynamically maintained: each new piece of evidence updates $P(H \mid -)$, making support a real-time relation that tracks the epistemic state.

Open questions

• Whether the Bayesian account of support (probabilistic relevance) is the right formal foundation for the support relation in a non-probabilistic setting — whether support can be given a formal definition in terms of the Heyting algebra structure of the history fiber without invoking probability.
• Whether Toulmin’s support hierarchy (data supports claim via warrant, backed by backing) corresponds to a layered structure in the fiber — whether each layer of Toulmin’s argument maps to a different nucleus application, with backing corresponding to saturation nucleus ($\sigma$), data to transfer nucleus ($\Delta$), and the warrant to the commutation condition.
• Whether there is a topological account of support — whether $E$ supports $H$ iff $E$ is in the interior of a neighborhood of $H$ in some topology on propositions, making strong support the condition that $E$ is close to $H$ in the epistemic topology.
• Whether Carnap’s lambda continuum of confirmation functions corresponds to a family of nuclei parametrized by $\lambda$ — whether the interpolation between pure frequency-based and pure prior-based confirmation corresponds to a nucleus interpolation in the Heyting algebra.